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PROCEEDINGS OF THE 1984 WORKSHOPSPECTRAL THEORY OF STURM-LIOUVHILE DIFFERENTIAL OPERATORS*

Held at Argonne National laboratory

May 14 -June 15, 1984

Hans G. Kaper and Anton Zetti, editors

Gail W. Pieper, technical editor

Mathematics and Computer Science Division

December 1984

*This work was supported by the Applied Mathematical Sciences subprogram ofthe Office of Energy Research, U. S. Department of Energy, under contract W-31-109-Eng-38.

DISTRIBUTION OF THIS DOCUMENT IS UNU

TABLE OF CONThNTS

Abstract v

Preface vii

List of Participants ix

Schedule of Lectures xi

Riccati Transformations and Principal Solutions of Discrete Linear Systems- Caivin D. Ahlbrandt and John W. Hooker 1

Oscillation and Spectral Properties of Weakly Coupled Elliptic Systems-W. Allegretto 13

Potentials Having Extremal Eigenvalues Subject top-Norm Constraints-- M. S. Ashbaugh and E. M. Harrell I 19

Indefinite Sturm-Liouville Problems-F. V. Atkinson and D. Jabon 31

Sturm-Liouville Problems with Indefinite Weight Functions in Banach Spaces-Harold E. Benzinger 47

Interlacing Property of Eigenvalues of Sturm-Liouville Boundary ValueProblems-J. Boersma, Hans G. Kaper, and Man Kam Kwong 57

A Krein Space Approach to Dirichlet and Dual Dirichlet Inequalities Associatedwith Sturm-Liouville Operators

-R. C. Brown 61

Spectral Properties of Seliadjoint Ordinary Differential Operatorswith an Indefinite Weight Function

-B. Caurgus and H. Langer 73

Linear Relations in Indefinite Inner Product Spaces-A. Dijksmna and H. S. V de Snoo 81

Spectrum of Selfadjoint and Positive Operators with Compact Inverse- I. Reckinger 91

Asymptotics of Eigenvalues of Variational Elliptic Problems withIndefinite Weight Function-.. Reckinger nd H. El Fetnassi 107

iii-

A Nonoscillation Theorem for Second-Order Linear Equations-S. G. Halvorsen, Man Kam Kwong, and A B. Mingarelli 119

Some Problems of Transport Theory-R. J. Hangelbroek 123

Asymptotic Behavior of Semigroups-J. Hejtmanek 131

Some Exteasions of Results of Titchmarsh on Dirac Systems- D. B. Hinton and J. K. Shaw 135

Semigroups Generated by Ordinary Differential Operators-Mark A Kon 145

Problems Concerning Orthogonal Polynomials and SingularSturm-Liouville Systems

-Allan M. KraLL 151

Spectral Theory of Elliptic Problems with Indefinite Weights-Michel L. Lapidus 159

J-Symmetric Differential Systems-Heinz-Dieter Niessen 169

Pointwise Equisummability of Elliptic Operators-Louise A. Raphael 181

The Essential Spectrum of a Class of Ordinary Differential Operators-Bernd Schultze 187

Dirac Systems with Oscillating Potentials and Absolutely ContinuousSpectra

- J. K. Shaw ad L1. B. Hinton 195

iv

Abstract

This report contains the proceedings of the workshop on "Spectral Theory ofSturm-Liouville Differential Operators," which was held at Argonne during theperiod May 14 through June 15, 1984. The report contains 22 articles, authoredor co-authored by the participants in the workshop.

Topics covered at the workshop included the asymptotics of eigenvalues andeigenfunctions; qualitative and quantitative aspects of Sturm-ILiouville eigen-value problems with discrete and continuous spectra; polar, indefinite, and non-selfadjoint Sturm-Liouville eigenvalue problems; and systems of differentialequations of Sturm-Liouville type.

DISCLAIMERThis report was prepared as an account of work sponsored by an agency of the United Statesbilityoee eir theanynwarranty,express ormet nor any agency thereof, nor any of theirem pl foye e a kesrany, w ra t , x r or im plied, or assum es any legal liability or respheirbility or d te accuracy, com pleteness, or usefulness , of any binfo mai nyp or responsi-process disclosed, or represents that its use would not infri ration, apparatus, r .

ence heren t specific commercial product, process, or e ric eby t d n e rights. Referkmanufacturer or otherwise does not necessarily constitute or imply its endorsemntrademarkand opinions of avorng by the United States Government or any acy t herement,recom

n d o p i n i o s o f ra u t o r p r e e h e r e i n d o n o t n e c e s s a r i l y t a g e n c y t h e r e of T he eo f h eUnited States Government or any agency tereof.o efet hs o h

Preface

This report contains the proceedings of the workshop "Spectral Theory ofSturm-Liouville Differential Operators," which was held at Argonne NationalLaboratory during the period May 14 through June 15, 1984. The workshop wasorganized by the Mathematics and Computer Science Division (MCS) as part ofits activities in applied analysis. Twenty-six mathematicians, nine of them fromabroad, took part in the scientific activities. These proceedings are the per-manent record of the research done at or stimulated by the workshop.

The objectives of the workshop were (1) to encourage the transfer of theoreticalresearch results to the domain of computational mathematics, and (2) to iden-tify open research problems in the area of Sturm-Liouville eigenvalue problems.The format of the workshop was chosen to emphasize communication andcooperation. Each participant was given the opportunity to present a summaryof his or her past or current work, but most of the time was spent on discussionsand joint research.

Each week the workshop focused on a particular theme. The five themes were(1) asymptotics of eigenvalues and eigenfunctions; (2) qualitative arid quantita-tive aspects of Sturm-Liouville eigenvalue problems with discrete spectra; (3)qualitative and quantitative aspects of Sturm-Liouville eigenvalue problems withcontinuous spectra; (4) polar, indefinite, and non-selfadjoint Sturm-Liouvilleproblems; and (5) systems of second-order Sturm- Liouville equations. For eachweek one participant was invited to act as program coordinator.

The main financial support came from the University of Chicago Fund forArgonne Activities. The MCS Division generously supported the activities of theworkshop.

Following this preface is a list of names and addresses of all colleagues who tookpart in the workshop. We express our gratitude to these colleagues and espe-cially to those who contributed manuscripts to the proceedings.

Special thanks are due to Gail Pieper, technical editor of the MCS Division, whoso ably handled all the work involved in the production of these proceedingsfrom rough draft manuscripts to the end product that we have before us. Wealso thank our colleague Jim Cody for his help in making the necessary physicalarrangements for the workshop. Finally, we offer our sincere appreciation toPaul Messina, MCS division director, for encouraging us to organize thisworkshop and for generally stimulating the proper environment for creativeresearch within the division.

The organizing committee:

F.V. Atkinson (Toronto)Hans G. Kaper (Argonne)Man Kam Kwong (DeKalb)Tony ZettI (DeKalb)

December 1984

vii I

List of Participants

Calvin D. AhlbrandtDepartment of MathematicsUniversity of MissouriColumbia, MO 65211

W. AllegrettoDepartment of MathematicsUniversity of AlbertaEdmonton, Canada T6G 261

Mark AshbaughDepartment of MathematicsUniversity of MissouriColumbia, MO 65201

Harold E. BenzingerDepartment of MathematicsUniversity of Illinois1409 W. Green StreetUrbana, IL 61801

Richard BrownDepartment of MathematicsUniversity of AlabamaUniversity, AL 35486

Branko CurgusPrirodno-matematickifakultet Sarajevo

Vojvode Putnika 45/W71000 SarajevoYugoslavia

H. S. V. de SnooMathematisch InstitutRijksuniversiteit GroningenPostbus 8009700 AV GroningenThe Netherlands

Jacqueline Fleckinger41 Rue Boyssone31400 ToulouseFrance

Gotskalk HalvorsenDepartment of MathematicsUniversity of TrondhE im-N.T.H.7034 Trondhheim-N.T H.Norway

Rutger J. HangelbroekDepartment of MathematicsWestern Illinois UniversityMacomb, IL 61455

B. J. HarrisDept. of Mathematical SciencesNorthern Illinois UniversityDeKalb, IL 60115

Johann HejtmanekMathematisches InstitutUniversitat WienStrudlhofgasse 4A-1090 WienAustria

Don HintonMathematics DepartmentUniversity of TennesseeKnoxville, TN 37996-1330

John HookerDepartment of MathematicsSouthern Illinois UniversityCarbondale, IL 62901

David JabonDepartment of MathematicsUniversity of ChicagoChicago, IL 60637

Hans G. KaperMath. and Computer Science Div.Argonne National Laboratory9700 South Cass AvenueArgonne, IL 60439

Mark KonDepartment of MathematicsBoston UniversityBoston, MA 02215

Allan M. KrallMcAlister Bldg.The Pennsylvania State UniversityUniversity Park, PA 16802

Man Kam KwongDepartment of MathematicsDeKalb, IL 60115

ix

Michel LapidusDepartment of MathematicsUniversity of Southern CaliforniaLos Angeles, CA 90089-1113

Angelo B. MingarelliDepartment of MathematicsUniversity of OttawaOttawa, Ontario KiN 9 B4Canada

Heinz-Dieter NiessenAn der Broelhecke 1D-5010 BergheimWest Germany

Louise RaphaelDepartment of MathematicsHoward UniversityWashington, D.C. 20050

Bernd SchultzeFachbereich 6 MathematikUniversitaet EssenUniversitaetstrasse 1-3 4300Essen 1West Germany

J. K. ShawDepartment of MathematicsVirginia TechBlacksburg, VA 24061-4097

Tony ZettlDepartment of Mathematical SciencesNorthern Illinois UniversityDeKalb, IL 60115

x

Schedule of Lectures

Monday, May 14

2:00pm B.J. Harris (Northern Illinois U., DeKab)"Asyrmptotics of Sturm-Liouville Eigenvalue Problems"

Tuesday, May 15

10:00am J. Hejtmanek (U. of Vienna, Austria)"Asymptotics of Linear Semigroups"

2:00pm H.G. Kaper (ANL)"Asymptotics of Eigenvalues and Eigenfunctionsof Indefinite Sturm-Liouville Problems"

Wednesday, May 16

10:00am H. Benzinger (U. of Illinois, Urbana)"Well-Bounded Operators and Semigroups"

2:00pm A.B. Mingarelli (U. of Ottawa, Canada)"Asymptotics of Sturm-Liouville EigenvalueProblems with Indefinite Weights"

Thursday, May 16

2:00pm J. Fleckinger (U. of Toulouse, France)"Asymptotics of Eigenvalues of DifferentialOperators with Indefinite Weights"

Tuesday, May 22

10:00am A. Zettl (Northern Illinois U., DeKab)"The Essential Spectrum of Non-SelfadjointSturm-Liouville Operators"

1:30pm Man Kam Kwong (Northern Illinois U., DeKab)"Oscillation of Sturm-Liouville Equations"

2:15pm A.B. Mingarelli (U. of Ottawa, Canada)"Lower Bounds for the Spectrum of Sturm-LiouvilleOperators with an Almost Periodic Potential"

Wednesday, May 23

9:30am J. Fleckinger (U. of Toulouse, France)"Eigenvalues of Schroedinger Operators"

2:00pm M. Ashbaugh (U. of Missouri, Columbia)"Stability of Resonances"

xi

Thursday, May 24

10:00am G. Halvorsen (U. of Trondheim, Norway)"Limit Circle Criteria for Sturm-Liouville Operators"

2:00pm R.C. Brown (U. of Alabama, University)"Sum and Product Inequalities with Some Applications"

Friday, May 25

10:00am J. Hooker (Southern Illinois U., Carbondale)"Principal Solutions of Discrete Linear Systemsand Riccati Transformations"

2:00pm C. Ahlbrandt (U. of Missouri, Columbia)"Principal Solutions of Discrete Linear Systemsand Riccati Transformations, cont'd"

Monday, May 28

2:30pm A.B. Mingarelli (U. of Ottawa, Canada)"Lower Bounds for the Spectra of Differentialand Difference Operators"

Tuesday, May 29

2:00pm A. Zettl (Northern Illinois U., DeKab)"Bounds on the Infimum of the Essential Spectrumfor Ordinary Differential Equations"

Wednesday, May 30

2:00pm D. Sorensen (ANL)"Numerical Solution of Riccati Equations"

3:30pm C. Ahlbrandt (U. of Missouri, Columbia)"Riccati Differential Equations (Periodic andConstant Coefficients)"

Thursday, May 31

9:30am G. Halvorsen (U. of Trondheim, Norway)"Sharp Bounds for Solutions of Sturm-Liouville Equations"

11:00am Excursion to CP-5

2:00pm M. Ashbaugh (U. of Missouri, Columbia)"Potentials Having Extremal Eigenvalues Subjected top-Norm Constraints"

xii

Friday, June 1

10:00am M. Kon (Boston U.)"Sturm-Liouville Semigroups"

2:00pm L. Raphael (Howard U., Washington D.C.)"Equisummability of Eigenfunction Expansions"

Monday, June 4

10:30am A.B. Mingarelli (U. of Ottawa, Canada)"A Survey of the Regular Weighted Sturm-Liouville Problem:The Non-Definite Case"

Tuesday, June 5

9:00am B. Schultze (U. of Essen, W.Germany)"The Essential Spectrum of a Class of Ordinary DifferentialEquations"

10:30am R.C. Brown (U. of Alabama, University)"Some Non-Standard Inequalities Associated withSecond-Order Sturm-Liouville Problems (and theIndefinite Methods Needed to Prove Them)"

Wednesday, June 6

9:00am Man Kam Kwong (Northern Illinois U., DeKab)"An Inequality for Hermitian Operators and Applications"

10:30am H. de Snoo (U. of Groningen, Netherlands)"Self-Adjoint Relations in Indefinite Inner Product Spaces"

Thursday, June 7

9:00am R.J. Hangelbroek (Western Illinois U., Macomb)"Some Problems of Transport Theory"

11:00am Excursion to IPNS

2:00pm M. Lapidus (U. of Southern California)"Eigenvalues and Eigenfunctions of Elliptic BoundaryValue Problems with an Indefinite Weight Function"

Friday, June 8

9:00am H.-D. Niessen (U. of Essen, W.Germany)"J-Symmetric Differential Systems"

10:30am H.G. Kaper (ANL)"A Regularizing Transformation for a Class ofSingular Sturm-Liouville Problems"

xiii

Monday, June 11

10:30am Man Kam Kwong (Northern Illinois U.. DeKalb)"Oscillation Theory for Systems ofSturm--Liouville Differential Equations"

Tuesday, June 12

9:00am R.C. Brown (U. of Alabama, University)"A Survey of Some Results Concerning theDirichlet Index"

10:30am B. Curgus (U. of Sarajevo, Yugoslavia)"Indefinite Sturm-Liouville Problems andHalf-Range Completeness"

Wednesday, June 13

9:00am A. Xrall (Penn State U., University Park)"Boundary Conditions for Systems"

10:30am A.B. Mingarelli (U. of Ottawa, Canada)"Some Questions in Oscillation Theory"

'Thursday, June 14

9:00am W. Allegretto (U. of Alberta, Edmonton, Canada)"Oscillation and Spectral Properties forWeakly Coupled Elliptic Systems"

10:30am E.L. Lusk (ANL)"The Automated Reasoning Effort at Argonne"

Friday, June 15

9:00am K. Shaw (VPI&SU, Blacksburg, Va.)"Dirac Systems with Long- and Short-Range Potentials"

10:30am D. Hinton (U. of Tennessee, Knoxville)"On a Dirac System of Titchrnarsh"

xiv

RICCATI TRANSFORMATIONS AND PRINCIPAL SOLUTIONS OFDISCRETE LINEAR SYSYEMS

Clvin D. AIlTbrandt *John W. Hookert

Abstract

Consider the second-order line? matrix difference equation

(i)L[X,,]_= -A(C,_.1AX, 1) + AX,= = 0, m= 1,2,3,...,

where C and A are given sequences of positive-definite, Hermitian rxr matrices,X, is an rxr matrix, and A is the forward difference operator AX, = X,+1 - X,. LetX, be a sequence of nonsingular rxr complex matrices. Then ifWn=(C,-AX,-i- 1)X 11 , we have

(ii) L[X,] = X R[ WJ]X,, n = 1,2,3,...,where

R[W,] = -W.,1 + W,[W +C, 1 ]~ _C,1+A,

a Riccati matrix difference operator.

We give a definition of principal and antriincipal, or recessive and dominwat,solutions of (i)' and use the relationship (ii) to prove the existence of principaland anti-principal solutions of (i) and the essential uniqueness of principal solu-tions.

1. IntroductionDiscrete-time linear systems and related discrete matrix Riccati equations

arise in a variety of applied problems, in particular, in discrete linear optimalcontrol and filtering problems (cf. Vaughan [1970]). We discuss here the con-cept of principal (sometimes called recessive, or subdominant) solutions of alinear vector difference equation

l[z,] = -A(C _1A,- 1) +Az~= 0, n = 1,2,3,..., (1.1)

where A. and C, are given sequences of positive-definite, Hermitian rxrmatrices with complex entries. A solution of (1.1) is a sequence z, of r x1 com-plex vectors satisfying (1.1), and A is the forward difference operator defined by

AU, = U,+1 -U,

for any sequence of matrices or vectors U,.

Instead of working directly with the vector operator 1[z,] of (1.1), we find itmore convenient to make use of the related matrix operator

L[X ] = -A(4.-.1 X.- 1) + AX,, (1.2)

where X, is an rxr matrix, n = 0,1,2,.... We will also make ue of a discrete

'University of Imssouri, Columbia, Missouri 85211.tSouthern Illinois Universuty, Carbondale, Illinois 62901.

1

2

Riccati matrix operator which will be defined in Section 2. In Section 3 we defineand discuss existence of principal and anti-principal solutions of L [X] = 0. Theessential uniqueness of principal solutions is discussed in Section 4. In Section5, some further properties of solutions of the Riccati equation R[ W ] = 0 aredeveloped. Section 6 connects this work with previous transformation theory fordifferential operators.

Our results on principal solutions are, for the most part, discrete analoguesof results for differential systems discussed by Hartman [1957], Reid [1958,1963], Ahlbrandt [1972], and others.

An alternative form for the equation L [X,] = 0 is

L[X] = -CX C+1 - 1X,.-. 1 + BX, = 0, (1.3)

where B,1 = C + C,..1 + A, n=1,2,3....From (1.3) it is clear that, given aninteger M a 0, the solution of L[A] = 0 for given initial matrices XM and AXy(or, equivalently, X1 and X+ 1) is defined for all n z 0 ani is unique. Indeed, therestriction to non-negative integers is unnecessary, and solutions may bedefined for all integers n by use of the recurrence relation (1.3).

Before proceeding, we state for reference some elementary properties ofthe forward difference operator A. If A and B, are sequences of rxr matrices,then

A(4B) = AAB, + (AA)B1 .1 = A4+1AB + (AA)Bn, (1.4)

AA = AN+ - Ay, (1.5)t=M

and

AAB= ANBN+1 - Ay- 1B 2 - (AAg_ 1)B4, (1.6)t:Y t=M

an analogue of "integration-by-parts" which follows readily from (1.4) and (1.5).By using (1.4), one can also easily show that the operators L[z,] and L[X ]satisfy discrete analogues of the Lagrange identity for differential operators;hence, we say that these operators are formally self-adjoint (cf. Hartman [1964:385]). We also state without proof the following variation-of-parameters lemma.

LEMMA 1.1. If X, is a solution of the homogeneous matrix difference equation

AX,- 1 + AnX = 0, (1.7)

and X, is nonsingular for all n z M -1 for some M>0, then the solution of

AY-1 + AnY =B, n &M ,

with initial value Y- 1. is

Y = X (X 1Y-1 + $Xi~\B ), n ;-! M -1,

t=2

adhere the summation equals Oifrn = M-1.

In the discussion below, r xr matrices with complex entries will always bedenoted by capital letters, I is the rxr identity matrix, and mm,n,M,N,i,j willalways denote non-negative integer indices.

2. A RIccati Difference Operator and Prepared Solutions of L[X] = 0.Throughout this paper, we assume that the coefficients of the operator

L[X,] = 0 satisfy the following condition:

3

Aand Care Hermitian and positive definite for all n z 0. (2.1)

The concept of a prepared (self-conjoined, isotropic) solution of a linearself-adjoint system of differential equations has been employed by many writers[Coppel, 1971; Hartman, 1957; Reid, 1958]. An analogous concept for differenceoperators follows.

For an arbitrary ordered pair of sequences Un and Vl of rxr complexmatrices, we define a "bracket function"

(UVJ = UC,_-1AV_ 1 - [C_1AU.1]JV,

=U-1Cn-1 V - U C-1 In-.1,

where U, denotes the transpose conjugate of Un. It is easily verified that

SU,V = - V,,Un O' (2.2)

It is also readily verified that if Un and V are solutions of L [X ] = 0, thenU, iV' = 0 for all n. Hence we have the following lemma.

LEMMA 2.1. For any solutions Un and V of L[X ] = 0, Um,Vm is constant.

DEFINrION. A sequence of rxr complex matrices is prepared, or isotropic, if

JX,XI = 0 for all n a M for some M z 0, (2.3)

or, equivalently, if X_1Cn-1X is Hermitian for n z M.

In preparation for the development of a principal solution of L [X,] = 0, wefirst introduce a Riccati difference operator. By analogy with differential sys-tems (cf. Reid [1972]), if X is a sequence of nonsingular'- xr matrices, oneexpects a matrix identity of the form X: L [X] = XR [ W ]Xm for some non-linearoperator R. Now

L[X] = -A(C..1A n.1) + AX

= -CnAXA + C- 1 X.-1 + A X,

= [-(CX)x,- + (4 _1X,- 1)X, 1 + A]X,. (2.4)

We let

Wn = (C 1AX..1 )X~11, (2.5)

which is analogous to the usual Riccati transformation w = cz'/x for thedifferential equation -(cx')' + pz = 0. (For other discrete analogues of the Ric-cati transformation, see Gautschi [1967], Hooker and Patula [1981], Hooker,Kwong, and Patula [1984], and Arscott [1981].) Then (2.4) becomes

L[Xn] = (-Wm+l + W Xn-A.1X 1 + An)X. (2.6)

Now W = C,- 1X, 1X4~ 1 - C- 1X.- 1 X~11. Hence, XmX,-1 = C11(Wm + C.-1 ), soX,. 1 XA-V = (W + C,-1)-'C_1. Substitution into (2.6) gives

L[X, ] = (-W 4. 1 + Wm(Wm + C- 1)-1Cn. 1 + An )X.

Hence,

whLrXe] = XR[W ]X, (2.7)where

4

R[7W] = -W+1 + WY(W, + Cn_ 1)~1C,_1 + An, (2.8)

a Riccati matrix difference operator of the type that occurs in problems ofdiscrete linear optimal control and filtering [Vaughan, 1970].

THEOREM 2.1. Assume condition (2.1). Given an integer M > 0, let X, be theunique solution of 'he initial value problem

L[X,1 ] = 0, X- 1 = I, C_ 1AX- 1 = I. (2.9)

Then X, anddAX, are nonsingular for all n M-1, W, defined by (2.5) is posi-tive defnite for all n z M, D defned by

Dn = XC,Xn+1 (2.10)

is Hermitian and positive definite for all n & M-1, and X, is a prepared solu-tion.

PROOF. From the initial conditions we have X(_ 1 = I and AX,1 -= Xli - XH-1 = C~l . Hence,

X =I+ C! 1 , (2.11)

so X, is Hermitian. Then

D- 1 - Dn- 1 = Xn 1C,- 1X, -X,'C,- 1 X,-1 = i Xn,X ,a constant matrix by Lemma 2.1. But

AX,,X = XC-1AXM- 1 - [C, 1AX- 1]'X

=XI -IX = 0,

since Xy is Hermitian. Therefore, D, is Hermitian for all n z M-1. Hence, X, isa prepared solution.

Thus we have X2 -.1, X 1 nonsingular, and WV = I is positive definite. AlsoDy-1 = X. 1C- 1Xy, so from (2.11) we obtain

Dg..1 = C 1 (I + Cjj1 ) = C. 1 + I,

so D_1 is positive definite. We then proceed by induction, assuming that X_1and Xk are nonsingular and Wk and Dk_- are positive cd.efinite. Now for Wk+1defined by ( 2.5), we have R[Wk ]= 0 by (2.7). Thus

W+1 = W(W + C-1)-1C- 1 + 4

= (C-F1 (fWk + Ck- 1)W')- 1 +Ak

= (C1i1 + Wk') 1 + 4,so Wl1 is positive definite.

To show that Dk is positive definite, consider

X Wk+lXk =Xk(C Xk)X X

= XCkAk+1 - XCkX.

Then

Dk = XCkX+l = X Wk+1Xk + XkCDXk,

so Dk is positive definite and, furthermore, Xk+1 is nonsingular, which completesthe induction. Finally, since W,+1 = (CAX,)X,~1 is positive definite for all

5

n z M-1, AX, is nonsingular for all n z M-1, which completes the prjof.

We will return in Section 5 to further relationships between solutions ofL[X,] = 0 and corresponding solutions of R[ W ] = 0.

3. Principal Solutions of IX,.] = 0.We now make use of Theorem 2.1 to establish the existence of principal

(sometimes called recessive, subdominant, or distinguished) solutions ofL[X~] = 0. The definition we use is analogous to that of Hartman [.957] forSturm-Liouville differential operators. It stems from the property

f(cu2)-dt = oc,which characterizes principal solutions of scalar self-adjoint differential equa-tions (cf. Hartman [1964]). Principal solutions for second-order and nth-orderscalar difference equations have been discussed by Patula [1979] and Hartman[1978], respectively. For computation of principal solutions of scalar differenceequations, see Gautschi [1967] or Olver and Sookne [1972]. Our discussion ofprincipal solutions is similar to that of Hartman [1957] and Reid [1958].

DEFINITION. Aprincipal (or recessive) solution of L[X] = 0 is a solution X, that isnonsingular for n z M, for some M z 0, and that satisfies the conditions

X,CXi+1 is positive definite for n L M, (3.1)

and

k *(XQCtX+ 1)-k -l as n - m (3.2)i=M

for every unit vector k. A solution is anti-principal if (3.1) is satisfied and

2k *(X* X+1)-'kt=M

converges for every unit vector k.

LEMMA 3.1. Assume condition (2.1) and let X, be prepared solution of L [Xn ] = 0which is nonsingular for al, n z M -1 for some M > 0. Then every solution Y ofL[Y ] = Oforn M-1 is given by

Yn = X(P + 2 (X 1C- 1X)-1'Q), n M-1, (3.3)ti=M

for constant matrices

P = XxI 1 Yg_-1 and Q =(Xy-1,YM-1=X,Yd. n > M-1, (3.4)

lhere the summationin (3.3) e duals 0 if n = M -1.

Conversely, if P and Q are given constant matrices, (3.3) defines a solution

of L[Y,] = 0Oforn M-1.

PROOF. Given X and Y as stated, we know

(X,,Y) = X'C_1 Y-1 -[C -1AXy Y =,

a constant matrix (i.e., Q is independent of n). Since X is nonsingular forn z M-1, we can write

Cn1AY_1 - X,~[C-1X ]Y = X*Q.

Hence

6

AY-1 - LC-1[(C-1Ax,_1)X-1]'Y = C,~11 X *-'Q. (3.5)

Since X is a prepared solution of L [X,] = 0, X is a solution of the homogeneousequation related to (3.5) (i.e., substituting X7 for Y makes the left-hand side of(3.5) equal 0). Then by variation of parameters, Lemma 1.1, the solution Y ofthe non-homogeneous equation (3.5) can be written as

Y =AnX(XI11YM-.1+ ic4X-\ -c1Q),

which is of the form (3.3) with P = X! 1 Y_ 1. Conversely, given P and Q, onemay verify that (3.3) is a solution of L[ Y ] = 0, which completes the proof.

LEIMMA 3.2. The solution Y, of L[YR] = 0 given by (3.3) is a prepared solution ifand only if

PQ = QP. (3.6)

PROOF. We need to show that Yn' 1C,-1 Y is Hermitian for n a M and, by Lemma2.1, it suffices to show this for n = M. From (3.3) with n = M, we have

Y'-1CM-1Y, = Y-iCj-.XP + Y-lCM,1XM(X IC-1XM)- 1Q. (3.7)

Since Il-1 = P'Xy- 1 ( from (3.3) with n = M--1), (3.7) can be written

Y1-1C-1YH = P*XX- 1C_1XMP + P*Xl-1CX_1X(X;_1C._lXI)- 1Q. (3.8)

Since X, is a prepared solution, the first term on the right in (3.8) is Hermitianand the second term equals P'Q. Thus Y- 1 C _1Y is Hermitian if and only ifP'Q is Hermitian, which completes the proof.

We now proceed to establish the existence of principal and anti-principalsolutions of L[X4] = 0, assuming Condition (2.1). If X = X is a prepared solu-tion of L[A}= 0, which is nonsingular for n z M-1, for brevity of notation wedefine

0, if n = M-1S(X)=(3.9)i X/1C_1i)1-if n aM

Then the solution (3.3) given in Lemma 3.1 can be written

Y = X (P + S (X) Q), n a M-1. (3.10)

In particular, let P = Q = I in (3.10), and let X, be the solution of the initialvalue problem (2.9). Then (3.10) becomes

YR = X,(I + Sn(X)), n LM-1, (3.11)

where, by Theorem 2.1, the solution X is prepared and S7 (X) is a positivedefinite, increasing sequence for n M (where matrix inequality is defined inthe usual sense of positive definiteness).

By Lemma 3.2, Y is a prepared solution. Also, Y is nonsingular for alln z M-1, since YM_1 = X,- 1 = I and S (X) is positive definite for n > M. Thuswe may reverse the roles of X and Y in Lemma 3.1, which then tells us that

X = Y(e + S(Y)), n maiM-1, (3.12)where E=Yj11lly1 = I and = Y,X J, a constant matri". By (2.2),

7

so (3.12) becomes

X = Y (I - S(Y)).

Hence

I = XX= Xn-IYn(I - Sn(Y)). (3.13)

Substituting (3.11) into (3.13), we obtain

I = (I + S (X))(I - Sn(Y)). (3.14)

Thus I - Sn(Y) is the sequence of inverses of the increasing, positive-definitesequence I + S, (X), and I + S (X) > I for n ? M. It follows that I - S (Y) is adecreasing, oositive-definite sequence satisfying 0 < I - S (Y) <I for all n ?M.Hence S,(Y) is an increasing sequence with a positive-definite limit S.( Y) s I asn-o. Now define the sequence

Z, = Y(S.( Y) - S( Y)), n aM -1. (3.15)

This is of the form (3.3) with P = S..(Y) and Q = Y,Z = -I, so by Lemma 3.1,Z, is a solution of L[Zn] = 0. We will show that Z, is prepared and recessive.

Since Sn (Y) is Hermitian for all n > M, S..(Y) is Hermitian; hence, byLemma 3.2, Zn is prepared. Also, S..(Y) - S,(Y) is positive definite and Y isnonsingular for each n a M, so Zn is nonsingular. Thus Zn satisfies thehypotheses of Lemma 3.1, and we may write Yn in terms of Zn as

Y = Z(P + S(Z)Q), n z M -1, (3.16)

where, by (3.4), P = Z1.11YM_1 = S.. (Y)~1 and Q = fZY, . By (2.2),Q = ZY = - Yn,Zn = I, so (3.16) becomes

Yn =-Zn (S..(Y)-1 + Sn(Z)), n M-1.

Proceeding as in Steps (3.13) and (3.14), we obtain

I = (S..(Y) - Sn(Y)(S. 1 (fY) + Sn(Z)). (3.17)

But S..(Y) - Sn(Y) is positive definite and decreasing, and tends to 0 as n-,w. Itthen follows from (3.17) that S(Z) is increasing for n a M-1, and for every unitrxl vector k, k*Sn(Z)k is positive and increasing as n increases, andlimf k Sn(Z)k = oo. Therefore Z is a principal solution of L [Zn] = 0, byn+"definition.

Similarly, from the discussion above, we know that Sn (Y) is a positive-definite, increasing sequence for n ;M, with a positive-definite limit S..(Y) s I,and it follows that Yn is an anti-principal solution. We have thus established thefoiiowing theorem.

THEOREM 3.1. Assume Condition (2.1). Then L[X ] = 0 has principal (recessive)solution and an anti-principal (dominant) solution.

4. Reid's Construction of Recessive SolutionsIn the matrix differential equation case, Reid [1958] gave a construction of

the principal solution at o by means of a limiting case of a solution of a two-point boundary value problem. Reid's construction for the system

U'=AU+BV, V'=CU - A'V

started with a solution Ut (z), Va (z) defined by

U,,(s) =I, Vg t)= 0,

8

He showed that V(s) had a limit V ,.(s) as t became infinite. Then(U, ..(z), V, ..(z)) was a principal solution at oo. Constructions of this type havebeen used in the construction of recessive solutions of three-term recurrencerelations [Gautschi, 1987; Olver and Sookne, 1972]. In particular, such methodshave been employed in numerical evaluation of Bessel functions J (x) from theirrecurrence relation -4.+ 1 (z) - J,_1(z) + 2n/ z J,(z) = 0.

We now use the same idea to construct a recessive solution at M. Of course,the previously constructed Z, is recessive, but for any nonsingular matrix K,Z,K is also recessive. One application of the following construction is to showessential uniqueness of Z,, i.e., uniqueness up to nonsingular constant multi-ples.

Let M and N be integers with M < N. For Z constructed as in Section 3,define S,(Z) as before. Then S,(Z) is increasing in n, positive definite forn Z M and (Sft(Z))- 1-+0 as n -'oo. We define X(M,N) as the solution defined bythe two-point boundary value problem

X-1(M,N) = I, XN(M.N) = 0. (4.1)

If such a solution exists, it must have the form

X(MN) = Z(P + S(Z)Q), n z M-1. (4.2)

But the choice

I = Z.. 1 P, Q = -(SN(Z))~ 1 P (4.3)

uniquely determines X (M.N) as

X,(M,N) = Z(I - S(Z)[SN(Z)j~1)Z 11 (4.4)

with the consequence

X,(M,N)-'ZZ11 as N -.o. (4.5)

Since P'Q = -P'[SN(Z)]~1 P is Hermitian, X,(M,N) is prepared for each N andsince

X (M,ao) = ZK for K = Zll , (4.6)

X (Mao) is prepared and recessive at co. We now show that the recessive solutionat oo is essentially unique. Let Y be any other recessive solution at w. Withoutloss of generality, assume M is sufficiently large that Y is nonsingular forn z M-1. Replace Z in the previous derivation of X3 (M,ac) by Y for the con-clusion

X(M ,0) a YY11 = ZZi1 (4.7)

i.e.,

Y = ZnH for H = Zj!1 YM- 1. (4.8)

THEOREM 4.1. The recessive solution at m is unique up to nonsingular constantmultiples and is determined as

lim Xn(M,N)N-w

where X (MN) is the unique solution of the boundary value problem. (4.1).

5. 1accati SolutionsWe now relate the recessive solutions to a solution of the Riccati equation.

9

THEOREM 5.1. Suppose that X, is any recessive solution of L [X,,] = 0 with X,nonsinguLar for n z M -1. Then, for n M,

Wn = (C- 1AXn- 1)X,11 < 0, (5.1)

Wn+1 = An + [C,' 1 + WWI]-', (5.2)

and

WWI = -Cn-31 + ( W+1 - An)-'. (5.3)

Consequently,

W > -C-1. (5.4)

PROOF. Without loss of generality, assume X_1 = I. LetW,(MN) = Cy-. 1X,-(M,N). Use the summation-by-parts formula (1.6) on

0 = t X{L(X ] =-t xi'(C- Xi-1) + X*A Xi

to obtain

-W(M,N) = (X41(MN))'Ci-tX -(M,N)t=M

+ X (MN)A ,X 1(M,N).

In particular,

-WM(MN ) ZX(M,N )AMX(M,N ).

Let N-+o for

-W(M,,o) XJ AMXMX > 0.

Hence W7 < 0 for all M. Use 0-L[Xr]to obtain

-C, (tX)X, 1 + Cn- 1(Xn- 1)X, 1 + A, = 0

and

W,+ 1 -An = C- 1( ,x-1)X~ 1 = WnX-1X-.

But

W, = C,_ 1AX,_.. 1X,-i 1 = C-1[X,X,11 - I]

and X 11 = C,11 W, + I , which is nonsingular. Thus

W, 1 - A, = WR(C-1W71, + I)-'

= RW',(W, + C1)-1C .= [c~1h(W, + C-i)WW~1]-

= [W' + C W1]-1.But W.,1 - A,, <0 since W4.1 < 0. A, > 0. Consequently,

W,-' + C~ 1 < 0,

W1W + Cl 1 = (Wn+1 -An )~1

10

and

W-1 = -Cn 1 + (WR+1 -An)~ <-C-1 <0.

8. A Connection with'Transformation TheoryA general discussion of transformation theory for linear differential opera-

tors was presented by Ahlbrandt, Hinton, and Lewis [1982]. Lemma 3.1 above isrelated to an analogous transformation theory for difference operators.

Let X, be a prepared solution of (1.3) with X,, nonsingular for n M-i. Avariable change of the form

Y = X, V, n zlM-1 (6.1)

induces the operator identity

X,L[Y] = -A(D_1AV..1) + E,1V a [Vn], n M. (6.2)

where DR = X,C~XR1 and En = X,L[Xn] = 0. Hence L[YR] = 0 if and only ifthere exists a constant matrix Q such that

D_.V _1 = Q. i M. (6.3)

Thus, by summing (6.3) and using (1.5), one obtains

Vn = V- 1 + (Sn(X))Q, (6.4)

which gives relation (3.3). Choice of n = M implies that Q is the indicatedbracket function.

References

Ahlbrandt, C. D. 1972. "Principal and antiprincipal solutions of self-adcjointdifferential systems and their reciprocals." Rocky Mountain J. Math. 2:169-182.

Ahlbrandt, C. D., Hinton, D. B., and Lewis, R. T. 1982. "Transformations of secondorder ordinary and partial differential operators." Proc. Royal Soc. Edin-burgh 92A:31-49.

Arscott, F. M. 1981. "A Riccati-type transformation of linear difference equa-tions." Congressus Numerantium 30:197-202.

Coppel, W. A. 1971. "Disconjugacy." Lecture Notes in Mathematics 220,Springer-Verlag, New York.

Gautschi, W. 1967. "Computational aspects of three-term recurrence relations."SIAM Rev. 9:24-82.

Hartman, P. 1957. "Self-adjoint, nonoscillatory systems of ordinary, secondorder, linear differential equations." Duke Math. J. 24:25-36.

Hartman, P. 1964. Ordinary Differential Equations. John Wiley, New York.

Hartman, P. 1978. "Difference equations: disconjugacy, principal solutions,Green's functions, complete monotonicity." Trans. Amer. Math. Soc. 246:1-30.

11 v

Hooker, J. W., an,' Patula, W. T. 1981. "Riccati type transformations for second-order linear difference equations." J. Math. Anal. Apple. 82:451-462.

Hooker, J. W., Kwong, M. N., and Patula, W. T. 1984. "Riccati type transformationsfor second-order linear difference equations II." J. Math. Anal. Apple. (toappear).

Olver, F. W. J., and Sookne, D. J. 1972. "Note on backward recurrence algo-rithms." Math. Comput. 26:941-947.

Patula, W. T. 1979. "Growth and osciLation properties of second order lineardifference equations." SIAM J. Matl. .ial. 10:55-61.

Reid, W. T. 1958. "Principal solutions of non-oscillatory, self-adjoint lineardifferential systems." Pacific J. Math. 8:147-169.

Reid, W. T. 1963. "Riccati matrix differential equations and non-oscillation cri-teria for associated linear differential systems." Pacific J. Math. 13:665-685.

Reid, W. T. 1972. "Riccati differential equations." Mathematics in Science andEngineering, 86. Academic Press, New York.

Vaughan, D. R. 1970. "A nonrecursive algebraic solution for the discrete Riccatiequation." IEEE Trans. Automatic Control 15:597-599.

13

OSCILLATION AND SPECTRAL PROPERTIES OF WEAKLY COUPLEDELLIPTIC SYSTEMS

W. Allegretto*

AbstractWe establish the connection between the nonoscillation of the weakly coupled el-liptic system id = -M + Ai and the finiteness of the negative spectrum S_(L) ofthe associated Friedrich's extension L. We next show how criteria for the finite-ness of S.(L) can be obtained from comparison with scalar equat ons and fromcomparison with systems of ordinary differential equations. As an application, wethen give an extension to of Kneser's classical nonoscillation theorem.

Let G be an unbounded domain in R", with n > 2 for convenience, andsmooth boundary 8G. Consider first in C (G) the expression

go= -Ago 4-qro

with q a real function. Here A is the Laplace operator and q is assumed tosatisfy the following conditions:

(i) q cLa(S) with a = a(S) > n for any bounded subdomin S of G, and

(ii) for all z E8G there is a neighborhood R such that q E C(R), # = 1(R) > 0.

These conditions may be relaxed somewhat, but they are simple and allow aunified presentation. Let B be the form associated with t in the usual way:B(rp,p) = f G (D )2 + go 2. Following Glazman [1965], we term B (or 1)nonoscillatory iff there exists a neighborhood N of m such that if ScNnG, Sa bounded domain, then B(Sp,9 ) z k (o,p) with k = k (S) > 0, SE Co (S). Thisis clearly a localized property. In the middle to late 1970s, it was shown thatB was nonoscillatory iff there existed fjr= in L2 (G) such that B(p,p) z 0 ifpoLi" in L2 for j = 1,...,r and SEC (G). From this it followed that if L wasthe Friedrichs extension of I and S_(L) denoted the negative spectrum of L,then S_(L) was finite iff B was nonoscillatory [Allegretto, 1981; Moss andPiepenbrink, 1978]. Hence the finiteness of S_(L) was determined near thesingularity: co. One could change the boundary conditions, coefficients,domain G, in a compact set without affecting this property. Observe, howev-er, that these results do not determine when S-(L) is finite in any specificcase. For this purpose oscillation or nonoscillation criteria are needed.There is a considerable amount of literature on this subject; we refer in par-ticular to the books of Swanson [1968] and Kreith [1973] and to the more re-cent survey articles [Swanson, 1975 and 1978].

Motivated by these considerations, we recently looked at a related prob-lem: the case of a weakly coupled system. Consider now the expression

Lt = -fi + AI, (1)

where A = (aj) is an mxm symmetric matrix and 21 = (u1 , . . . , u2 )r. Forconvenience, assume that aw satisfies the same regularity properties as q

*Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada TG 2G. Theauthor wishes to thank the organizers at Argonne National Laboratory for the opportunity to partici-pate in the Workshop on Spectral Theory of Sturm-Liouville Differential Operators.

14

for i,j = .... ,m.If (1) decomposes into smaller subsystems, then what follows can be ap-

plied subsystem by subsystem. We thus consider explicitly only cases wherethis does not happen. Specifically, it suffices but is not necessary, to as-sume that there exists a permutation a of , such thata,( 0 q.(t+ 1), a,(m.i() are not identically zero near infinity, i = 1...,m-1.Whether A is irreducible at any specific point is irrelevant. Nonoscillation isdefined in the obvious way, and BL are as before. The following result wasrerentiy established [Allegretto, 1984], and the proof will appear elsewhere.

THEOREM 1. Assume Tl = 2, and set q = (-a 12 ,(a22 -a 11 )/ 2), Q = q/ Ig I ( I|isthe Euclidean norm) with domain Q = J |q (x) # 0. Suppose there exists aneighborhood N ofan such that Q(x) IxENndom(Q)j is inside an arc of angles n. Then S.(L) is finite iff B is nonoscillatory.

It is not known if the result is valid without specifying the relativebehavior of the aq near cm. Note that if a12 is of fixed sign near , then thecondition on Q is satisfied.

These remarks indicate that, as was the case for m = 1, it is of interestto establish oscillation and nonoscillation criteria for (1) as a tool for lookingat S.(L). Observe, however, that we are not concerned here with deter-minants of prepared matrices nor with h-oscillatory solutions (see Kreith[1973] and Swanson [1968, 1975, and 1978] for clarification of these con-cepts). As far as the relative behavior of ag is concerned, note that ifB1(O,0) z B(0,$) for all EC (G) and S...(L) < cm, then S..(L1 ) < ec, where L1is the operator associated with B1, regardless of whether or not thecoefficients of L1 satisfy the conditions at infinity. This leads naturally to acomparison with the scalar equation -Au + pu, where u is the least eigen-value of A. Reversing these arguments leads to a comparison with -Au + duand S.(L) = criteria, where 6 is the largest eigenvalue of A. Such scalarcomparisons can obviously be optimal in some cases, but they may bemisleading in others. We illustrate this remark with the following theorem,based on considerations of Martynov [1965] for ordinary systems and the re-lated problem of spectrum discreteness.

THEOREM 2. Let m = 2. Assume X1,X2 are the eigenvalues for A with associat-ed normalized (in R m ) eigenvectors 01,02. Let p(x) measure the rotation of

01,02: --- (01) = 02 and -8-(02) =- 1x for k = 1,...,n. Assume

that A ,1 ,p are smooth and for some positive constants e,6 with e2 < 1/ 2 theinequalities

X1+ Iop - 62- 4_ x I- [1-2e2]

Xs+I|pI2(1-?- - 12(Ap)2 22

hold near infinity. Then I is nonoscillatory and S.(L) is finite.

PROOF. Let iECo (G) be given, and set = ai + 902. A simple calculationshows

= f [( -a_)2 + ( - )2] + a2 [X1 + IVp I2]{ tA

15

+ #2[x + |Vp 2] + fE[a -- 9- ]2a- -

= I1 + 2.

Now the divergence theorem yields

I2 = f-E4P - __ -- 2aflp.

Estimating this in the usual way leads to

B(,#) af(1-2s2)IVa|j2+ (X1+ IVpI2|-6 2)a2

+ jVp2 + f#2[x2 + IVp 12 ( 4) (A)]

Since the right-hand side represents a diagonal system, we merely applyKneser's theorem to each component.

Observe that if tp = 0, we may choose 6 = 0. In any case, if I|Vp I and2 - X1 are sufficiently large (depending on p) near cc, then X 1 may be chosenarbitrarily, so that the scalar comparison fails. Analogously, we have the fol-lowing theorem.

THEOREM 3. If one of the following conditions holds near infinity, then B isoscillatory and S.(L) is infinite:

a. X1 + jVp1 2 + 62 41z 12 (1+2E2);

b. x+ |Vp 2(1+3) +C(p)2 4Iz I2

where E, > 0, c < -(n -2) 2.This shows that, once again, one of the eigenvalues may be chosen arbi-

trarily. Consequently, the scalar comparison fails again.To indicate how further criteria may be obtained, we introduce the fol-

lowing notation. We write A ;B iff b so. for all ij, while A L B signifiesthat A - B is nonnegative definite. Clearly A B and A z B are differentconcepts. Assume G is an exterior domain.

THEOREM 4. Let a 0, i j . Assume (A {A} a.E Cfor convenience,and let . be the associated form. If H is nonoscflatory, then S(L) isfinite.

PROOF. Observe that . nonoscillatory implies that B is nonoscillatory, since(|si, ll) s B( s1,|s1) s B(p,) by the sign condition, where

I|I = + 0-. We now apply the general version of Theorem 1; see Allegret-to [1984].

Let Co-R denote the radial Co functions, and assume at1 are radialfunctions. The obvious choice is adg(r) s inf [ay(z)].

lz1=r

THEOREM 5. Let aj s 0, i ? j. If 1 Co-R is nonoscillatory, then . Ci' isnonoscillatory.

PROOF. Under these conditions there exists a radial vector 77 > 6 such that

16

-AV + 2 = 6 near infinity. From this it follows that Ca' is nonoscillato-ry; see again Allegretto [1984].

Theorems 4 and 5 show that criteria for the nonoscillation of ordinarysystems may be used to obtain finiteness criteria for the negative spectrumof the partial differential operator L. This is a classic way of dealing with theproblem if in. = 1 (see, again, Kreith [1973] and Swanson [1968]). The signcondition ac- s 0 may be modified by transformations, but this may meanthat inf should be replaced by max in some cases. This will happen, e.g., if

I:Il=rm = 2 and aC1 2 0.

We observe that the proof of Theorem 5 also establishes the connectionbetween positive supersolutions of the homogeneous equation and the finite-ness of S_(L). From this observation, finiteness conditions may easily be es-tablished by selecting a > a priori. As a very special example, we formu-late a Kneser theorem.

THEOREM 6. Let m = 2, a12 1 0, G an exterior domain. If there exists a con-stantc, -m <.c < cc, and a neighborhood N of m in which

ll+c l x--+2 -i C 4c2 -(n -2)24|z|

then S.(L) is finite.

PROOF. We merely select O = (ra,rP)T with a = 2-n and = a + c, r = |1.

The assumed inequalities then imply -All + AOl 9 and the result follows.

One obvious choice for c in Theorem 6 is c = 0, but clearly otherchoices of c may be more advantageous in cases where ail, a 12 , and a2 2 growat different rates.

It would be most desirable to obtain nonoscillation theorems that takeadvantage of the rotation of the eigenvectors, as was done in Theorems 2 and3. It is not clear, however, how this can be done in any generality for largem, without the calculation of the eigenvalues.

References

Allegretto, W. 1981. "Positive solutions of elliptic equation in unboundeddomains." J. Math. Anal. Apple. 84:372-380.

Allegretto, W. 1984. "Positive solutions and spectral properties of weaklycoupled elliptic systems" (in preparation).

Glazman, I. M. 1965. Direct Methods of Qualitative Spectral Analysis of9fgular Differential Operators. Davey and Co., New York.

Kreith, K. 1973. Ocillation Theory. In Lecture Notes in Mathematics, Vol.324. Springer-Verlag, Berlin.

Martynov, V. V. 1965. "Conditions for discreteness and continuity of thespectrum for a fourth order selfadjoint system of differential equa-

17(\$

tions." Differ. Urav. 1:1578-1591.

Moss, W., and Piepenbrink, J. 19978. "Positive solute.-is of elliptic equations."Pacific J. Math. 75:219-226.

Swanson, C. A. 1968. Comparison an m C citlation Theory of LinearDifferential Equations. Academic Press, New York.

Swanson, C. A. 1975. "Picone's identity." Rend. Mat. 8:373-397.

Swanson, C. A. 1978. "A dichotomy of PDE Sturmian theory." SIAM Review20:285-300.

19

POTENTIALS HAVING EXTREMAL EIGENVALUES SUBJECT TO p-NORM CONSTRAINTS

M. S. Ashbaugh*E. M. Harrell II**

Abstract

We consider the Sturm-Liouville operator Hy = d+ V(t) on certain subsets of

the real line with various selfadjoint boundary conditions. We find the optimalupper and lower bounds for the eigenvalues of Hv when the potential V obeys aconstraint of the form VIIp, s M. We characterize the extremizing potentials inthose cases where they exist. Analysis of this one-dimensional problem is facili-tated by interpreting it in terms of a classical oscillator.

1. Introduction

In this paper we address the problem of finding optimal bounds for theeigenvalues of the operator

-d2Hy = dt2+ V(t) (1.1)

on certain subsets Q of the real line (finite interval, half-line, line) with a varietyof boundary conditions subject to p-norm constraints on the potential functionV. To be more precise, having fixed an interval, a set of boundary conditions,and an index k z 0, we find optimal upper and lower bounds for Ek (V) where V isallowed to range over the set S = VELP())I I IVIIp s Mi. Here Ek(V) denotes the(k +1)th eigenvalue of Hy as defined by the min-max principle [Reed and Simon,1972-79]. These bounds depend on S only through the constant M and, as willbe made clear shortly, give upper and lower bounds for E k (V) in terms of |IVljy.

Our interest in such problems was first stimulated by a problem list of A. G.Ramm [1982] in which the problem of maximizing E0 (V), where Hy acts on afinite interval, has Dirichlet boundary conditions, and V is subjected to a 1-normconstraint, was posed. In particular, in an earlier paper [Harrell, 1984], themaximization problem was analyzed for Ek (V) on a finite interval with variousselfadjoint boundary conditions, while laying the foundations for a solution tothe problem with general p-norm constraints and also for multidimensionalproblems, i.e., for Hy = -A + V(z) acting on a set OcR', d 2, with suitableboundary conditions. Much of the groundwork for the present study was laid inthat paper, and henceforth we shall refer to it as article I. In a paper currentlyin preparation, we shall give the results of our investigations into the multidi-rensional case, as well as further material and some of the proofs dealing withthe one-dimensional case. The multidimensional case turns out to be closelyrelated to the problem of best constants in Sobolev's inequality and certain non-linear elliptic partial differential equations which have been the subject of muchcurrent interest [Brezis and Nirenberg, 1983; Lions, 1982].

*Department of Mathematics, University of Missouri, Columbia, Missouri 65211. Work supportedby a Summer Research Fellowship granted by the Research Council of the University of Missouri -Columbia.

*School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160.Work par-tially supported by USNSF grant MCS 8300551 and an AlfredP. Sioan Fellowship.

20

Following the publication of Ramm's problem list, several other authorssolved the problem posed above and, in some cases, pursued generalizations,restrictions, or related problems of their own. Solutions of which we are awareare those by Essen [1983], Farris [1982], and Talenti [1983]. Talenti, in particu-lar, solved not only the problem posed by Ramm but also the problem of minim-izing E0 (V) under the same hypotheses and of minimizing E0 (V) under the con-ditions V;e 0, 11V11, = U, and IIVI1.. = B. The extremizing potentials that Talentifinds have more than a passing resemblance to those found by Krein [1955] inhis investigation of a similar problem for the equation of the vibrating string,y" + Xp(z)y = 0 on [0,] subject toy (0) = y (L) = 0.

Independently of this, there accumulated over the last 15 years or no abody of literature among workers in ordinary differential equations giving lowerbounds for the operator Hy in terms of a given p-norm of V. The relevantpapers are those by Everitt [1972], Eastham [1972-72], Evans [1981], and Veling[1982 and 1983]. Each of these authors obtained a lower bound for Hy acting onL2(0,cc) of the form -c I i11 where c and a are constants depending on p. Eachhad the correct exponent a = 2p/ (2p-1), but Veling was the first to find theoptimal value of the constant c. All of these authors dealt with a Dirichet boun-dary condition at t = 0 and, to varying extents, certain other standard boundaryconditions. In particular, Veling [1982] gives the optimal lower bound of theform -cjI V1J for Hy on L2 (0,oo) with either a Dirichlet or Neumann boundary con-dition at t = 0. Also, Veling [1983] states the optimal bound for Hy on L2(f).Not surprisingly, there is a close connection between the three bounds dis-cussed by Veling.

There is yet another line of work that is closely related to our current inves-tigation. This work has been pursued in the mathematical physics community inan effort to get accurate bounds on the number of bound states of a Schr6dingeroperator and the slightly more restricted problem of obtaining optimal condi-tions for absence of bound states. The work most closely bearing on our own isthat of Glaser, Martin, Grosse, and Thirring [1976], Glaser, Grosse, and Martin[1978], and Lieb and Thirring [1976]. These papers treat problems by methodsthat are similar in many respects to our own, though since they have somewhatdifferent objectives, our results are largely disjoint from theirs.

Finally, in a f orthcomin book by Trubowitz [1984] the problem of extremiz-ing E (V) for Hy acting on L (0,1) with Dirichlet boundary conditions and with Vsubjected to a 2-norm constraint is posed and its solution is outlined in hints.One finds in this case that the extremizing potentials have explicit representa-tions in terms of elliptic functions. We shall see shortly that the case p = 3 alsoleads to elliptic functions and, moreover, that qualitatively the solutions in thecase of general p are very much the same. This situation is brought out mostclearly by discussing the general problem in the context of classical mechanics.It is also worthy of note that elliptic functions arise in the problem of minimizingresonance widths within a suitable class of potentials [Harrell and Svirsky, 1984]and that the potentials for which Hill's equation has precisely one nonvanishingfinite instability interval are elliptic functions [Hochstadt, 1976].

2. General Remarks

Since many of our arguments are not special to one dimension, we find itappropriate to include them in our longer paper [Ashbaugh and Harrell, 1984]and only to summarize them here. In addition we present those results of Har-rell [1984] on which we base our current analysis.

In any problem involving maximizing or minimizing a functional, one isimmediately confronted with the following questions:

21

1. (Semiboundedness) Does the appropriate supremum or infimum exist?

2. Can we find (or estimate) this value?

3. (Existence) Is there an optimizing function, i.e., a function at which thefunctional attains its sup (inf)?

4. (Characterization) What are the optimizing functions?

5. (Uniqueness) Is there a unique optimizing function?

General results [Ashbaugh and Harrell, 1984] give affirmative answers to ques-tions 1, 3, and 5 in most cases of interest. Exceptions for questions 3 and 5 doarise and will be discussed at the appropriate point. Our main thrust in thispaper will be toward answering question 4 and, to a lesser extent, 2. It willtranspire that our answer to question 4 will often answer question 5 as abyproduct. This is because our approach to characterization is to study theequation

-u" (sgn u)u jI(P+1)/(P-1) = Eu (2.1)

whiih, together with appropriate boundary conditions, was shown in article I(with the + sign only, for maximizers) to be a necessary condition fort Iu j2/(P-1) to be an optimizing potential for p > 1. (For additional commentson the sense in which this equation holds and on the domain on which it holds,see Ashbaugh and Harrell [1984].) Thus, for instance, if we already haveexistence and can show that equation (2.1) has only one solution of the requiredtype, then uniqueness follows immediately.

One further remark about the formulation of our problem seems appropri-ate here. While the requirement that the potential function V be locally L isoften regarded as the weakest reasonable condition (see, for example, the com-rerts in Eastham and Kalf [1982: p. 4]), we have occasion to consider the opera-tor H,, where represents a Borel measure. As pointed out to us by BarrySuion, this provides a reasonable operator since one can show that is a rela-tively form-compact perturbation of Ho = -d2 / dt2 using Fourier transforms. Infact, for H. acting on L2 (R) in Fourier transform space, the kernel of(Ho-'-1)MA(Ho+1)-,

K(k 1,k 2 ) = (k? +1)~92(k1 -k 2 )(k&2+1)-, (2.2)

is easily shown to be Hilbert-Schmidt since A is a bounded continuous function.(Essentially we are defining the operator HA by means of quadratic forms inFourier transform space.) The cases where H. has other domains are handledsimilarly by suitable choice of "Fourier transform." Allowing V to be a measureis crucial to the eigenvalue minimization problem when p = 1 since the ball ofradius M > 0 in L1 has no extreme points, but it is easy to see that an eigenvalueminimizer must be an extreme point using the Rayleigh-Ritz inequality. Thuswhen p = 1, minimizing potentials do not exist. However, if we allow V to lie inthe larger class of all finite Borel measures, then we can obtain an existenceresult. For example, as exhibited by Talenti [1983], the minimizing potential fora finite interval with Dirichlet boundary conditions is a centered 6-function. Withslight modifications the above relative compactness argument also applies toVELA, 1 <p ! 2. This observation is useful in the one-dimensional case sinceour general methods and results handle only p z 2.

Even after restricting attention to the one-dimensional case, there arequite a variety of problems to be considered. First, one can consider the

problem either of maximization or of minimization over a setS = IVELP(fl) j |1V11, s M. Since by the min-max principle it is easy to showthat a maximizing (minimizing) potential satisfies V;a 0 (VT 0) and ||VII, = M, itis a small step to consider what we shall call the mistre problem of minimizingwithin the class V z 0, ||VI| = M (maximizing within the class V 0, 1|V, = M).We will see, in fact, that the misere problems do not have extremizers and thatthe optimal bounds are the appropriate V = 0 eigenvalues. Second, one has thethree choices of domain to consider: finite interval, half-line, and line. Third,one can impose a variety of boundary conditions at the finite endpoints of thedomain. Those with which we shall deal are Dirichet, Neumann, separated (i.e.,au(t%) + fu'(t{) = 0 where t is an endpoint), and "compact-support" boundaryconditions. Since this last terminology is not standard, we explain: These arethe boundary conditions one gets at L if one allows -o < t < m but requires Vto have support in the interval [-1.,L]. In particular, they take the form

u'(-t) = t -~u(tl).

Lastly, one can concentrate on any eigenvalue Ek(6') for k = 0,1,2,.... Theground state E0(V) is perhaps the most interesting, and in fact we can get moredetailed results about it (partly because more tools are available for studyingit). The ground state is also unique compared to higher states in that for a givenproblem certain results will hold for the ground state but for no excited states.For example, the finite-intervalp= 1 maximization problem has a unique max-imizer for E0 (V) but not for E(V), k z 1 [Harrell, 1984]. As a second example,on R with p > 1 there exists a ground-state minimizer (unique up to transla-tions), but minimizers for the higher states do not exist. However, the generalmethod and viewpoint presented here lend a degree of unity to the various casesand problems outlined above. In particular, the method applies to a largeextent equally to the ground and excited states.

3. The Classical Oscillator ViewpointWhile we chose time as the independent variable with the classical oscillator

interpretation in mind, we find it convenient here to set forth other standardnotations from the perspective of classical mechanics. For a modern and morecomprehensive discussion of classical mechanics, we refer the reader to therecent book by Thirring [1978]. By viewing equation (2.1) as Newton's equationfor motion in one dimension (u represents position), we can identify the classi-cal potential energy as

W(u;E) = 12 U t(&)iu 1R/-'). (3.1)

Note that the quantum energy E appears as a coefficient in this classical poten-tial. A first integral for this system is given by

(d 2+ W(u;E)=h, (3.2)2 dt

where we have let h denote the classical (total) energy of our oscillator. Ourconvention for the ambiguous sign in all equations - (2.1) and (3.1) thus far - isthat we take upper signs when considering maximization problems and lowersigns when considering minimization problems.

Though we will refer to the above equations as describing an oscillator, forcertain choices of the sign referred to above and the sign of E one will not haveoscillations or will have oscillations only for suitable initial values. For the mostcommon boundary conditions (Dirichlet, Neumann) only the truly oscillatorysolutions will enter, but with more complicated conditions other solutions cansometimes come into play.

23

We will refer to the curves given parametrically by (u(t), u'(t)), where usolves equation (2.1) as trajectories in phase space. Of course, the oscillatorysolutions referred to above are just the closed orbits in phase.space. In phasespace, separated boundary conditions (Dirichet and Neumann included) can beviewed geometrically as the condition that a trajectory start on a given linethrough the origin and end on a second line through the origin (possibly thesame) at a specified later time. When the interval is finite, we choose it as[0,1], 1 > 0, or sometimes [-1,1]; for the half-line we choose [0,0).

4. Minimization on the Line and the HaLf-Line

We begin our detailed discussion with these cases since from the classicaloscillator viewpoint the constant h must be 0, which simplifies the analysis. Alsothese are the cases that have drawn attention previously. Now since u is an L2

solution to Hyu = Eu, where V = - I u 12/(P~)ELP, we can be sure from the

theory of Schrodinger operators [Reed and Simon, 1972-79; Richtmyer, 1978]that u and u' go to 0 as t goes to w. Thus on infinite intervals our only concernis with classical oscillator solutions having total energy h = 0, and we need onlysolve the equation

L( dug2 + 1Eu2 + (P-1 )u2P/(P-1) = 0. (4.1)2 dt 2 2p

This equation is readily integrated, with the result that

u(t) = ( )(P-1)/2 sechP~1[--(t-c(p-1 p-1(4

and hence

V(t) = E sech2[ -- ~ (t -c ) (4.3)p-1 p-1

Here c is the constant of integration. For the minimization problem on the line,it represents the expected fact that a minimizing potential cannot be uniquebecause of translation invariance. For half-line problems, the constant wouldhave to be chosen so that u satisfies the boundary condition at the origin. Weshall see shortly that this has the interesting consequence that no minimizersexist for certain choices of the boundary condition. But first let us finish ourdiscussion of the standard cases.

For the full line minimization problem one can compute

I IVII _ P ( -E )(2p~-1/ 2

)P1 _K, B(p,),(4.4)

or, solving for E,

E = -[ -1 ')] 2 /(2P-1) I1I/(P-). (4.5)pP B(p,. )

Here B(p,X) represents a beta function in standard notation. This formula isthat given by Veling [1983] except for a misprint of (1-6)tT/"-) as (1-6)1/(-4).

For the half-line problem with Neumann boundary condition one must takec = 0 in equation (4.3). The computation can be carried out as before, yielding

F' = -22/(2-1) [( -1 -' ]/(2-1) | Vp/(p-1, (4.6)pPB(p,

again agreeing with a result of Veling [1982].

We now consider the general boundary condition

24

U'(0) = muu(0) (4.7)

From equation (4.2) tis reduces to

m = V-tanh(V~c/(p-1)) (4.8)

which has a solution for c if and only if v~E> ImL. Holding E fixed, we seethat as m -+/~T from below, c ->, and that as m -+f-v~ from above, c f-o.Thus as min-- our sech2 -potential well translates off to the left, "leaving"the positive half-axis, and as m-+/ it translates to the right into the positivehalf-axis. We can better understand what is happening here if we note that thepotential V = 0 with boundary condition (4.7) has a negative eigenvalue atE = -m. 2 ifin < 0. Thus a fixed E < 0 will not be minimal for the operator Hy onL2(0,oo) with boundary condition (4.7) for m <0 until m increases to -vE/. Atthat value of m, E will be minimal for IIVI = M = 0. For Iml <V<-E , E will beminimal for IVJJ, fixed as required by equations (4.3) and (4.8). One could writethe relation betwen E and IIVlI for this range of m in terms of the incompletebeta function, but we refrain from doing so here. When m exceeds v~E, one nolonger has a minimizing potential, but a minimizing sequence of potentials iseasily constructed by taking a sequence of Vs given by equation (4.6) with c'sgoing tom and suitably modified on [0,1], say, to meet the boundary condition att = 0. This latter situation also includes the case of Dirichet boundary condi-tions. In these cases the value E in equation (4.5) is a strict lower bound for theground state and hence also for the operator Hy.

We close this section with some cursory remarks about higher eigenvalues.To obtain a minimizing sequence of potentials for a higher eigenvalue, one"pastes on" more sech2 -potential wells out near infinity. The modificationsrequired in the pasting can be shown to have vanishing effect as the spacingbetween consecutive wells is sent to infinity. We note that the potentials in theminimizing sequence for the k-th eigenvalue approach k-fold degeneracy, i.e.,the first k eigenvalues come together in the limit. The appropriate eigenfunc-tion in this case is much like the potential (to the power (p -1)/ 2) except thatwe flip its sign each time we paste on a new piece; on [0,co) we also must rescalethe left-most bump so that its L2 norm is the same as all the others. As an illus-tration one obtains the bound

E1( V > -2-2/(O -1) ([(p -1)"~ '«/ -1) ap,2/ (2 -1)pPB(p,)

in the case of the second eigenvalue of Hy acting on L2 (R).

To those familiar with high-energy physics, there is more than a passingsimilarity between the above construction of minimizing sequences and the con-struction of a multiple instanton configuration. We also remark that the sech2

form of our potential is precisely a soliton solution to the Korteweg de Vries(KdV) equation. There is an extensive literature detailing the intimate connec-tions between the KdV equation and the Schr6dinger equation; we content our-selves with noting that the article [Lieb and Thirring, 1976] presents some par-ticularly pertinent observations of P. Lax.

5. Minimization on a Finite IntervalWhen one seeks to find eigenvalue minimizers on a finite interval, one must

consider equation (3.2) wth all allowed values of the classical energy h. Weadopt the following strategy in this discussion: with fixed p > 1 and interval [0,1],we pick a possible optimal eigenvalue E and choose suitable boundary condi-tions; then we look for those values of h that allow u to meet the boundary con-ditions at t = 0 and t = L; and finally we determine the value M = I|Vt|p for whichV = -ju 1/(P -1) is a possible minimizer. If at the end of this process we have

25

only one candidate, then, having already proved existence of a minimizer [Ash-baugh and Harrell, 1984], we can conclude that we have found the unique minim-izer. Even if we find several candidates, the existence result guarantees that atleast one of them will be a minimizer. Existence of minimizers on a finite inter-val when V is allowed to range over the class of Borel measures satisfyingfdI l s M is shown in our longer paper. This result handles the minimizationquestion when p = 1.

We begin our discussion by considering Dirichet boundary conditions andtaking E < 0. Then the only h's for which Dirichet conditions can be met areh > 0, and the time required for one excursion (half the period of the orbit) is

h 1

T(h,E)/ 2 = Jf[h -W (u;E)]-*du, (5.1)0

where ui represents the positive turning point of the motion, i.e.,W(u1 ;E) = h, ui > 0. To see how T(h,E) varies with h we eliminate h in favor of

u1 while noting that the mapping h -'it1 is an increasing function from (0, m) onto(u 1 , co) where u imin satisfies 0 = W(u 1,s;E). One has

U1

T = 2vf [W(u1 ;E) - W(u;E)]~*du (5.2a)0

= 2, f [E(u2 -u 2)/ 2 + (p-1)u4P/(P1) -u p/(p~/ ) p] du

= 2Vgf[(p -1)u i4'P~ (1-s2P(P-1))/ 2p + E(1-s2 )/ 2]~ds, (5.2b)

where we changed variables to s = u/u1 to arrive at the last expression. Thusone sees that T decreases from cc to 0 as h increases from 0 to cc. Since toaccommodate the (k+1)th eigenvalue Ek we need

(k+1)T(h,Ek)/2 = 1 (5.3)

to be satisfied, we see that any E < 0 can be a minimal (k +1)-th eigenvalue forany k z 0. A similar analysis leads to the same conclusion when E = 0. WhenE > 0, one finds that the period T decreases from 2r// to 0 as h increasesfrom 0 to m. Thus if E > (k +1)2r2/ 12, then E cannot be a minimal E, whereas ifE ! (k +1) 2rr2 /1 2, it will be attainable as a minimal Ek. If one notes thatE (0) = (k + 1) 2rr2/ 12, the reasonableness of these conditons is apparent. Actu-ally, to complete this discussion, we must look at the equilibrium solutions, i.e.,the critical points in the phase plane. These solutions are exceptional in thatthere is not a fixed period associated with them. For the above, the only criticalpoint solution of relevance is u = 0, which is trivial to analyze.

With Neumann boundary conditions the same considerations apply for theorbits and their periods as discussed above. However, there are additionalorbits having h <0 to be considered in the case of E < 0, including anotherequilibrium solution corresponding to the minimum of W(u;E). This compli-cates the indexing of the eigenvalues somewhat, but Sturm's theorem on nodesof eigenfunctions suffices to sort things out. The orbits considered previouslylead to candidates for minimal EA, k a 1, under the condition

kT(h,Ek)/ 2 = 1, (5.4)

and the newly considered orbits lead to candidates for E0 since they give node-less solutions. Again any E ! 0 can be a minimal Neumann Ek, k z 0, but for

28

E > 0, E >k2 rr2/ 12 precludes E from being a minimal E and E : k2 ir2/L 2 allowsit. That all allowed E's are actually assumed as minimal E's for some choice ofM = VI, follows from continuity considerations which are taken up by Ash-baugh and Harrell [1984].

Other choices of separated boundary conditions at t = 0 and t = I will forceus to consider more complicated conditions than (5.3) or (5.4) for meeting theboundary conditions. In fact, trajectories that are not closed orbits will evenenter: the appropriate point of view is that we need to find those trajectoriesthat take time I to pass from one line through the origin to a second linethrough the origin in phase space. Periodic or antiperiodic boundary conditionslead back to the same orbits as were discussed in the Neumann case, as doseparated boundary conditions of "periodic type": u'(0)= mu(0), u'(1) = mu (L), mER

6. Marimization on a Finite IntervalThe analysis of the maximization problem differs only in detail from that of

the minimization problem. The most significant difference is that the potentialW(u;E)is now upside down; in particular, W-+-ao as u-'o. This has the effectthat for all standard boundary conditions only K 0 need be considered. Byanalyzing T(h,E), one finds in this case that 27r/vT s T(h,E) < m for the per-missible values of h. Thus E < (k + 1)2 r 2 / 12 implies that E cannot be anextremal (k +1)-th eigenvalue for the Dirichlet problem whereasE z (k +1)2 7/ 12 can be. As should be clear, the discussion of this problemparallels almost exactly that of the previous section, so we conclude it here.

7. Misfre ProblemsWe turn now to a brief discussion of the misdre problem, that of minimizing

(respectively, maximizing) a given eigenvalue when V is constrained to the classS = V| V a 0,1V6= M; (resp., S = JVI V s 0,V6=4 M). We shalliconfine themajority of our remarks to the case of the ground state for Dirichlet boundaryconditions which we shall denote by E(V).

We begin by considering the minimization problem with V z 0 where OcRd isbounded and has smooth boundaries. The case of unbounded domains for thisminimization problem is of no interest since E(V) (as defined by the min-maxprinciple) is then always 0 = E(0). We shall show that (1) E(V) > E(0) for allVES and (2) infE(V) = E(0). Thus there is no V that is a minimizer for this

misfre problem. To obtain (1), we simply use the Rayleigh-Ritz inequality for -Awith Oy, the normalized ground-state eigenfunction of Hy, as trial function:E(V) = (0y,(-A+ V)#y) = (#y,-Aiy) + f V0Y v 2> E(0). To prove (2), note thatsince the ground state, 00, of -A on 0 wih Dirichlet boundary conditions goes to0 on &0 and since 80 is smooth, we can nind a sequence of sets B c0 satisfying(i) sup ISoI s 1/fn and (ii) 0 < IB, I < K, K a constant independent of n. Then

B,with V = M|BI-VPX and again using Rayleigh-Ritz, we compute

E(V) < (00,(-A+V)#0 ) = E(0) + fI|#2MIBI-l/P s E(0) + MK- 1/~p/n

2 ,

which goes to E(0) with increasing n.The problem of maximizing over S = Vj V 0, |1V Il = MH, p z 1, is more

difficult to analyze, but leads to much the same result. That E(V) s E(0) isagain a consequence of Rayleigh-Ritz or, more precisely, the min-max principle.When E(0) is in the discrete spectrum, this inequality is strict; in any event,there is no VES for which -A + V(z) has E(0) as an isolated eigenvalue of finite

27

multiplicity. This, together with the fact that supE(V) = E(0) (to be shown

shortly) shows that this misere problem also lacks an optimizer in the sense ofE(0) being attained as a point in the discrete spectrum. In some cases, E(0) isattained by E(V) for some VES; but in cases where E(V) is in the discrete spec-trum for all VES, this cannot occur. All problems where 0 is bounded and thecases 0 = Rd, d = 1 or 2, fall in this latter class. If 0 is unbounded, one can con-struct a sequence JV3 of potentials in S having E(V)-0 = E(0) by usingV, = M IB, PYB where the sets Bc0 satisfy IB, +-o. Here we have usedwide but shallow square wells in our construction. For bounded domains thisavenue is not open to us, so we shall use narrow and deep square wells. We picka sequence of balls B c0 with IB. I 0 0 for all n and IB I -0. Then for p > 1and Vn = -M IB, ~WPX , we have IIVn I = MIB I1ie -0 as n-+w; and using thefact that our lower bound for E(V) in terms of 11V11 1 goes to E(0) as 11V11 1-+0 [Ash-baugh and Harrell, 1984], we see that E(V)+E(0). We remark that thissequence works equally well for 0 unbounded but has the drawback that it doesnot cover the case p =1. The essential observations in the above discussion arethat for 0 unbounded there is a sequence J V, in S also lying in L with II V,11..-+0and that for p>1 and arbitrary 0 there is a sequence V J in S also lying in L 1

with IIV II1-.0. These observations would also have sufficed in dealing with thernisdre minimization problem except for the case p =1. Indeed, except for thiscase, the argument given above could have been concluded just by choosing theB 's so that |B j-+0.

To complete the discussion, we need to treat the case of a bounded domainwhen p = 1. Just as in the misere minimization problem, our argument nowrests on our choice of Dirichlet boundary conditions. The idea is to take asequence rj approximating a 6-function located on 00 and argue that forV= -Mv7n, we have (On, V,#O)-+0 as n -' where On represents the normalizedground state for -A + V. However, here we shall give a proof only in the case ofdimension d = 1. In this case we may take 0 = [0,1], 1 > 0, and we define asequence of potentials V = -Mn)Qo.n. By standard methods found in any ele-mentary quantum mechanics textbook, one could give an explicit argumentshowing that E(V)-E(0) = n2/1 2 . Instead, we note that E(V) is the first eigen-value of the three-dimensional problem for -A + VR (r); we remark that this iswhere we make use of the Dirichet boundary condition. As a function in three-space, we have jIV I1 = Mn 3Tr (1/n)3 = 4irM/ 3n2-+0 as n -+ and thus, by the

result alluded to above, E( W)-'E(0), where the 0 represents the 0 potential onthe ball of radius I in R'. But, passing back to one dimension, we haveE(0) = n2/ 12, which completes the proof. Finally, we remark that except whenp = 1, Dirichlet boundary conditions were not needed; in particular, the lastargument works for arbitrary boundary conditions imposed at t = 1.

Acknowledgments

We would like to thank Toni Zettl, Hans Kaper, Gotskalk Halvorsen, andAngelo Mingarelli for helpful remarks, particularly in regard to the existingliterature. We also thank Barry Simon, John Piepenbrink, Hans Weinberger, andJoe Conlon for useful discussions. We are grateful to Jurgen Gerlach for a sug-gestive numerical study of square wells in one dimension. Finally, it is a pleas-ure to thank Hans Kaper for the opportunity to participate in the Sturm-Liouville workshop.

28

References

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Eastham, M. S. P. 1972-1973. "Semi-bounded second-order differential opera-tors." Proc. Roy. Soc. Edinburgh 72A:9-16.

Eastham, M. S. P., and Kalf, H. 1982. Schrndnger-type Operators with Cbntinu-ous Spectra. Pitman Publishing, Boston.

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Glaser, V., Grosse, H., and Martin, A. 1978. "Bounds on the number of eigen-values of the Schrbdinger operator." Commun. Math. Phys. 59:197-212.

Glaser, V., Martin, A., Grosse, H., and Thirring, W. 1976. "A Family of OptimalConditions for the Absence of Bound States in a Potential." Studies inMathematical Physics. Ed. E. H. Lieb, B. Simon, and A. S. Wightman.Princeton University Press, Princeton, pp. 169-194.

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Krein, M. G. 1951. "On Certain Problems on the Maximum and Minimum ofCharacteristic Values and on the Lyapunov Zones of Stability." Amer. Math.Soc. 7Trnslations, Ser. 2, Vol. 1. American Math. Soc., Providence, pp.163-187 (trans. of Prikl. Mat. Meh. 15:323-348).

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Veling, E. J. M. 1982. "Optimal lower bounds for the spectrum of a second orderlinear differential equation with a p-integrable coefficient." Proc. Roy. Soc.Edinburgh 92k95-101.

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31

INDEFINITE STURM-LIOUVILLE PROBLEMS

F. V. Atkinson*D. Jabon'

Abstract

We study regular Sturm-Liouville problems

Ly=-(py')'+qy = Xry on[a,b]with separated, selfadjoint boundary conditions, where r is assumed to changesign and L is not assumed to be positive definite. We look primarily at a specialcase, which seems to give some insight into the general case, obtaining boundsfor the so-called Richardson numbers of our problem and presenting the resultsof numerical calculations of the spectrum of our problem.

1. IntroductionWe consider problems of the following form:

Ly = -(pu')' + gy = Xry on (a, b), (1)

where

a ,b are finitep,q,r: [a,b ]-R

p(z)> 0 for almost all zE[a,b]

p-1 = -,qr EL [1,b]p

measures E[a,b]: r(z) > D >0measures E[a,b]: r(z) < 0>0r(z) 0 almost everywhere.

A solution must satisfy (1) as well as separated selfadjoint boundary conditions

A,y (a) +Ap (a)y'(a) = 0

By (b) + Bap (b)y'(b) = 0. (2)

Such problems are called "indefinite" because the operator L is not required tobe positive definite. As the following proposition shows, however, indefinite prob-lems differ from the definite ones only for finitely many eigenvalues, a fact firstproved by Richardson 11918].

PROOSITION 1. For problem (1), (2), if I X I is sufficiently large, the number ofzeros of the eigenfunctions associated with two successive eigenvalues differsby precisely one.

"University of Toronto.?Student Resident Associate, Argonne National Laboratory, Argonne, Illinois 60439.

32

PROOF. Definet6(z,xa)by

tany = 4(3)py.

with a(a,X) fixed. (This is the Prufer transformation.) We find that

'' = p-1cosZ6 + (Xr -q )since (' .(4)

If r > 0, we can conclude that 6 is increasing in X. for fixed z E(a,b).Without assuming that r > 0, we argue from (3) that

0A = ((py')y.\ - (py')a y 3/ ly2 + (py')2j ( ).(5)

Here y,py' are fixed at z = a. We find that the numerator in (5) isb

fryed. (6)

Hence ,6 is increasing in X if (6) is positive.

For suitable boundary conditions, in particular for (2), a solution of (1), (2)will satisfy

b b

Xfry2dz = f1py'2 + gydz. (7)a a

Hence, if the quadratic form

fpy2 + gy2 dp (8)

is positive for nontrivial differentiable functions satisfying (2), then '(b,X) isincreasing when X is a positive eigenvalue and decreasing when X is a negativeeigenvalue.

In general, the form (8), when diagonalized, may have a finite number ofnegative squares, the number of which may be ascertained by replacing r in (1),(2) by a nonnegative weight function, e.g., jI , or 1. If N is the number in ques-tion, then for at most N eigenvalues of (1), (2) we can have

't#(bX) < 0. (9)

The result follows because y (z) vanishes if and only if 't(z) is an integral multi-ple ofir. //

In light of Prop. 1, we may define

a+=inizERIVX>z, X(b,X)>0j

X-J= supjzERVX <z, x0(b,X) < 0j. (10)

We may interpret these quantities as follows: X+ is the smallest number suchthat the real eigenvalues greater than X+ behave as in a "typical" Sturm-Liouvilleproblem; that is, an eigenvalue is uniquely associated with its oscillation number(the number of zeros of the associated eigenfunction). X- is interpreted simi-larly. .

Note that if r(x) > 0, for all z c[a,b], then as was shown in the proof ofProp. 1, a+ X= a = -ac. If r is as in (1) and L is positive definite, A+ = X- = 0. Inthe indefinite case, the determination of these numbers is a very significantproblem; Mingarelli [1982] has quite appropriately called these quantities "theRichardson numbers" of the problem (1), (2). We can determine what we believe

33

to be a sharp bound on X+ and X for the following specific problem:

Ly = -y " + gy = Ary on [-1,1]

y (-1) =y (1) =o0(11)

r (z) = sgn(z) =-1 if z<0lif z<0

q(z) = go, goER

Note that for go >4, the operator L in (11) is positive definite; we are thus

interested in the case where g i<-. We now proceed to show that for (11),

with go < - =,+s Igo) - -(hence, X- z -I qo1 + by symmetry).4' 4 4

LEMMA1. In (0,1) let

y"~= -y , y< -4

y (0) = 0, y'(z) >O in [0,1].?hen f y2(t)dt < hy2(1).

PROOF. Suppose first that s 0. Then

0 < y', zE(0,1],and so

0 s y (z) s zy (1), z E[0,1].

Hence f y2 (t)dt s 2(1). Suppose next that3

0< < T.Then

y (z) = Csinkz, 0!9 z s 1

C>0,O'<k< IT

fy2(t)dt = Cfsinektdt = C2(1- sin2k).o o2 2k

Since 2L sin2k > cos2 k , we have the result. //

LEMMA 2. In (0,1) let

y= -.Uy,,> 0

y(0)= 0, y (1)y'(1) <0.

hen f y2(t)dt > y2(1),

34

PROOF. It is sufficient to suppose that

y(x) = sinkz, k > 0.

Since y (1)y'(1) > 0, we have

sin2k < 0.

But

fy2(t)dt = 2 (1- sin2k),

and here <0. Hence

fy2

(t)dt > 1> 1y2(1). //

PROPOSITION 2. Ifin(11) , go < - - thenX+ o--

PROOF. Let y satisfy (11), i.e.,

y"' + (X-go)y = 0, 0:9 z s 1

y" + (-X-qo)X = 0, -1!c z <0

withy(-1) = y(1) = 0.

If X > -go- -, then in (-1,0) we have -X-q 0 < -, and so applyingLemma 1 to the interval (-1,0), with shift of origin,

fy(t)dt < 1y2(1).-1 2In (0,1), we use Lemma 2 -replacing (0,1) by (1,0) -and have

fy 2 (t)dt >Fy"(1).

Hence, for an eigenvalue X such that

X> -qo -

we have

fy2(t)(sgn(t))dt > 0.

Now if -t(zA) is defined by

= 0

tani(zA) = y(zX)y'(zA)

where y (zA) is a nontrivial solution with y (-1,A) = 0, we have

35

(2=aQ= fy2(t)(sgn(t))dt.

Hence for eigenvalues X such that

X> -q 0***r2

we have a>s . //

Another approach to this problem (namely, the determination of X+ for(ii)) is the following one, which is less precise but slightly more general.

PROPOSITION 3. Let y (zX) be an eigenfunction of

y" + (ar+q)y = 0, y(a) = y(b) = 0 (12)(r,q as (1),(2))'with zeros

a=zo<z 1 < --- <z=b.For each j = 0,...k -1, Let there be a solution u* of

uj"+ (pjr+q) = 0such that u1j(z) > 0in(z3 9 ,z 1+]. Then X'for (12) is Less than X.

PROOF. The proof follows immediately from the Sturm comparison theorem byconsidering the function z3 = 1 -on [z,,zj 1+]. ///

lLj

If we specialize to the case where for some c E(a,b), r(z) > 0Din (c,b), thenwe have the following proposition.

PROPOSITION 4. For some &, letu" + ( r+q)u = 0

have a solution positive in [a .d]for some dE(c ,b ). For some X' > , lt there bea solution y = 0 with y (a) = 'and also a zero in (c,d). Then Xa < X (i.e., theeigenvalues for which X > X :* re uniquely associated with their oscillationnumber.) / / /

PROPOSITIoN 5. The above will be fufilled if

r + q s 0 in (a,c)

qr+q :0in(cd)

(d-c)su pr+q

(d -c ) infVXr+q>r. //(c.4

If we specialize further to the case

y" + (-)+q )y = 0in (-1,0)

y" + (X+q) = y = 0 in(0,1)

y(-1) = y(1) = 0. (13)

38

For some M, M > g, we assume that12q(z) s M in (-1,0)

-M sqg(z) sM in (0, ) .

There is no lower bound on q (z) in (-1,0), and no bounds in ( n-,1).

PROPOSITION 6. For (13), X > 13M.

PROOF. This results from the choice = 2M. Here

a =-,b = 1, c = 0, d =2

Thus,

(d -c)su 4pr g ' v'47J =Ss2%/3V 2

( d -c )finl A'~+~qg > 2M E = n(c.25

as required. .1/

This gives a less precise bound for X' in problem (11) (for g0 < --- ),4'

namely, 131q0 .Finally, we investigated problem (11) numerically using the VAX 11/750

computer in the Mathematics and Computer Science Division at ArgonneNational Laboratory. We wanted to understand how the spectrum changes as gochanges. We calculated a great many eigenvalues and eigenfunctions (a selec-tion of eigenvalues is given in the appendix) by solving the dispersion relation for(11)

Vq-X cosVJg~X sin-~g +X + -q +A cos'T+ msin-X = 0

in the complex plane; the eigenfunction is then given by

Csin[v-~X(1 +z)]__s__~-~Xif-1szs0

Csin[ -q+X(1-z)] if O !g z s 1

sinv'-q +X

C an arbitrary constant.The 20 diagrams at the end of this paper summarize the behavior of the

spectrum. An "x" denotes a multiple eigenvalue (necessarily of algebraic multi-plicity two). The arrows indicate how the eigenvalues changed with decreasing q(g is always a constant function). The diagrams are informative; they give us aclearer view of the complexity of indefinite problems. In the region between X-and X*, real eigenvalues tend to coalesce and then "go off" into the complexplane, only to return to the real axis and once again go in opposite directions.The amazing part of the process is that the associated pigenfunctions gain azero when the eigenvalue again becomes real. This phenomenon explains theirregularities one commonly sees in the oscillation numbers of eigenvalues ofindefinite problems. The fact that the eigenvalues "come back" to the real axisalso is the reason for the difficulty in predicting the number of non-real eigen-values and the oscillation numbers of the real eigenvalues.

37

In summary, two fundamental issues that remain before us in the study ofindefnite Sturm-Liouville problems are the determination of A+ and X-("Richardson numbers") and the determination of the precise number of non-real eigenvalues and the oscillation numbers of the real eigenvalues. The exam-ple we have considered reveals many of the complexities involved in theseissues.

References

Mingarelli, A. B. 1982. "indefinite Sturm Liouville Problems." Lecture Notes inMathemnctics, Vol. 946. Springer-Verlag, New York.

Richardson, R. G. D. 1915. "Contributions to the study of oscillation propertiesof the solutions of linear differential equations of the second order." Amer.J. Math. 40:283-316.

38

~-41.9093 < q <-47x (0)

(3) -m3)

(0)

>0)

(3)-

/ )

q = ~-41.9093

(3)

1 < <7r2-4..0934

F' x x' A x

-17, 2

4(3) (3)

F' F' F'5'

(3) (3) (3)~ff- - I -

(3) (3)

(3)x(3) (3)- ..- (3)

(3) (3) (3) (3)

AFqL AWRL

(3).,(3) (3).-(3)

39

K N KI

4(2) (cl)

2

(o) (i) (2)

I(0)

"- (I) -(2)

A LI R R

(0)

1

-. 2 <q <~-6.402(O)

Il(o)

4

VW

Alk ARL

=mew-

(2)-- (i)- (O)- -wa j - ( ) - 2)x

~- 6.402 < q <I

(2)- 1)-

I

APL M PI KI

4

(I) (I)

K K X K P

AM -OM m - qw A1

(1)

-21.99604 < q <4

(0)

(2) -

() K

(0)

- i -

40

q:-7 2

(2) (i) (2)

= 572

(2) (1) (2)

A

(2)+ (I)-'(I) (I)- .() -- 2)

t/)o

41

q= -21.99604

(2)=Now GEIM

<q <~-21.996044

(2) . 12) -2) (2)-42)

4(3) (2) (2)

F' F'ohm u b R

-9.2

(3)- (2"2)

(2) (2) (2)

(2)_______ ______ ______ ____________ _____ _____ -

(2)w

(2) (2) (3)

G - - -

(2ma ) (3)

(0)

V V V I

~-32.076 <q<'r~-31.00

(3)-. (2N

ow -32.076

(3)

4w'2 q<- 32.076 (0)(3)-

(0)

(2)

X( 0 )

qs 2

(3)

T 0)-4.m(3)

(2) (3)

j()

I (0) (0)

I SAM

-3)

t(0)

X( 0 )

(3)I U

(3)

Ii(O

(2"-2)

4% A

-7IF- -vv-

x (o)

43

Some eigenvalues for the problem-y- gyqiJ=Asgn(z)y y(-1)=y(1)=0

for varying q : 0.

9 Re(X) im(X)

0.00.50.751.01.52.02.252.302.352.402.452.482.4652.4672.4674011012.4870122.472.53.04.05.05.551656.07.08.09.09.59.89.869

12.012.22212.312.3222.3312.33512.33612.33712.33812.3412.35

5.59344.88304.50854.11693.26342.20571.48341.2981.0840.818840.414870.270470.153950.062920000000000000007.36098.29269.05599.318919.51659.68089.73599.85979.86969.86969.8700

0000000000000000.000820.352370.566132.219483.516054.16864.362804.450124.417574.027143.071672.115970.949550.08912

-0.4996 x10--0.1049 x10-4-0.1606 x10-6

0.1948 x10-6-0.3019 x10-4

0.1404 x10-60.1015 x10-*

-0.2 x1040.13290.23090.4801

44

Q

13.0014.0015.0016.0017.0018.0019.0020.0021.0021.5021.0021.9021.9521.9621.9721.9821.9921.99521.99621.9960321.9960421.996521.997522.022.2032.0732.07632.0732.076532.07732.07832.07932.0832.0932.1032.2032.3032.4032.50

Re(X)

9.873659.829869.73929.60579.42649.189998.874848.43377.74287.181368.66806.433856.295966.266208.235598.2041006.1716406.1550326.1516786.1515776.1416578.0032445.8851235.70690.7693241

29.2662429.5450729.2662429.6087729.6089529.80853929.608393129.60824229.6067329.6052229.5899529.57442229.5586329.542570

Im(X)

3.311414.96845.93876.54216.87176.95386.77186.24765.121264.0122.76262.0151.429681.271631.086800.8577380.52940030.22028:=40.042955510.02104900.0115332

a0a0fto

*0

0.07363690.12211310.18408650.22991540.26801380.5112460.67128131.5254052.0429392.4476152.7889968

4rL

9

32.8032.7032.8032.9032.9533.0034.0035.0036.0037.0038.0039.0040.0041.0041.5041.8041.9041.90541.907541.90841.90941.909141.90241.909341.9093541.909441.9095

Re (A)

29.526242329.50965529.492771629.47582229.46894329.458192729.2676529.04411128.780510128.46573528.081490227.594957126.937831925.91560025.033100124.05075123.33729423.320500623.29282923.287166923.27570723.27455023.273392323.272252423.271651723.229116423.160789

Im(A)

3.08814263.3583603.6005853.82535193.931514944.03396495.56988166.53688937.161008447.51589257.62011157.45120426.92149955.73225584.48038082.7893580.950858830.658385290.434510570.37289940.198740290.139138500.13600400.091511910.057520700

47

STURM-LIOUVILLE PROBLEMS WITH INDEFINITE WEIGHT FUNCTIONS

IN BANACH SPACES

Harold E. Benzinger *

Abstract

Methods of Paley and Wiener and of Levinson for investigating the convergence ofnon-harmonic Fourier series are applied to the problem of the convergence ofhalf-range expansions for Sturm-Liouville problems with indefinite weight func-tions in LP spaces for p o' 2.

1. IntroductionWe consider a simple example of a regular Sturm-Liouville problem on

[-rrr], with a weight function that changes sign. The completeness of eigen-functions and the convergence of eigenfunction expansions in the Banach spacesLP(-ir,rr) and in subspaces are considered, 1 !p < m. For the case that p = 2,this and other problems have been considered by a number of authors [Beals,1979 and 1981; Kaper, Kwong, Lekkerkerker, and Zettl, 1983; and Kaper, Lek-kerkerker, and Hejtmanek, 1982]. Their results have been obtained using thetheory of selfadjoint operators. Lacking this tool when p s 2, we use heremethods from the theory of functions of a complex variable, particularly resultsfrom Paley and Wiener [1934] and Levinson [1940] on non-harmonic Fourierseries.

Let a > 0 be a given constant, and let

w (z)= ~4' ~"CG .(1.1)W(X) 1, O<zsrr1

We consider the boundary value problem consisting of the differential equation

-u" =XJw(z)u, -rz s ir, (1.2)

and the boundary conditions

U(-') = 0, (1.3)

U(r) = 0. (1.4)

A solution of the boundary value problem is a function u in C[-irr], satisfyingthe boundary conditions, such that u' is absolutely continuous and u" is inLP(-rr) fora fxed p , 1 s p < o .

2. Eigenvalues. Eigenfunctions, and Green's FbnctionIn (1.2) we make the substitutions

X=pQ, o=cap. (2.1)

Let AB be constants, and let

'Department of Mathe:-.Atics, UniversityL J inois, Urbana, llinois 61801.

p(zp) = Asinh(n+z), (zp) = Bsinp(,r-z).

Then go satisfies the differential equation on [-r,O] and satisfies the boundarycondition (1.3) at -r, while * has the analogous properties for [Ox,]. Thus thefunction

P) = z zr, ' fsz , (2.3)

is an eigenfunction if AB can be selected to satisfy the interface conditions

p(O,p) = (p), (2.4)

p'(0,p) = 0'(,p). (2.5)

Let

D(p) = asinpircoshar + cosprsinhar. (2.6)

A direct computation shows that nontrivial A.B satisfy (2.4), (2.5) if and only if

D(p) = 0, (2.7)

so that X = p2 is an eigenvalue if and only if (2.7) holds, and the eigenfunctionsare scalar multiples of

' ninhr nz ,-irz<0u(zp) = _sipp n-x , 0. (2.8)

To determine the location of the eigenvalues, define y by

cosr = a/ (1+a2)M, sinyr = 1/ (1+a2). (2.9)

Then 0 <-, < 1/2 and

D (p) = (1 +a2))I [cosynrsinprcoshrr+sinrrcosrfsinhu7r]

= (1/ 2)(1+a2 )3 e'"[sin(p+7)7r+e-2"sin(p-7)1r]. (2.10)

Using identities such as coshiz = cos, we have

D(ip) = (i/2)(1+a2)% eP"[cos(a-7))rr-e-Pwcos(a+7)7r]. (2.11)

THEOREM 2.12. The eigenvalues = p of the problem (1.1)-(1.4) are real andsatisfy the asymptotic relations

ph = k -y+0(e-Zwa), k -. , (2.13)

ap.. = -i[k+7++0(e-kwa)], k fm. (2.14)

PROOF. Consider (2.13). The sequence Jk -y arises as the zeros of sin(p+y)r.Applying Rouche's theorem, we see that the error term in (2.13) arises from theterm e-2c"sin(p-')ir in (2.10). The estimate (2.14) follows from similar con-siderationsapplied to (2.11).

To see that the A* are all real, we note that the spectrum is independent ofthe particular LP space being considered. and in the case that p = 2, i- is thesequence of eigenvalues of a selfadjoint operator on a Hilbert space [Kaper,Kwong, Lekkerkerker, and Zettl, 1983: Theorem 2.1]. This completes the proof.

For f in LP(-rrr), we consider the nonhomogeneous differential equation

(z.2)

49

-u" = Xw (z)u+f , (2.15)

along with the boundary conditions (1.3), (1.4). As is the case for a weight func-tion that does not change sign, the solution is given by

u(z,X) =fG(z,t,X)f (t)dt ,(2.16)

provided X is not an eigenvalue. Green's function G(,t ,X) is given by

1_(_,_)u (t, -r<t<z<trG(z,t,X) = W(X) ut(z,X)uz(t,X) ' -rr<<t<tr' (2.17)

where u1 (z,X) is a solution of (1.2) on [-.rr] satisfying the boundary condition(1.3), u2 (z,X) is a solution on [-rrir] satisfying (1.4), and W(X) is their Wronskian:

W(A) = u1( -,a)u2'( ',X) - u'( -,X)u 2( ',X). (2.:.8)

One choice for u1 ,u2 is (X = p2 ),

sinho(r+z) -r<z <0(219ul(z,P) = sinhanrcospz+acosharrsinpx 0<z<lr (2.1)

U2( ,P f sinprcoshaz -(1/ a)cosprsinhoz -r<z<0(2.20)u2(P) =sinp(ir-z) O<zczr 2.0

Then

W(X) = -pD(p). (2.21)

3. Completeness of the genfunctions

DEFINITION 3.1. A sequence 19 is complete in a Banach space X if the set offinite linear combinations of the po is dense in X.

DEFINITION 3.2. The sequence fOk j is a basis for X if for each f in X, there existsa unique set of scalars Caf (f )3 such that

Sa *(f)po=f.

REMARK 3.3. In the literature on problems with indefinite weight functions, theterms "full-range completeness" and "half-range completeness" are customarilyused to refer to basis properties of eigenfunction systems [Kaper, Kwong, Lek-kerkerker, and Zettl, 1983: pp. 21, 25]. In this paper we use "completeness" inthe sense of Definition 3.1.

Let jpa , k = 1,t2, .- denote the eigenfunctions of (1.2)-(1.4), normal-ized so that (wpk,9pk) = 1.

THEOREM 3.4. The sequence f pk j is complete in L(-tr,r) for 1 5 p <z.

The proof (given below) is based on Benzinger [1973: Theorem 1.1]. For anyf in LP(-r,ir), the function u(zX) defined by

u(zX) = fG(zta)w(t)f (t)dt (3.5)

is a merorrorphic function of X, with poles at the eigenvalues X and residue(wf ,k)pk(Z) at 4 [Neumark, 1963: p. 33]. If pk is not complete in the dual

50

space L4(-r,tr) (pq = p+q), then there exists f s 0 in LP such that (wf ,Pb) = 0for all k, and therefore u(z,A) is an entire function of A for axed z. Using (2.':7),we show below that is (z,A) is then bounded in A for each z, so u is constant in Afor each z. This is inconsistent with the differential equation unless f = 0, yield-ing a contradictin. Applying this argument to the adjoint problem T' (which isformally the same as T), we obtain the completeness of {pa ijin LP, 1 p < .

LEMMA3.1. Let 6 > 0 be given, and let XA= p2 . For Imp z 0 and Ip-pk Ia 6, wehave

G(z,tX) = O(1/p), IpI -+.

PROOF. From (2.17), (2.19), and (2.20), we see that pD(p)G(z,t ,A) is a linear com-bination of exponentials eP*, where z = z (zt) is taken from the rectangle

|RezI s ar, IImz I srr.Also, D(p) is a linear combination of these exponentials, where z is taken onlyfrom the vertices of the rectangle. Thus for each p,z,t, the dominant exponen-tial in pDG does not exceed the dominant exponential in D(p). Factoring thedominant exponential out of D(p), and using jp-pi, I 6, the result follows.

PROOF OF THEOREM 3.4. Let T denote the linear operator defned by

Tu, = (1/w(z))(-u.")on the domain consisting of functions is in C[-irir] such that u' is absolutelycontinuous, u" is in LP, and the boundary conditions are satisfied. Then for f inL ,

R(XT)f = -G(z.f ,X)w(t)f (t)dt,

and from (3.6),

IIR(AT)II = O(1/p)as I +I -. o on an increasing family of circles that are uniformly bounded awayfrom the eigenvalues of T. Thus the conditions of Benzinger [1973: Theorem 1.1]are satisfied.

4. Half-range ExpansionsFrom (2.8) we see that for 0 ! z s ir, the eigenfunctions are scalar multiples

of sinpk (vr-z). We consider here the completeness and expansion properties ofthese functions in L (0,7r), using only those pb that are real zeros of D(p). Thusfrom (2.13),

Pk = k -7+ a, a = O(e-eawa), k +==. (4.1)

There is no loss of generality in replacing i-z by z, so we consider

f.'z)=5sipkz, k'z1. (4.2)

Let

Pk kz1p. = 0 k=0 (4.3)

-p .nk i-1

The expansion properties of Yfe J, k a on [0,r] are related to those of

51

rk (s) = { k 0-' 2... (4.)pa()=en k =0, 1, 2, -a -(.4)

on [-r,]. Such problems of non-harmonic Fourier series are considered inPaley and Wiener [1934], Levinson [1940], and Young [1980].

Denote by Pk (z), k=0,t1, - - - those functions (if they exist) that arebiorthogonal to 9o j:

(f,,p1 ) = fa (x)jI(z)dz = 5ki. (4.5)-ff

Let X denote any of the Banach spaces LP(-r), 1 s p < co. Note that as aconsequence of . = -, we have

o.a(z) = SPk(-Z). (4.6)

LEMMA 4.7. If ifr is complete in X and if the biorthogonal sequence jij exists,then

-()= 'ft'(-Z).

PROOF. Let 'w1 (z) = ' f(-z). Using (4.6), we obtain, for any k ,j, that

( ,w .. )= 0,

so from the completeness of j j1, ( -'- = 0.

Let

k(z) = pak(z)-g.-k(z), 0sz sr,k z 1. (4.8)

*k (z) = 'il (z) -k-(z) 0 s x s rr, k z 1. (4.9)

Using (4.6), (4.7), we have

a f~az)~y (~dz= d5 kjy -J.. = 6bj, k,jm1.

Thus

fk(z) = (1/2i)Iak(z), (4.10)and if 1ki exists, then functions biorthogonal to ft also exist:

gk (z) = 2i+k (z). (4.11)

LEMMA 4.12. Assume pk is complete in X and (ok exists. If f is functiondefined on [0,], and if F is its odd extension to [-ir.ir], then

(f ,g)fk = $ (F,ip,)oP.k=1 k=-N

PROoF. From (4.6), (4.7), and the oddness of F, we have

(F,k)9k + (F,'p.)k = (f,gk)fk. k z 1.

THEOREM 4.13. The set 17, i is complete in LP(-rr), 1 s p <o.

PROOF. Let vk = k-y for k a 1, and v.. = -v, for k s -1. Let n(t) denote thenumber of 0 k satisfying Iv |IS t. Then

52

n(t) = 2[t+ 7 ]+1,

where [- ] is the greatest integer in z. Let

N(r ) = ( n (t)}/ t ]dt.

Then

N(r) = 2r+27logr+0(1), r-+O,

so

lien sup[N(r)-2r +(1/p)logr ] > -m.

For 1. <p <, this condition is sufficient for e"'* to be complete in LP(-rr)[Levinson. 1940: p. 6; Young, 1980: p. 118]. Clearly, completeness in L(-r,r)follows. The completeness of ie "l is transferred from the completeness ofSe tvk* since E .,plj-vk I< < [Young, 1980: p. 132].

CoROLLARY 4.14. The system If , k z 1 is complete in LP(0,r), 1 s p < .

PROOF. Letf be in LP(O,ff), and let F be its odd extension to [-r,r]. Given e > 0,there exists a finite sum

FN(z) = chk-N

such that

F(z) = FN(z)+r(z),

where iIrIt<e. Using (4.6),FN (--z)=

-N

Since F is odd,

-F(z) = Fn(z)+r(-z),

so

F(z) = (1/2)[FN(z)-FN(-z)] + (1/2)[r(z)-r(-z)].

Clearly Ij(1/2)[r(z)-r(-z)]| < r, and the remaining term is a linear combina-tion of fk's. The result follows upon restricting to (0,r).

We next consider basis properties of ifkJ. Let 1 <p < c, p-1+q - 1 = 1,

r = min(p,q), (4.15)

and let

ak(z) = e . (4.16)

THEOREM 4.17. Let 1 < p <w. The wsystemrrak I is a basis for LP(-rrr), equivalentto the ordinary Fourier system ie* , provided

yT < 1/ (4+tan(r/ 2r)). (4.18)

PRooF. Let ek (z) = e , and let

53

ck (f) = (1/ 2r) ff(t)'E(t)dt.

Consider the operator A defined on LP(-ir,r) by

Af = ck(f)ak(Z) (4.19)

= co(f )+ei7" ck(f )ek(z)+et" ck(f )ek(z).

Let C:Lp+LP denote the conjugate function mapping

Cf = -ii(sgnk)ak (f )ek (z). (4.20)

We have ICI = tan(r/ 2r) [Pichorides, 1972]. Let H:LP-.LP be defined by

Hf = 2ck(f)ek = (1/ 2)[co(f)+f +iCf ]. (4.21)0

Then

(A-I)f = (e~"-1)(Hf -co(f ))+(es?--1)(f -Hf),

so

||( A-I)f ||p 2 n[|H||p+1].

If IIA-II 6 <1, then A is invertible and Aek = at. Now

IIHIs, (1/ 2)[2+IICII] = 1+(1/ 2)tan(ir/ 2r).Thus A is invertible if (4.18) holds.

COROLLARY 4.22. If (4.18) is satisfied, then there exists a sequence (fg (z) inLT(-r,7r) such that (a,fl) = 6 f.

PROOF. The sequence is

fl = A-'e.

We note that since tanyrT = 1/ a, condition (4.18) requires that a be selectedsufficiently large. A further transplantation of this basis property to thesequence lpk can be achieved again at the cost of selecting a sufficiently large.From (2.10) and an application of Rouche's theorem, we see that for asufficiently large, supip -vk I can be made arbitrarily small. Let a = p, -vk.Then a = O(e "), so C. I converges, and this sum can be made arbi-trarily small.

THEOREM 4.23. For a sufficiently large, f pg is a basis for LP(-7,r) equivalent tothe Fourier system fe.

PROOF. Let

Bf = 2(f , )ok = (f ,#4)4(z)e*a*.

Since

54

If =(f,#A:)a,

we have

(I-B)f = (f ,#k)ak (z)(1-ea"),

and then

||(I-B)f ||, s K,( |aM \)||f ||.

Thus, for a sufficiently large, I-BIp < 1, so B is invertible and

pk = BAek, p = (BA) 1 ek .

COROLLARY 4.24. For a sufficiently large, fk j is a. basis for LP(0, r), 1 <p <-.

For - s z s 0 and negative eigenvalues, the eigenunctions are scalar mul-tiples of

sinhiak (ir+z),where, using (2.14),

4 = k +(1/ 2)+7+z, a = O(e-"), k-.ao. (4.25)Thus we consider

g (z) =sinakz, 0sz sir.

The completeness of fga in L (O,ir), 1 !p <a* follows as before, since this isindependent of the size of y. The analysis of basis properties presented heredoes depend on 7. In this case, to make at sufficiently close to an integer, it isrequired that y be close to 1/ 2, i.e., a should be sufficiently small. Presumablya more refined technique would remove this dichotomy.

References

Beals, R. 1979. "An abstract treatment of some forward-backward problems oftransport and scattering." J. Funct. Anal. ?<:1-20.

Beals, R. 1981. "Partial-range completeness and existence of solutions to two-way diffusion equations." J. Math. Phys. 22:954-960.

Benzinger, H. E. 1973. "Completeness of eigenvectors in Banach spaces." Proc.Amer. Math. Soc. 38:319-324.

Hangelbroek, R. J. 1976. "A functional analytic approach to the linear transportequation." Transport Theory and Statist. Phys. 5:1-85.

Kaper, H. G., Kwong, M. K., Lekkerkerker, C. G., and Zettl, A. 1983. Full- andHalf-range Theory of Indefinite Sturm-LiouvWile Problems. ArgonneNational Laboratory Report ANL-83-76.

Kaper, H. G., Lekkerkerker, C. G., and Hejtmanek, J. 1982. Spectral Methods inLinear Transport Theory. Birkhiuser Verlag, Basel, 1982.

551 e

Levinson, N. 1940. Gap and Density TheoreCTSs. AMS, Providence.

Neumark, M. A. 1963. Lineare Differentialoperatoren. Akadernie-Verlag, Berlin.

Paley, R. E. A. C., and Wiener, N. 1934. Fourier Transforms in the ComplezDomain. AMS, Providence.

Pichorides, S. K. 1972. "On the best values of the constants in the theorems ofM. Riesz, Zygmund and Kolmogorov." Studia Math. 44:165-179.

Young, R. M. 1980. An Introduction to Norharronic Fourier Series. AcademicPress, New York.

57

INTERlACING PROPERTY OF EIGENVALUESOF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

J. Boersma*Hans G. Kaper**

Man Kam Kwong t

Abstract

This note contains the proof of an interlacing property of the eigenvalues ofboundary value problems described by the Sturm-Liouville differential equation-y"+ q(t)y = ay on[o ,e).

Recently, Van Duin, Boersma, and Sluiter [1984] studied the problem ofone-dimensional wave propagation in a stratified inhomogeneous medium with arefraction index of the symmetric Epstein type: e,.(z) = 1 + A sech2(z/ 2). Pos-sible TM-modes for this problem are given by the egenfunctions of a Sturm-Liouville problem on the half-line:

Ty = -i' + q(t)y = Ay on [o,oo),

y(0)cosa + y'(O)sina = 0. (1)Here, acE[0,n/ 2] is fixed and q :[o , m)-R is smooth, strictly positive, and rapidlydecaying at infinity. The differential expression t and the boundary conditiondefine a selfadjoint differential operator in L2(0,oo), whose spectrum consists ofthe semi-axis [0,) and a finite (possibly empty) set of eigenvalues14(a): k = 0,1,...,n j, with

-cc < M(a)< N(a) < <A,(a)< 0. (2)

In this note we shall prove an interlacing property of the eigenvalues Xk (a). It isnot unlikely that this property can be found somewhere in the vast literature onSturm-Liouville boundary value problems, but we have not been able to locate it.

THEOREM. The eigenvalues have the following interlacing property:

Xk(n/2) < 4(0) < .Ak+I(n/ 2), k = 0,1,2,.... (3)

We shall need the following lemma, which is a stronger version of the Sturmcomparison theorem.

LEMMA. Let P and Q be given on [a,b), and let P(t) & Q(t) for all t E[a,b ). Sup-pose u and v satisfy the equations

u" + P(t)u = 0, v" + Q(t)v = 0,respectively, on [a,b). If u andv do not vanish on (a,b ), and if

*Department of Mathematics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindho-ven, the Netherlands.

"Mathematics and Computer Science Division, Argonne National Labratory, Argonne, IL 60439.Work supported in part by the Applied Mathematical Sciences subprogram of the Office of EnergyResearch, U.S. Department of Energy, under contract number W-31-109-Eng-38.

tDepartment of Mathematical Sciences, Northern Illinois University, DeKab, IL 60115.

58

u(a) v(a)

then

u'(t s '(t for all t E(a,b).u(t) 'u(t)

REMARK. If v (a) = 0, it is understood that v'(a)/v (a) = lsim v'(t )/v(t) =Then the assumption u'(a)/u(a) s v'(a )/v (a) is trivially satisfied.

PROO'. Without loss of generality, it is assumed that u and v are positive on(a, b) and satisfy the condition

u'(a)v (a) - u(a)v'(a) s 0.

From the differential equations for u and v it is readily found that

u"v - UV" = -(P-Q)uv.

Then by integration from a to t, one has

u'(t)v(t) -u(t)v'(t) = u'(a)v(a) -u(a)v'(a) -f (P-Q)udt s 0

for allt E(a,b). Divide by u(t )v (t ). Then

u st) ' t)for all t E(a,b). /

PROOF OF THEOREM. The proof consists of three steps.

Sep 1. (0) # (r/ 2) for any k, L. That is, the eigenvaues in the cases of Diri-chlet and Neumann boundary conditions are different.

Let y, be the eigenfunction associated with the eigenvalue (0). Then yrhas k zeros in (0,m) and y (t)-+0 as t +ca. Let the zeros of y1 (t) be t ,t2,...,t-.Let y be the solution of the boundary value problem

'ry = 4 (O)y on [0,m=),

Y(0) = 1, y'(0) = 0. (4)

It follows from Sturm's separation theorem that y has one zero in (0,t 1), and onezero in (t,t. 1) for each i = 1,2,...,k-1. Suppose y does not vanish in (tf,co).Then

y s y ( = +me.y(tk) yj(th)

Hence, according to the lemma,,tsya' ,(t)L for all t > tk .

3(t) ye(t)

Integration over [tk + it] yields the ineqLL'tL.'

y3(t ) ya(-' for all t z.. +1.y (t+1) y (t+1)

Because y (t)-"0 as t f'a, it follows that

59

y(t )-+ as t fcc.

Next,

d _ar c t a n_ __|_=_|_ _= | c oa s t .( _ )_ | -+wa s t + e .

dt ya a+ yb +y&

Hence,

arctan 0 |(-+ f as t -'e.ya (t)

However, j(t )/y(t) is bounded for all t z tk + 1, so certainly

I arctan I <(t)

Here we have arrived at a contradiction, so we must conclude that y vanishes atleast once in (tb,m). From Sturm's separation theorem, it follows that y cannothave more than one zero in (tk,=), so y vanishes exactly once in (tk,m). Thus wehave shown that y has exactly k+1 zeros in (0,cc); let these zeros be denoted by

Finally, compare y with the auxiliary function u(t) = exp[tV - (] (recallthat 4 (0) < 0 ), which is a solution of the equation

u" + A(0)u = 0.

ThenUI,'+i) '(k+T1)

h< =77 .u Ik+1) yk7+1)

Hence, according to the lemma,u't) Lfor all t >k +1.u(t) y(t)

Integration over [Tk + 1,t] yields the inequality

~ tU(t ) = exp[(t -_k+1-1)v- )JY] for all t Zk+ +1.y A;+1+1) u(Tk+1+1)

Consequently, y is unbounded as t -f.

Since eigenfunctions tend to zero as t -cm, it is clear from (4) that Xk(0) isnot an eigenvalue under the Neumann boundary condition y'(0) = 0. Thereforewe have the inequality X*(0) X(ir/2) for any k, L.

Step 2. A*(0) < Ak+1ir/ 2).

Let g be the solution of the boundary value problem

ry = )y on [0,m),

Y (O) = 1, y'(0) = 0, (5)

where X <A*(0). It follows from Sturm's oscillation theorem that g cannot havemore zeros than y on (0,a), so y has at most k+1 zeros. If g has less than k+1zeros, then X is certainly different from Xk..(r/ 2). Suppose g has k+1_zeros,denoted by tf,t2,...,t . Then tk+1 > k1, so neither y nor y vanish on (t+ 1, ).Because

60

y(4+,1) <g'(4a. 1)=y! (tk.i) g (t,)

it follows from the lemma that

4 U for all t > th+1.y(t) g(t)

Integration over [[4l + t1,t] yields the inequality

() _4(L) for all t t fe + 1.y (tt~i+1) g(tk~+1 )

Because y(t) is unbounded as t -+oo, it follows that

I g(t) |Ia= as t -ee.

Hence A # X+ 1 (ir/ 2). This result, together with the result of Step 1, implies theinequality

Xk(0) <Ak+i(I/ 2).

Step 3. A*(Tr/ 2) < k (0).

Let yj be the eigenfunction associated with the eigenvalue Ak (in/2). Supposethe inequality were not true, i.e.,

(ir/2) a (0).

Then it follows from Sturm's oscillation theorem that yua has at least as manyzeros in (O,=) as y, which, as we have seen, has k +1 zeros in (O,oo). But here wehave a contradiction, because y has exactly k zeros in (0,mo); thus

Ak(n/ 2 )<Ab(O). ///

References

Van Duin, C. A., Boersma, J., and Sluyter, F. W. 1984. 'The TM mode in a planaroptical wave guide with a graded index of the symmetric Epstein type," submit-ted to J. Opt. Soc. Amer.

61

A KREIN SPACE APPROACH TO DIRICHLET AND DUAL DIRICHLET INEQUALITIESASSOCIATED WITH STURM-LIOUVILLE OPERATORS

R. C. Brown'

Abstract

By embedding the quadratic form associated with a second-order symmetricdifferential expression defined on [1,ac] in a Krein space, we derive Dirichet ine-qualitites as well as a new class of "Dual Dirichiet inequalities" when the potentialmay have an unbounded negative part.

1. IntroductionLet I be the real interval [1,m), L, (I) the space of complex valued func-

tions, Lebesgue integrable on compact subintervals of I, and L2 (I) the Hilbertspace of (equivalence classes of) complex valued functions having square inte-grable modulus with inner product [f ,g ]: = ffg and norm ||f 1|: = (f I f 12).PoSPi are real measurable functions onI satisfying the "minimal conAitions":Po > 0 a.e., pl.pi E4,.(I). M[y] denotes the symmetric differential expression-(p ')'+pity, and L,L0 the maximal and minimal operators corresponding to Min L (I) (for precise definitions of L,L0 as well as further details concerning theelementary Hilbert space theory of symmetric differential operators, seeKauffman, Read, and Zettl [1977] or Naima-k [1988: Chap. V].

We are concerned in this paper with inequalities similar in fo m to

fpoly'l 2+piIyI2 iifyI 2. (1.1)

fj(-p z1 )'+p I y zA supi fZ p# '+z P P 2:

poIl' I2+pI Iy 2 = 1; p# Z 1(1+) = 01j, (1.c)

where in (1.2) (zi,zZ) belongs to a class of suitable function pairs in L2 (I)xL2 (I).

Inequality (1.1) is the classical Drichlet inequality. It has had a long his-tory going back at least to R. G. D. Richardson [1910-1912] and Lichtenstein[1919], and in the past decade has attracted renewed iriterast on the part ofseveral authors (for an illuminating survey and bibliography, see the recentpaper of Everitt and Wray [1983]). Inequality (1.2), on the other hand, seemsrelatively new in spite of a natural relationship between it and (1.1).

When pi a e > 0, a satisfactory theory of (1.1) and (1.2) can be constructedin several ways. For example, it is not hard to show that the quadratic "Dirichletform"

t[y]: = fpoly'1 2+p1 y 12

defined on

D = yEL2 (I): y locally absolutely continuous ("AC") onI; t[y] < en

*Department of Mathematics, University of Alabama, University, Alabama 35488.

62

is closed in the sense -_f Kato [1980: p. 313]. Next the First RepresentationTheorem [Kato, 1980: Theorem 2.1, p. 322] gives the existence of a selfadjointextension T:Lot defned on a core of t with lower bound 1. T has domain

D(T): = jy ED(L): poy'(1+) = 0; t[y ] < cc .Since D(T) is a core of D, (1.1) holds on D as well as D(T). A somewhat differentbut equivalent point of view was introduced by Brown [1984]. First define the"maximal" operator L:DcL2 (I)-L2 (I) by

y= .(1.3)

Since t is closed and t[y ] = ILIjj2, L is closed and has closed rare. Hence bythe closed graph theorem L is bounded. Further |IL-11 = i. Additionalanalysis shows that T is the selfadjoint operator L*L which exists by a theoremof von Neuman [Kato, 1980: Theorem 3.24, p. 275].

One advantage of the latter approach is a convenient interpretation of the"dual" inequality (1.2). Since L' is the adjoint of L, it is dened on a certain sub-space of pairs (z ,z2 )EL 2(I)xL2 (I); for (z 1,z2 ) in this domainL (z21 z2) = -(pt z I)'+p z2. Equation (1.2) then states that the norm of the Hil-bert space pseudoinversett of L = |jLj-II= 1.

For a discussion of a number of other approaches to Dirichet inequalities,the reader may consult Everitt and Wray [1983].

We pass now to the fundamental problem addressed in this paper: What hap-pens to (1.1) and (1.2) when the potential p 1 is no longer positive and can takeon possibly unbounded negative values in a neighborhood of ms?

If t is closed and bounded below, reasonably straightforward argumentsfrom the theory of sesquilinear forms apply to (1.1) (e.g., see Brown [1984]).The situation, however, is more complicated with respect to dual inequalities.Since p P need not be real, (1.2) requires restatement and new methods of proof.In particular, the pseudoinverse analogy would appear to break down.

The primary goal of this paper will be to present a unified treatment of boththe Dirichlet and dual Dirichlet inequalities in the new setting where p 1 is notnecessarily positive or bounded below. We will do this, moreover, in a way that isconsistent with our earlier treatment. To this end, we shall introduce the power-ful apparatus of the theory of Krein spaces. Specifically, t or R(L) will beembedded in a natural Krein space. Using this device, we can extend the posi-tive coefficient treatment sketched above,

We briefly survey the contents of the paper. In Section 2, the machinery werequire is presented. The fundamental Krein space K and the new maximal andminimal operator analogues L1 ,I to (1.3) are defined. Several technical lemmasexplore the properties of t as a subspace of K especially with reference to theexistence/uniqueness of orthogonal projections upon it. Our results for themost part depend on a fundamental assumption "Hi" which relates the positiveand negative parts of t and which forces t to be closed. Towards the end of Sec-tion 2, we determine the adjoints L' and I. Precise statements of the new ver-sions of inequalities (1.1)-(1.2) are given and proved in Section 3 (Theorems 1and 2). We also state an additional theorem (Theorem 3) and some corollaries to

tT may also be viewed as the Fridichs eztnsion of the symmetric operator S Lo defined onfunctions of compact support such that poz'(1+)=0. See Kato [1980. p. 325].

TtIf T: H+H' is a closed operator with closed range between Hilbert spaces H and H', the pseu-douuwu ofT is the operator H'-.h giving the least norm solution of the equation Tz=g. For a gooddiscussion of pseudoizverses, see Luenberger [1989: Section 8.11].

63

show how other differential inequalities can be derived from dual inequalities.(To our knowledge, the inequalities presented here are new.) The last section ofthe paper investigates the situation when H1 is abandoned, and consequently theDirichlet form t need not be closed. Our results here are less complete than inthe previous sections. Nevertheless, we are able to derive some new propertiesof the ordinary Dirichlet inequality (Theorem 4) and also to prove a new dual ine-qualit, (Theorem 5).

Finally, we remark that for the sake of technical simplicity only inequalitiescorresponding to a second-order Sturm-Liouville operator defined on I aretreated in this paper. The reader should have little difficulty in supplying thetechnical changes required for the higher order case on R different interval, e.g.,(0,1].

2. Preliminary Theory Relating to Krein SpacesLet popi be as above. Define the sesquilinear form t[u,v ] in L2 (I)xL2 (I) by

t[u,v]: = y''+p apU

on D. Set

pi : = max(p1,0)

Pi~: = max(0,-p 1 )

and define

M [y ]: = -(Pas')'+P iy

t*[x]: = fpopi'+ptipuI

t-[i,v]: f=p1-,u.

so that t = - t-.Let C signify the space of infinitely differentiable functions with compact

support in I. Then C CD but D(L) need contain no nontrivial element of C .*We now assume the following hypotheses.

H1. For some 0 <0 < 1 and ai ilCoC, there exists a constant K > 0 (not depend-ing on o) such that

t~[4c] s KII|I2 + a3fpo Io'1- (2.1)I

Since 13 < 1 and t4 [p]> flpol p' 12, H1ft[] z -KA1PI1 2. Let i signify thegreatest lower bound of t. For a suitable translation of t, i > 0. We make thistranslation if needed, and continue to refer to the translated form by "t."

Summarizing, H1 implies the weaker principle IH.

" Cf cD(L) if po.p 1' and p i are L$ integrable on compact subintervals of 1.

64

H2. For alP in C(,.

t[ ] a pg||||12.

Examples of Hi:

1. pj1 has compact support.

2. p i is integrable.

3. p i is essentially bounded.

4. po = to pi = to anda!5f.

5. For every r > 0 there exists a positive continuous function f such thatt~e pit+tcf -1

Further sufficient conditions for H1 and its higher derivative analogues may befound in Brown and Hinton [1984].

3Let z~: = (zl,z 2 ,z3 ), z': = (z1 ',z2 ',z3 ') be elements of H: = XL2 (I). Define the

sesquilinear form

[Z~']r.= [z 1.z 1'] + [Ze z']-[Za,zs']. (2.2)

H endowed with the indefinite inner product (2.2) is a Krein space that we label"K" Clearly K= K(+)K- where (+) is orthogonal direct sum decomposition interms of (2.2) and

KcK = 1(z 1,z2 ,0);

KcK = (OOza.

K+,K~ are Hilbert spaces with respect to the inner products [ - , -] and -[ -, -]H.In the terminology of Bognar [1974: p. 71 they are "intrinsically complete" sub-spaces of K Given zEK we write z = z +z where a~EK4 and z-EK~. Theoperators P:K+r and P...:K+K- defined by Pi = z+ and P-~ = z- are calledthe fundamental projectors. Let /: = P-P-. Then J is the "fundamental sym-metry"; and

[z,$"]H: = [/']r

is the standard inner product for a Hilbert space structure in H.The next step is to embed t into K via the map

Sy'

ICI: = 1 *y,

IN y

for y ED, and to explore the Krein space attributes of t considered as "R(L)"If for z~EK there exists y ER(L) and v ER(L)1 such that z~ = y+v, we call y an

orthogonal projection of z onto R (L) and write y EP . In Krein spaces theexistence and uniqueness of orthogonal projections are difficult questions. Forexample. z may have nonunique projectors P1 z, P2 which differ by an "isotro-pic" element ;ER(L)nR(L)', such a p is also neutral, i.e., [cp]K = 0.

65

The proof of the following lemma (which is a collection of results concerningthe existence of projections) may be found in Bogn&r [1984].

LEMMA 1. Statements (i)-(iv) are equivalent:

(i) A projection P:K-R(L) exists.

(ii) R(L) is "orthocornplemented" (i.e., span <R(L), R(L) > = K

(iii) The functional d (y): = [ z-yVj. Z-]K considered for y E R (L) attains itsminimum for every z EK

(iv) If R(L) is a positive definite subspace of K and is "intrinsically complete"(i.e., a Hilbert space under the norm [ Ly ,Ly ]g), for every z'EK the functionaLS(y ): = [z ,Ly ]= is "intrinsically continuous" (or, in Bognar's nomenclature,R(L) is regular; see Bognar [1974: p. 71 and Theorem 9.2, p. 73]).

(v) For every z EK there exists y 0ED such that

[LboLyo]K = [Ilo 5 ] = supf 1((y) I 2:[Ii,Ly]x = 13,

and Lyo = Fc.

LEMMA 2. For all y ED

t [Y] zA pl yWI I. (2.3)

PROOF. By H2, (2.3) holds on C. Rewrite (2.3) restricted to C in the equivalentform:

fpoIy'I 2 +p pIy 2 f '+pr)hyI 2. (2.4)

Consider the differential expression

IMI[v]: = -(pov')' + (pi +pi ).and the maximal and minimal operators IL I, ILoI which I M I generates in L2(I).It is known [Kauffman. Read, and Zettl, 1977: Theorem 2.1, p. 23] that IM I islimit point at m. This in turn (see Brown [1984] or Kauffman [1979]) implies thatthe "Dirichlet index" of I M I +I - or the dimension Uf the space of solutions vsuch that t[v] < co - is 1. However, the minimal Dirichet index property isequivalent (again see Brown [1984] or Kauffman [1977] for a proof) to CL' being acore of tI: = ILtI + It-I. Since D It = D, we conclude that for every y ED and6 > 0, there exists cE C such that

f y -p I2 < d, fp i |yJ-co I2 < b

fpi Iy -9I 2 <6, fpoIy'-'l 2 <6.(2.5)

These inequalities and H2 establish (2.4) and hence (2.3) on D.

REMARK 1. Lemma 2 is a proof of the classical Dirichlet inequality on D whichconsists of functions satisfying the minimal smoothness and integrability pro-perties for the inequality to make sense (assuming I fp vI y 2<). Since theproof depends only on H2 and the fact that I M I{[y] has a Dirichlet index 1, itextends immediately to the case where M[y] is a quasi-derivative of order 2n

66

based on coefficients po,Pi,...pn satisfying minimal conditions if H2 is satisfiedand I L +1 has a minimal Dirichet index n. (Here jIL I is the maxiirnal operatorcorresponding to the quasiderivative I M I-[y]: = y]" based on the coefficientsfPo.Pi +pi-,... +p). The significance of the Dirichet index concept in theproof of Dirichlet inequalities seems first to have been pointed out in the paperof Bradley, Hinton, and Kauffman [1981].

LEMMA 3. H1iis true forcaU y in D.

PROOF. This follows at once from H1 and the approximation inequalities (2.5)established in Lemma 2.

LEMMA 4. The Dirichlet form t is closed.

PROOF. Since t+ majorizes t, the inequality

t+[y ] zp 11y11 (2.6)

holds on D. We show first that t+ is closed. If t[y, 7m]-+0 for sufficiently largem and n (2.6) =><y,> is a Cauchy sequence in L (I). Let y be the limit of<yn>. Since the maximal operator K determined by pay' is closed (see Brown[1983: Lemma 1]), y ED(K) and fpoy'-y,'I2=>O. Set 7: = ptl+1 and considerthe Hilbert space I?(I). Since t [yi-yim] > f p i(y rn) 2 and by (2.6), <y> isa Cauchy sequence in 4d(I) with limit . Since the norm in I?(I) dominates thatof L2 (I), y = '. Thus yED and t[y -yn]-+0, so that t+ is closed. Finally, t isclosed by Lemma 3 (see Kato [1980: Theorem 1.33, p. 320]).

LEMMA 5. Al the statements of Lemma 1 are true.

PROoF. It is enough to show that any single statement of Lemma 1 is true. Weconsider (iv). Statement (iv) asserts that

z(y): = fp wiy'Zi +p yfE -P-*yza

is continuous in the Hilbert space norm [Ly.Lg]#. (That R(L) is a Hilbert spacewith this definition of a norm follows from Lemmas 2 and 4.) We must showtherefore that L(y)-+0 => p (yn)-+0. Now

111y112 0t[y] = t*[y] -t-[y]

Also from Lemma 3,t[y Kly 11 + it[y]. (2.7)

From (2.7) and Kato [1980: Problem 1.2 and (1.2), p. 190] we obtain the esti-mates

t-[y]3!g (1-t)-1(K11l|+iI|LyII),

(1-)t[y]M KIyII + J|Ly11. (2.8)

Also by Cauchy's inequality

I|rPo(y,)1 s t+(yn) (Jjz il 2+IIz2|I 2) + t-(yn)'l1z31l. (2.9)If then L(y)-+0, rp(yn)-0 by (2.3) and (2.8), (2.9), and the lemma is proved.

REMARK. Inequality (2.3) of Lemma 2 assures us that projections upon RL) areunique, since R(L) has no neutral elements. Lemma 5 furthermore assures us of

67

the existence of a projection and that K = R(L)(+)R(L)+.

DEFINITION 1. If T: L2 (I)-.Kis a densely defined operator, T' is the set of pairs(a, T'a) such that [ Ty,a]= [y, T'a] for ally in D (T).

DEFINITION 2. We introduce the following additional operators: I'cL on

D:= ycD: y has compact support in L ;

L*: K+L 2(I) defined by

L+(z):= -(pd z)'+p iM z2-PjI- z3

on

D+: = RzEKp z1 is AC; L*(Z~)EL2 (J)

L+cL' on

I: = Rz ED+: p# z1 (1+) = O.

The proof of the next lemma parallels similar arguments in Brown [1983, 1984].Therefore we shall omit it.

LEMMA 6.

L" =_+; Lo = L'0*=_ L;

Lo =L; L, =L= .

COROLLARY 1. K= R(L)(+)N(I4) = R(l)(+)N(L+).

LEIMMA 7. L}IQ and Lo+L are selfadjoint extensions of Lo. Their domains are coresof and L They have the same lower bounds as L4 and L. FurtherD (LL!)= Ik and Dv'( TL) = D.

PROOF. By inspection L*I and 4L are exactly the operators i~io and io i investi-gated by Brown [1984: Lerma 5].

3. Dirichlet and Dual Dirichlet InequalitiesWith the machinery developed in the previous section, our treatment of ine-

qualities is similar to the positive coefficient case studied in Brown [1983].

THEOREM 1 (The Dirichlet Inequality). For all y ED(D0) the inequality

fpoly'12 +pjy |2 jhl(/e (3.1)

holds. Here jt and N2 are. respectively, the least elements in the spectrum ofq+L or LL4. Equality holds in (3.1) if and only if y is an eigenfunctioncorresponding to l or pg. Otherwise there is a sequence of functions <p, >such that

Jpo l ?' l + P i! l n 12 - p1(/2) f I Pn12_+O. (3.2)

PROOF. The inequality (3.1) follows from the First Representation Theorem andLemma 7. The equation (3.2) expresses the fact that the spectrum of a

68

selfadjoint closed operator is its approximate point spectrum.

REMARK 3. Recall that according to H2, y (pu) > 0. However, Theorem 1 is truefor any finite 1 or A2. To see this, simply subtract what necessary from bothsides of (3.1) to get the original pi or y. By the spectral mapping principlethese numbers are the infima of the spectra of the corresponding translates ofL+L or L4L .

COROLLARY 2. For alL y in D (I4L) or D (LI.)

f l-(p oy')' + p y |2 m (y2)(fpo ly'|e +1|y 1)

THEOREM 2 (The Dual Inequality). For alL zED+(D+)

f -(pg z1 )' + p8t"Z2 - P Ze 1 a g( )supi I[z,L(Lo)y]rl 2;I y

IIL(I)yII = 1. (3.3)Equality holds in (3. 3) if and only if I mod N(L (L+)) where Lg is an eigert-function corresponding to 1(p) of L4 (L4L' ). Otherwise there is a sequence<z,> of elements in D(L4) or D(I4L+) such that

IiM f |-(p Z 1n)' +1 Za. - Pik Zs 2 -p(A)sup I[znL(I4)y ]l e;n-y

IIL()y II|= 1 = 0.

PROOF. Let P be a projection on R (L) which, in view of Lemma 5, exists and isunique. Then for any z EK z = P +(I-P)z where (I-D)z~EN(L) (Corollary 1).By Corollary 2

II+z1I2 = = II+PI 2 z | P 1|12.The equation (3.3) now can be obtained by applying (v) of Lemma 1. To provethe second statement, observe that

I !1L|Q112 2 = 1IIle12 [L iLe - iA ,Ly] = 0

L.4(Lg) /=1Q.Hence Ig is an eigenfunction of K corresponding to l and z = Ly modNV(I ).The last statement depends on the same principle of the indentity of the approx-imate point spectrum and the spectrum of a selfadjoint operator that we used inTheorem 1 to justify a similar statement.

Since the argument is the same for the inequality involving z in D+ and A2,we omit it.

THEOREM 3. For all aED(LI) or D(ILL)

fpol(L')'| 2 + Pt !L~zI 2 i- IL~z l2

a 4(p,)2 sup i i [.L(L,)y]1 I2 : |lL(4I)yl = 1;.V

69

PROOF. By Theorem 1 IIL(z )II2 z p lvjLz jl2. Now apply Theorem 2.

COROLLARY3. For auo vinD(I0L) orD(L+4 )

(fpo jV' 12+piI'J2)f (pot') I2 j(p)pIi' 2)2.

PROOF. In Theorem 2 take z 2= zsQ = 0, z 1 = v, and y = vII v L~.

COROLLARY 4. For alU y in D(LL) or D(L+L0)

fp 0 1M [y ]')2 + p I IM [y ] 12 a y2( )( fpoy j, +p I l ).

4. Addendum: Inequalities without H1

In spite of the Krein space methods employed in this paper, our resultshave so far been a not-unexpected generalization of the positive coefficient case.In particular, the Friedrichs extensions LQL and LI0 we have constructed havebeen of a familiar type. Further, as should be clear from Section 2, the funda-mental assumption behind our results is Hi. H1 states that the form t- is rela-tively bounded with bound <1 with respect to t on an appropriate subspace.Together with H2 this implies that t is a positive definite intrinsic closed sub-space of K Furthermore, a unique orthogonal projection exists upon t con-sidered as R(L). Once this point has been reached, the remaining argumentsare standard ones.

Suppose now that we assume only H2. Then Lo is bounded below on theintersection of its domain with Cfo , but t need not be closed. As we have seen, itis still possible to prove the Dirichlet inequality in D (or IO) (Lemma 2). How-ever, the number pe has no spectral significance. The difficulty dies in the factthat although t is closable, t itself need not be closed; furthermore,' . need nothave the same structure as t. Thus the inequality not only may be defined on alarger set than D but it may be a "different" inequality. To illustrate these ideas,we appeal to an example of Kalf [1978]. Let I = (0,.), po = 1, pi = -1/ 4x2_

0. Then

'1[y] = fly'-y/2A(4.1)

where l is the least element of the spectrum of the Friedrichs extension havingdomain fy¬D(L): f y'-y/2|I <c . Now (4.1) => inequality (2.3) on D, but Dis strictly contained in DC).*

It would therefore be an interesting problem to determine necessary andsufficient conditions that t be closed and when it is not, to determine the clo-sure, especially when M[y] = y[2"1, n>1. We can also ask what the dual inequal-ity corresponding to (4.1) is.

In the remainder of this section, we show how something can be salvagedusing H2 rather than H1, together with a technique that incorporates the nega-tive part of the potential into the weight of a certain weighted symmetricexpression. Although our results do not coincide with Kaf's, they do yield newinequalities and some new properties of known inequalities, and thus may be ofindependent interest.

" This example has singular point 0 and so does not ft into our setting. However, it is easy toconstruct similar examples with singularity at cc.

70

THEOREM 4. Assume H2; then for all y ED(DI)

t[y ] a pl(pa)||y112. (4.2)

Equality holds in (4.2) if and only if y is an eigenfunction of Mh[y ] in D with theaddt~inal properties

(i) poy'(1) = 0 (poy(1) = 0).

(ii) poy' is AC.

(iii) f 1 (p + Mi(/Ae))~ 1 |-(poy')'+piy 12<

Otherwise there exists a sequence of functions <in> in D(D0 ) satisfying (i)-(iii)such that

ifpoIopn'I 2 +pIk ,iI2 - 1 (Me)I I2 = 0.

PROOF. (4.2) is (2.3) of Lemma 2. Rewrite the inequality in the form (2.4). Letw: = 1-+ pl. Then w > 0 a.e. and w EL (I). Define M[y ]:= w [-(pay')' + ipy]. Let LW, Lo be the associated maximal and minimaloperators in LW(I). Since t is closed on L2(I), it is closed on L(I).

We are now in the positive coefficient case with respect to the weightedexpression M.. We define L'o, L0,L: IW(I)-+L2 (I)xL 2 (I)L+, I1 as in Brown [1983]and repeat all the arguments of that paper. By Theorem 2 of Brown, we end upwith 2.4) - and therefore (4.2) - in D (the Friedrichs extension14L L (I)-+LJ(I) has lower bound 1) and discover that properties (i)-(iii))express the fact that yED(U+L).

The same reasoning may be used for the inequality on Db.Corresponding to Theorem 4 is a dual inequality that is a special case of

Theorem 3 in Brown [1983].

THEOREM 5. For all (z 1 ,z 2 ) IX(D*) the inequality

,/(P i-+tatpW))-1I l-(P# z i)'+p 13 +22z aansupl| p f p#1'1 V t

+pti z2gI 2:jIL(LI)yI| = 1 (4.3)

holds with conditions for inequality as in Theorem 2 above or in Brown [ 1983:Theorem 3].

REMARK 4. Since Theorems 4 and 5 are true a fortiori if H1 is assumed, (4.3) is anew dual inequality that is valid in all circumstances. Moreover, an extremal yfor the ordinary Dirichlet inequality (Theorem 1) will always have the additionalweighted integrability property (iii) of Theorem 4.

References

BognAr. J. 1974. Indefinite Inner Product Spaces. Ergebnisse der Mathematikund ihrer Grenzgebiete, Band 78. Springer-Verlag, New York.

Bradley, J. S., Hinton, D. B., and Kauffman, R. M. 1981. "On the minimization of

71!

singular quadratic functionals." Proc. Roy. Soc. Edinburgh 87A:193-208.

Brown, R. C. 1983. "A Von Neumann Factorization of Some Selfadjoint Extensionsof Positive Symmetric Differential Operators and its Applications to Ine-qualities." Ordinary Differential Equations and Operators. Ed. W. N. Everittand R. T. Lewis. Lecture Notes in Mathematics, Vol. 1032. Springer-Verlag,New York, pp. 68-92.

Brown, R. C. 1984a. "The Dirichet index under minimal conditions." Proc. Roy.Soc. Edinburgh. Section A (to appear).

Brown, R. C. 1984b. "A Factorization Method for Symmetric Differential Opera-tors and its Applications to Dirichlet Inequalities and to the DirichletIndex." Diferential Equations. Ed. I. W. Knowles and R. T. Lewis.Mathemat-ics Studies, Vol. 92. North-Holland, pp. 61-70.

Brown, R. C., and Hinton, D. B. 1994. "Sufficient conditions for weighted inequali-ties of sum and prod . -t form." J. Math. Analysis and Applications (toappear).

Everitt, W. N., and Wray, S. D. 1983. "On Quadratic Integral Inequalities Associ-ated with Second-order Symmetric Differential Expressions." OrdinaryDifferential Equations and Operators. Ed. W. N. Everitt and R. T. Lewis.Lecture Notes in Mathematics, Vol. 1032. Springer-Verlag, New York, pp.170-223.

Kauffman. R. M. 1979. 'The number of Dirichlet solutions to a class of linearordinary differential equations." J. Differential Equations 31:117-129.

Kauffman, R. M., Read, T. T., and Zetti, A. 1977. The Deficiency Index Problemfor Powers of Ordinary Differential Expressions. In Lecture Notes inMathematics, Vol. 621. Springer-Verlag, New York.

Kalf, H. 1978. "A characterization of the Friedrichs extension of Sturm Liouvilleoperators." J. London Math. Soc. (2) 17:511-521.

Kato, T. 1980. Perturbation Theory for Linear Operators. Springer-Verlag, NewYork.

Lichtenstein, L. 1919. "Zur Variationrechnung." Kgl. Ges. Wiss. Nach. Ma.th.-Phys. Xl., pp. 161-192.

Luenberger, D. G. Optimization by Vector Space Methods. John Wiley, New York.

Naimark, M. A. 1968. Linear Differential Operators, Part I. Ungar, New York.

Richardson, R. G. D. 1910-1912. "Das Jacoobische Kriterium der Variation-rechnung und die Oszillationseigenscharften linearer Differential gleichen2 Ordnung." Math. Ann. Part 1 (1910) 68:279-304; Part 11(1912) 71:214-232.

73

SPECTRAL PROPERTIES OF SELFADJOINT ORDINARY DIFFERENTIAL OPERATORSWITH AN INDEFINITE WEIGHT FUNCTION

B. &argvs *H. Langer**

Abstract

Spectral properties of the equation L (f) - arf = 0 with an indefinite weight func-tion r are studied in L , 1 . The main tool is the theory of definitizable operatorsin Krein spaces. Under special assumptions on the weight function, for theoperator associated with the problem, the regularity of the critical point infinityis proved. Some relations to full- and half-range expansions are indicated.

1. Basic Properties

1.1. We consider the formal differential expression 1(f) of order 2n on theinterval (a,b), -m s a <b s +oo;

1(f): = (-1)"(pof ())(n) + (-1)n-1(p1 f ("-1))(- 1 ) + ... +pf ,

where the functions Po,...,Pn are real, Po > 0 a.e. on (ab) and 1/pa,pi,...,p ELl.(a,b ). The exact meaning of (f) under this general assumption isthat of the qguasi-derivative of order 2n (see Krein [1947] and Naimark [1968]):1(f ): = f [". We study the spectral properties of the equation

1(f) - arf = 0, (1.1)

where the real weight function rE4jo (a,b) is indefinite, that is, the setsA+: = z: r (z)>0J, A: = fz:r (z)<0J are both of positive Lebesgue measure. Forthe sake of simplicity we assume that r 0 0 a.e. on (ab). The problem (1.1) is

1called regular if - < a < b < cc and -, p1,...,pf, rEL1 (a,b); the boundary

point a (b) is called singular if a = - (b = cm) or at least one of the functions1-Pa . . . , p,.r is not summable at a (b, respectively).

By 4 we denote the Krein space [BognAr. 1974; Langer, 1982] of all measur-

able functions f on (a,b) such that fIfI|2rI|dz < co, equipped with the

indefinite and definite inner productst i

[fg]: = ffrdz and (fg): = ff I|r Idz, resp. (1.2)a a

Evidently, the operator J

(Jf )(z): = (sgnr(z))f (z) (zE(a,b))

is the fundamental symmetry connecting the scalar products in (1.2).

By Dp we denote the set of all f ELr4 which vanish identically in neighbor-hoods of a and b and have absolutely continuous quasi-derivatives up to order

'Department of Mathematics, University of Sarajevo, Yugoslavia. The author thanks Argonne Na-tional Laboratory and, in particular, Hans Kaper for supporting his stay at the workshop.

"Department of Mathematics, Technical University, Dresden, German Democratic Republic.

74

2n -1 such that

f [2"] = l (f)= jr ig

with some g EL 2. On D we define the operators Bin andAO : D(Au i~): = D(B i~): = DP

Bbnf : =g itl(f) =|rjg, gE 4

and A : = JBin. Evidently, A Jnf = g if and only if for f ElF, g EL . we havel(f) = rg. It is easy to see that the definition of these operators is correct. Theclosure of Agn in 42 exists; it is denoted by Ami and called the minimal opera-tor associated with the problem (1.1). It is easy to see that Amin and Ami areHermitian with respect to the inner product [-,-], that is, they are Hermitian inthe Krein space 4 (for the definition of Hermitian and selfadjoint operators inKrein spaces, we refer to Bognar [1974] and Langer [1982]).

1.2. Recall that an inner product on a linear space L is said to have a finitenumber c of negative squares, if it is negative definite on a i-dimensional sub-space of L and there exists no (a+1)-dimensional subspace with this property. Inthis paper we study the problem (1.1) under the following assumptions (Al) and(A2).

(Al). The inner product "-, - , defined on IF by

f ,g;: = [AJnf .g ] ( = fp -i1f(I)UTdx)j=o a

has a finite number of negative squares.

PROPOSITION 1. 7he condition (Al) is satisfied in each of the following cases:

(a) The problem (1.1) is regular.

(b) For each singular boundary point a or b of the problem (1.1), thereexists a' E(a,b) or b' E(a,b) such that the inner product [- , - is nonnega-tive definite on the set of all functions f ElF which vanish outside of (a ,a')or (b',b), respectively.

To prove the first statement, we observe that [Af ,g] = (BnJ ,g)(f ,g EY) and use M. G. Krein's results that Bin is bounded from below and thatan arbitrary selfadjoint extension of Bn in Lf. has discrete spectrum (seeKrein [1947J and Naimark [1968]). The second statement follows if we use thedecomposition method of I. M. Glazman [1967], restricting APn to all functionsf ED with the property f (b') = f ['1(b') = - -- = f[t2-'](b') = 0 (if, for example,b is singular), and use statement (a).

1.3. The Hermitian operator Ai in the Krein space 42 has selfadjoint exten-sions in 42. In fact, A is a selfadjoint extension of Ami if and only if the operatorB: = JA is a selfadjoint extension of the Hermitian operator Bn in the Hilbertspare L r.. Therefore the selfadjoint extensions A of Ami are completelydescribed by boundary conditions at a and b which are the same for A andB = JA and which can be found, for example, in Naimark [1968]. Now we formu-late the second assumption.

(A2). For some (and hence for all) selfadjoint extensions A of Amn in 42, theresolvent set is nonempty.

75

We mention (cf. Daho and Langer [1977]) that this condition is equivalent toeach of the following:

(A2') For some (and hence for all) AEC, the range R(A,i~ - XI) is closed.

(A2") For some selfadjoint extension A of Ain and for some AEC, the rangeR(A - XI) is closed.

PROPOSITION 2. The condition (A2) is satisfied in each of the following cases:

(a) The problem (1.1) is regular.

(b) For each singular boundary point a or b of the problem (1.1) thereexists a'E(a,b) or b'E(a,b) such that the weight function r is of constantsign a.e. on (a,a') or (b',b), respectively.

Here the statement (a) is a classical result, and (b) follows again from anapplication of Glazman's decomposition method [Glazman, 1967].

2. Definitizability of the Selfadjoint Extensions

2.1. Recall that a selfadjoint operator A in a Krein space K is said to bedefinitizable [Langer, 1982] if p(A) 0 and there exists a polynomial p suchthat [p (A)f ,f ] 0for all f ED(Ak), where k is the degree of p.

THEOREM 1. Suppose that the operator A nin in Section 1.1 satis es the conditions(Al) and (A2). Then every selfadjoint extension A of Ai~ in 4 is definitizable.

Indeed, it is easy to see that for such a selfadjoint extension A the innerproduct [Af ,g] (f ,g ED(A)) has a finite number of negative squares. Hence wecan apply Langer [1982: 1.3(f-)], and the statement follows.

Suppose, for example, that the problem (1.1) is regular. Then we have forf ,g ED(A)

b

[Afg] = fpajf2fUdz n+b(fg), (2.1)j=0 a

where b(f ,g) ("the boundary form") is an inner product, depending for a regular(singular) boundary point only on the values of f ,g and their first 2n-1 quasi-derivatives at this point (in the neighborhood of this point, respectively). Thenumber of negative squares of the inner product, given by the first term on therighthand side of 2.1), coincides with the number of negative squares of theinner product [Arnif,g ] on D. Thus, the number of negative squares of[Af ,g] (f ,g ED(A)) is not greater than the sum of the negative squares of[A n f,g ] (fg E D) and of the boundary form b(f ,g) (f ,g ED(A)).

2.2. Here we suppose that the conditions of Theorem 1 are satisfied and A is anarbitrary selfadjoint extension of A ; I, in the Krein space 4. By A we denotethe number of negative squares of the inner product [Af ,g] (f ,g ED(A)). Thefollowing spectral properties of A are immediate consequences of thedefinitizability of A (see Langer [1982]).

(1) The operator A has at least AA eigenvalues A (counted according to theiralgebraic multiplicities) in the closed upper half-plane with the following pro-perty: If A > 0 (A < 0) there exists an eigenelement f of A corresponding to Asuch that [f ,f] 0 (f,f]z 0).

76

(2) The nonreal spectrum of A consists of pairs of isolated eigenvalues A,,;the linear span of the root spaces corresponding to these eigenvalues A in theupper half-plane is neutral and hence of dimension !5x .

We mention that. for any selfadjoint operator A in a Krein space the rootspaces, corresponding to two eigenvalues X, are orthogonal with respect to theindefinite inner product if A y, and skewly linked if A = X and a, are isolatedpoints of a(A).

(3) The operator A has positive and negative spectrum, both of infinite multi-plicity. If, in particular, a(A) is discrete, it contains infinitely many positiveeigenvalues sj' and infinitely many negative eigenvalues sj~, j = 1,2,..., and theroot spaces, corresponding to the real eigenvalues of A are nondegenerated withrespect to the indefinite inner product.

We denote the signature of the root space corresponding to the real eigen-value X of A by (_(X),,c+(X)); for an arbitrary eigenvalue X of A, its algebraicmultiplicity is denoted by v(X).

(4) If a(A) is discrete, we have

EK+(sj-) + EJC-(S +) + v(X) SrcA,

where the sign = holds if 0 is not an eigenvalue of A.We mention that these statements imply some results of Mingarelli [1983a

and 1983b].The spectral theory of definitizable operators in Krein spaces yields the

existence of a spectral function with critical points (see Langer [1982]) for A. Itcan also be shown that there exists a scalar or matrix spectral measure thathas, possibly, certain singularities; in a special situation this spectral measurewas considered by Daho and Langer [1977]. Moreover, expansions of arbitraryelements of L,. with respect to eigenelements or generalized eigenelements ci Ahold. However, they become more complicated than in the case of a positiveweight function as the integrals need a regularization at the singular criticalpoints of A (see Daho and Langer [1977] for the case of second-order operators).In Section 3 we shall show that these expansions are "nice" if the spectrum of Ais discrete, r has only finitely many turning points, and at these turning pointssome condition -going back to Beals [1984] -is satisfied. Recall that the pointsof AinA. are called the turning points of r.

2.3. The following result can also be proved by means of Glazman's decomposi-tion method, using Theorem 1 of Jonas and Langer [1979].

PROPOSITION 3. Suppose that the condition (A) Ls satisfied and that r has -nly afinite number of turning points. If the set A_ has a positive distance from allthe singular boundary points a or b, then a(A)n(-oo,0) is discrete with the onlyaccumulation point -co.

For a special differential operator, this structure of a(A) was established inMikulina [1971].

Finally, we mention that in the special case IA = 0 (that is, [Af ,f ] > Fif ED(A)) and Ou(A) the eigenvalues sf, j = 1,2,..., can be characterized bymeans of rminimax principles (see Phillips [1970] and Textorius [1974]).

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3. Regularity of the Critical Point Infinity

3.1. The turning point zo of the weight function r is said to be n-simple if thereexists an interval I0 around z0 such that for x El0 x0 j representation

r(X) = sgn(z-xo) -z -z0 ap(x) (3.1)

holds with some a >-3 and a function p: '

p(x): = p(x), z > x0 ,

p(x): = p-_(z), z < x0 ,

where p.(p-) is defined and of class C" on Ion[zo, ) (Ion(-',z0], resp.),0 # sgnp (zo) = sgnp_(x 0) and for the one-sided derivatives at x 0 we have

Pt(Zo) = P*(Zo) = ... = p"~4)(xo) = 0 if n > 1.

THEOREM 2. Suppose that the following conditions are satisfied:

1. The problem (1.1) is regular.

2. The weight function r has only a finite number of turning points thatare all n-simple.

3. There exists a 6 > 0 such that for each turning point x 0 of r we have

0< inf po(z) s suppo(x) < .Ix-x01- IZ-z0I

sc(s.b) ze(s.b)

Then infinity is not a singular critical point for every selfadjoint exten-sionA of Awn in L,2.

We shall only sketch the proof. Propositions 1 and 2 and Theorem 1 implythat A is deflnitizable. We show that for A there exists an operator W with theproperties given by Curgus [1984: Remark 3.6], and an application of a proposi-tion given in that paper Curgus, 1984: Proposition 3.5] yields the desired result.

To simplify the construction of W, we suppose a < 0 < b and that x0 = 0 isthe only turning point of r. Let 6 > 0 be such that (-6,6)C (ab) and Io in (3.1)can be chosen to be (-6,6).

We choose 2n mutually distinct points t1, .... , ta E(1,2) and define thefunctions

hy(x): = Q-- ) t. (xE[-2, s-], x 0).tfa p(-tyx)

By D we denote the yet of all functions f E42 which have an absolutely continu-ous (n -1)-st derivative and for which

6

fpolfC()I 2 IrIdx < co.G

Further, we choose 9EC"(a,b), which is constant in a neighborhood of zero,o(0) = 1, and supppc[- 2 ]2'

78

Now we define linear operators X, Y in L, as follows:

(X+u)(x): = u(z), z E(0,b ]

(X+u)(x): = Sp(z) tof tu(-t jz), ZE[a,0).

(Y~u)(z): = 0, z ea,0),

u(z) + Eaj (phju)(- , zE(Ob],j=1 ti

where al,.. . , af are reals to be chosen below. It is not hard to see that X+, Yare bounded in 4 . Moreover, the numbers a 1, ... , a2. can be chosen such thatX+, Y+ map D into itself. In order to see this with u ED, we form the first nderivatives of X~u on [a,0) and on (0,b]. Then X~u will have n-1 absolutelycontinuous derivatives on [a,b] if and only if for the first n-1 derivatives thelimits from the left and from the right at zero coincide, which is equivalent tothe equations

ait +1 = (- 1)k (k = 0,1,...,n-1). (3.2)1=1

A similar reasoning for Y yields the equations

Aajtj-a - = (-1)+i P+(0) (k = 0,1,...,n-1). (3.3)j=1 p-(0)

The system (3.2), (3.3) determines the numbers a, j = 1,2,...,2n uniquely. It iseasy to check that the operators X+, Y+ satisfy the relation X+ = YJ where*denotes the adjoint in L1.

In the same way, exchanging the roles of [a,0) and (0,b], operators X-, Y_with similar properties are defined. Finally, put

W: = YX. + Y-X-.

As in Curgus [1984: Remark 3.6], it follows that W is positive, bounded, andboundedly invertible in the Krein space 42. Moreover,

(Wu)(z)=u(z) if zE[a,b] (-2-, 2 ).

We mention that Xt, Y here do not necessarily have the property (a) given inCurgus [1984: Remark 3.6].

The set D[JA] (see Krein [1947] and Curgus [1984]) consists of those func-tions of D that satisfy the essential boundary conditions. As the function Wucoincides near a and b with u, it satisfies the same boundary conditions as u;hence WD[JA] C D[JA]. Thus W has all the desired properties.

The construction of the operators X, Y, follows [Beals, 1984: Lemma 1].

3.2. Under the conditions of Theorem 2 we denote by P,*1 the orthogonal projec-tion in the Krein space 4 onto the root space of A corresponding tos, j = 1,2,..., and by PO the orthogonal projection onto the (finite dimensional)span of the root spaces corresponding to the (possible) eigenvalue zero and toth nonreal eigenvalues of A.

COROLLARY. Under the conditions of Theorem 2 we have for arbitrary f E4

79

f = Pof + + 2P4.-f ,

,dwre both sums converge in the normof L r I.We mention that for all the points s with the property i+(st) = ,+(sf7) = 0

there can be chosen an orthogonal basis of eigenvectors efb inP1,t4 2 , k = 1,...,vf, j = 1,2,..., such that

j-*t .=1[eJA. , k]

The corollary contains, for example, the full-range expansions of the "regular"examples in Kaper, Kwong, Lekkerkerker, and Zettl [1984]. We mention that theabove construction of W and hence the statement of Theorem 2 can also beextended to some singular operators. This extension will be considered else-where.

3.3. Suppose now for a moment that (under the conditions of Theorem 2) wehave cA = 0 and Ofa,,(A). Then o(A) consists of the two sequences (sj'), (s),and we have *c_(s+) = is(s[) = 0, j = 1,2,.... Moreover, the subspace

c.l.s. JP,+42: j = 1,2,...3is a maximal nonnegative subspace of the Krein space 4L (see Bognar [1974] andLanger [1982]). If we denote by K the subspace

K,: =If E4: f(z)= 0if xEA-4and by P+ the orthogonal projection onto K} in 4, it follows that for arbitraryf +EKe. we have

f1+ = P+Pj.+f+. (3.4)j=1

where the series converges again in the norm of LFr i . This is an abstract form ofthe half-range expansion considered, for example, in Beals [1984] and Kaper,Kwong, Lekkerkerker, and Zettl [1984].

If ICA z 0 we consider for arbitrary f +EK.. the sum

E P+Pj.+f +.j x_(sj) =o

It converges in the norm of L f,1; however, it equals f + only for f +EK',. where Kis a subspace of K+ such that dimK4 / K,' < cc. To expand arbitrary elements ofK+, we have to add to the elements of P.R.4 , ic_(sj) = 0, finitely many ele-ments hk that are the projections onto K, ofroot vectors h* of A correspondingto the possible eigenvalue zero, the eigenvalues sf with ic(s -) > 0, #c(s-) > 0and to nonreal eigenvalues. A minimal set of such elements hk which have to beadded can easily be found from the condition that the linear span of these rootvectors and of c.l.s. JP, L, :.:(s,'S) = 0, j =1,2,... is a maximal nonnegative sub-space of the Krein space I,2.

References

Beals, R. 1984. "Indefinite Sturm-Liouville problems and half range complete-ness." J. Differential Equations (to appear).

80

BognAr, J. 1974. Indefinite Inner Product Spaces. Springer-Verlag, New York.

Curgus, B. 1984. "On the regularity of the critical point infinity of definitizableoperators" (to appear).

Daho, K., and Langer, H. 1977. "Sturm-Liouville operators with an indefiniteweight function." Proc. Roy. Soc. Edinburgh 78A:161-191.

Glazman. I. M. 1967. Direct Methods of Qualitative Spectral Analysis of SingularDifferential Operators. Nauka, Moscow (in Russian).

Jonas, P., and Langer, H. 1979. "Compact perturbations of deflnitizable opera-tors." J. Operator Theory 2:63-77.

Kaper, H. G., Kwong, M. K., Lekkerkerker, C. G., and Zetti, A. 1984. "Full- andpartial-range eigenfunction expansions for Sturm-Liouville problems withindefinite weights," Proc. Roy. Soc. Edinburgh 98A.

Krein, M. G. 1947. "Theory of selfadjoint extensions of semibounded Hermitianoperators and its application 1, II," Mat. Sb. 20 (62):431-495, 21 (63):365-404.

Langer, H. 1982. "Spectral Functions of Definitizable Operators in Krein Spaces."Functional Analysis (Proc., Dubrovnik, 1981). Lecture Notes in Maths mat-ics, Vol. 948. Springer-Verlag, New York, pp. 1-46.

Mikulina, 0. F. 1971. "Eigenfunction expansion of the 2nd order differentialequation with one turning point." Differencialnye Uravnenija 7 (2):244-260.

Mingarelli, A. B. 1983a. "Indefinite Sturm-Liouville Problems." Lecture Notes .nMathematics, Vol. 964. Springer-Verlag, New York, pp. 519-522.

Mingarelli, A. B. 1983b. "On the existence of nonsimple real eigenvalues for gen-eral Sturm-Liouville problems." Proc. Amer. Math. Soc. 89 (3):457-460.

Naimark, M. A. 1967-1968. Linear Differential Operators, Parts I and H. Ungar,New York.

Phillips, R. S. 1970. "A minimax characterization for the eigenvalues of a posi-tive symmetric operator in a space with an indefinite metric." J. Fac. Sci.Univ. Tokyo, Sect. IA Math. 17:51-59.

Textorius, B. 1974. "Minimaxprinzipe zur Bestimmung der Eigenwerte J-nichtnegativer Operatoren." Math. Scand. 35:105-114.

81

LINEAR RELATIONS IN INDEFINITE INNER PRODUCT SPACES

A Dijksma*H.S. V de ,n9oot

Abstract

The spectral theory of selfadjoint operators in Pontryagin spaces is welldeveloped; see Bognar [1974], lohvidov, Krein, and Langer [1982], and Krein andLanger [1971/72]. Even in Krein spaces there is a spectral theory for a class ofselfadjoint operators, namely, the definitizable operators; cf. Langer [1982].However, in a number of problems it is not sufficient to consider operators; in-stead one has to consider relations (multi-valued operators). The spectraltheory of such relations has been studied by Coddington, Dijksma, and de Snoo(among others), in the case of Hilbert spaces. In the more general case of Pon-tryagin spaces or Krein spaces, recent work has been done by Ricner [1982] andSorjonen [1978/79 and 1980]. Here we report on some results for relations insuch spaces. Details will appear in Dijksma and de Snoo [1984]. Also, we reporton some results of our joint work with Langer [Dijksma, Langer, and de Snoo,1984]. This last paper contains applications to boundary value problems. Otherapplications can be found in the area of "left-definite" eigenvalue problems; seeCoddington and de Snoo [1981] for the Hilbert space case, and also Langer[1972].

1. Sne General ResultsLet K be a Banach space and provide K2 with the usual topology. Let A cK2

be a linear relation or linear manifold. We define the set of points of regulartype by

7(A) = JXEC I (A -X)-' is a bounded operator;,

and the resolvent set p(A) by

p(A) = XEy(A) I R(A -X) is dense in K;.

It can be shown that 7(A) and p(A) are open sets. In addition, we assume that Ais a subspace, i.e., a closed linear manifold, such that p(A) 0 0. We define theresolvent operator RA:p(A)-[K] by

RA(X) = (A-X)-', XEp(A),

where [K] is the set of all bounded linear operators from all of K into K. Wehave the resolvent equation

RA (A) - RA ( ) = (X-p)RA (X)RA (A), ; .,Ep(A),

which implies that RA :p(A)-[K] is analytic. Note that

v(RA (X)) = A(o ), AEp(A),

*Mathematisch Institut, Rijksuniversiteit Groningen, Postbus 800, 9700 AV Groningen, the Neth-erlands.

tThe presentation of this note at the Workshop on Spectral Theory of Sturm-Liouville DifferentialOperators at Argonne National Laboratory, May-June 1984, was supported by the Netherlands Organi-zation for the Advancement of Pure Research (Z.W.O.).

82

where v(RA (A)) indicates the null-space of RA (A) and

A(o) = jgEK I og EA[.

REMARK. If 0 focC and R:0-+[K] is a pseudo-resolvent. i.e.,

R(X) - R( ) = (X-A)R (A)R (p), X,pEQ.

then there exists a unique subspace A zK2 such that

flcp(A), R = RA In.

Let p:C-+C be a complex-valued polynomial p (A) = c (a-a)k, acC, c 0 0.k =0

Then we define for the linear manifold A cK

p (A) = 2 c" (A -a)".k=0

If A cK2 is a subspace with p(A) ; 0, we have the following results which we statefor later reference:

(i) (p (A) is a subspace (closed linear manifold),

(ii) RA (A)"p (A)RA CU)" E[K], X,pEp(A) .

We define the semi-FredhLm sets 4 (A) by

@+(A) = EXEC I R(A-A) is closed, dimv(A-A) < o j,and

_(A) = EXEC I R(A-A) is closed, dimK/R(A-A) < a .

THEOREM 1. Let A cK be a subspace. Then

(i) $,(A) is open,

(ii) dimv(A-X) - dimK/ R(A-A) is constant on components of 4, (A), and

(iii) dimTv(A-X) and dimK/R(A-A) are constant on components of 4,(A) exceptat isolated points, where they are strictly larger than the said constants.

REMARK. Let U be a component of 4+(A). Then for all XE U

dim(v(A -A) n" R(A -A)"1)

is equal to the constant value that dimv(A -A) assumes on U, except at isolatedpoints. Hence it is clear that AE U is not an exceptional point if and only if

v(A -)cR(A - )", rn EN.

A similar remark applies to dimK/ R(A -A).

COROLLARY (Constancy of Deficiency Index). Let AcK2 be a subspace. Then

dimK/ R(A -A),

is constant on components of y(A), the set of points of regular type.

83

As to the proof of the theorem, it is possible to reduce the given result to acorresponding result for a pair of bounded linear operators, as given by Gohbergand Krein '1960] and by Kato [1966].

For a subspace AcKe, we define F(A) to be the set of all complex-valuedfunctions which are analytic in some neighborhood of a(A), the extended spec-trum of A. Note '(A) = a(A)u , when AA[K]; here a(A) denotes the spectrumof A: a(A) = Cp(A). If AcKe is a subspace with p(A) i 0 and AA[K], then v edefine for f EF(A)

f (A) = f ( )I -2 f (;)RA (X)dX,

where ' is a suitable contour, surrounding the extended spectrum c(A). Thenf (A)E[K] and has the usual properties. In particular, if aca(A) is a boundedspectral set of A, and

EA(a) = - 2 RA ()d,

where ' is a closed contour in p(A), such that a is inside ' and a(A)\a is outside', then EA (a) is a projection, the so-called Riesz-projection.

If A cKe is a linear manifold and M = [ )is a 2x2 complex-valued matrix,we define the linear manifold MA by

MA = fUxf +pg, -f +6g i ifg IEA;.

The Cayley and inverse Cayley transforms are defined by the matrices

- ] and [-i ,]

respectively.

If AcKe is a subspace with p(A) # and if M = P) is such that s 0 and

- -Ep(A), then

MA = - detM RA

In addition, if A E[K] and

f) +6 a #

then f EF(A) and f (A) = MA.

2. Krein Spaces and Pontryagin SpacesLet K be a linear space over C, and let [ , ]:KxK-+C be a sesquilinear form:

{gEK, f -ifg ] is linear,

Vf ,gEK, [f ,g ]= 9,f ,

which is non-degenerate:

[fg ] = 0 V EK=>f = 0.

Then (K,[ , ]) is called a Krein space if K = K, + K., where K.cK are linearmanifolds, such that

84

(K, t [, ] I<'xg ) are Hilbert spaces,

and

[K+,K] = 0.

In a Krein space the decomposition K = K + K_ is a direct sum. Let Pt:K-Ktbe the corresponding projections and put J = P+ - P... Then (, ):KxK-'Cdefined by

(fg) = [Jf,g]

is an inner product on K and (K,( , )) is a Hilbert space. In general, there aremany decompositions. If K = K'+ + K'_ with P't and J' is another such decom-position, then the norms corresponding to (,) and (, )', defined by(f ,g)' = [J' fg] are equivalent and dim K = dim K'i. We shall use the topologi-cal notions with respect to one of these norms, and hence with respect to all ofthem.

(K,[ , ]) is called a Pontryagin space if dimK <c or dimK_ < Do. In thesequel we shall always assume for Pontryagin spaces that 'c = dim K <c anddenote such spaces by H.

Orthogonality with respect to [ , ] is denoted by [ 4.; for L cK we define

L 1 = if EK I f[..L .

A linear manifold L cK is called non-nega.tive if [f ,f ] 0, f EL, non-positive if [f ,f ] s 0, f EL, and neutral if [f ,f ] = 0, f EL. Maximality withrespect to each of these notions is defined in the usual way. A linear manifold Lis maximal neutral if and only if L is non-negative and non-positive and maximalfor one of these notions. A linear manifold L is hypermaximal neutral if L ismaximal non-negative and maximal non-positive.

Example. In CP we consider a symmetric, non-degenerate form [ , ], given by

[fg] = g'Qf, fg ECP,

where Q is a symmetric, invertible operator. Let (a+,a) be the signature of [,];then a. + a_ = p. For a linear manifold L cCP we have

L maximal non-negative L non-negative, dim L = a.,L maximal non-positive L non-positive, dim L = a_,L maximal neutral e L neutral, dim L = min(a,,a_),L hypermaximal neutral L neutral, dim L = a+ = a_.

Note that

L neutral e L cL,L hypermaximal neutral e L = Li.

We have the following result. Let LcCP be a linear manifold. If

(i) L is maximal non-negative,(ii) L is maximal non-positive,(iii) L is maximal neutral,(iv) L is hypermaximal neutral,

then there exists a axp matrix M such that

(+) rank M = a,

85

with the following properties, respectively,

(1') Q = a+,

iii') = max(a+,,.),(iv') a =a=a

MQ-'Wm gO,MQ- 1M* o 0MQ-1m = og,MQ- 1M' =O ,

such that

(++) L = f ECj Mf= 01 .The matrix M is unique, up to multiplication on the left by an invertible oxa

matrix. Conversely, if we are given a axp matrix M such that (+) holds, andwhich satisfies (i'), (ii'), (iii'), or (iv'), and the linear manifold L is defined by(++), then L has the property (i), (ii), (iii), or (iv), respectively. These resultscan be applied to boundary value problems; cf. Coddington and de Snoo [1981]and Langer and Textorius [1982].

3. linear Relations in Krein SpacesLet K be a Krein space with inner product [ ,]. Let AcK5 be a linear mani-

fold. Then its adjoint A is defined by

A = f,gEK2 I [g ,h] =[f.k] for all h,kJEAj.

Note that

A+ = JA'J,

where A' is the adjoint in the Hilbert space (K,[J - , - ]) .

If AcK2 is a subspace, then we have

R(A - X) closed a R(A -~\) closed;

cf. Coddington and Dijksma [1978].

A linear manifold AcK2 is called

dissipative

symmetric

if lm[g , f ] 0, if,g; EA,

if lm[g ,f ] = 0, if ,g j EA, or equivalently if AcA ,

self adjoint if A = A+.

A linear manifold AcK2 is called

comtrac tive

isometric

unitary

if [gg] [f,f], f,gEA,

if [g.g ] = [f ,f], ifgEA, or equivalently if A- 1 cA+,

if A- 1 = A+.We have introduced the Cayley transform C,(A) of a linear manifold AcK 2,

C,(A) = Efg- f4, g- fl I fg /, yEC.

Note that for pEC\J, we have

-A is dissipativeA is symmetricA is selfadjoint

C,,,(A) i3 contractive,SC,(A) is isometric,

e C,J(A) is unitary.

86

REMARK. We may provide K2 with a new inner product, given by -i < , >, where< > is given by

<f ,g j," h,k j > = [g.,h] - [f ,k ], ,g j, ih,k j EK2 .

We denote orthogonality with respect to this inner product by < .>. For a linearmanif old AcKe, we then have

A+ = A>.

We observe that the notions dissipative, symmetric, and selfadjoint areequivalent to the notions non-negative, neutral, and hypermaximal neutral,respectively.

A similar remark may be applied to the notions contractive, isometric, andunitary if we provide K2 with the inner product

[f ,h] - [g ,k ], if ,g J, h,k j EK2.

Example. The above remark in connection with the example in Section 2 may beused to characterize maximal dissipative, selfadjoint, maximal contractive, andunitary linear manifolds in finite dimensional spaces. Consider the inner pro-duct space C 1[ , ]), where [f ,g] = g'Qf , f ,g EC1 as in the example of Section2. Let A c(C )2 be a linear manifold. If

(i) A is maximal dissipative (selfadjoint),ii) A is maximal contractive (unitary),

then there exist pxp matrices MN, such that

(+) rank (M:N) = p,

with the following properties, respectively,

(i') lm MQ-1N' s O(ImMQ-'N' = O),(ii') MQ-'M" - NQ-'N s $Q (MQ-'M" - NQ-'N" = O),

such that

(++) A = Nif ,g E(CP)2 I Mf + Ng = OI.

The matrices M,N are unique, up to simultaneous multiplication on the left byan invertible v xp matrix. Conversely, if we are given p xp matrices M,N suchthat (+) hole- and which satisfy (i') or (ii'), and the linear manifold A is definedby (++), then A has the property (i) or (ii), respectively.

In the rest of the paper we shall concentrate on symmetric and selfadjointlinear manifolds in Krein spaces (and in Pontryagin spaces). If A is a symmetricsubspace in K2 with p(A) 0 0, then we may choose pEp(A)\R, and Ch(A) is anisometric operator belonging to [K]. Conversely, if UE[K] is isometric andyEC\R, then the inverse Cayley transform F,( U) is a symmetric subspace, suchthat p.Ep(F,( U)). These and similar observations can be used to translate pro-perties from one situation to another. We shall not state further results in thisdirection, but we refer to Dijksma and de Snoo [1984].

A selfadjoint subspace AcK 2 is called deftnitizable if p(A) and if thereexists a real polynomial p such that p (A) > 0, meaning that

[g,f ]>0 for allif ,g lEp(A).

87

THEOREM 2. Let AcK be selfadjoint with p(A) i t, and aEp(A). Let p be a realpoLynomial of degree n. Then

p (A) &0 RA (a)" p( A)RA(a)" &0.

As a consequence we can prove, by means of the operational calculus, the follow-ing corollary.

COROLLARY. If the selfadjoint subspace A cX 2 is definitizable, then A-1 is alsodefinitizable.

It turns out that for definitizable subspaces a spectral theory can bedeveloped; for the case of definitizable operators we refer, for instance, toLanger [1982]. We shall not present the details here, but conclude with the fol-lowing observation.

PROPOSITION. Let K be a Pontryagin space and let AcK 2 be selfadjoint withp(A) # 0. Then Ais definitizable.

4. Symmetric Relations in Pontryagin SpacesWe now consider relations in Pontryagin spaces. We shall use the notation

116 for a Pontryagin space with k negative squares (k < co). First we note that ifA ci is a linear manifold with R(A -X) closed, then we have

dim H/ R(A -X) = dim v(A-X);

see Iohvidov, Krein, and Langer [1982: Theorem 3.4] or Bognar [1974]. Hence weobserve

4+(A) = 4_(A+),and the index

dim v(A-X) - dim v(A-X)

is constant on components of @i(A). A special case of this situation for non-densely defined multi-valued differential relations may be found in Coddingtonand de Snoo [1981: Theorem 7.3].

Let A c~i be a symmetric subspace. Then we may prove

C0 \u (A) c y(A),

and

R(A-X) is closed for XEC0 ,

where Co denotes C~uC-, and u, (A) denotes the point spectrum of A: the set ofall XEC for which there exists a non-trivial f such that f , Xf JEA.

If AcID is symmetric, then it is not difficult to see that v(A-X), for XEC, isa neutral subspace of Hk, and hence

dim v(A-a) <;k, XEC0 ;

see Iohvidov, Krein, and Langer [1982: Lemma 1.2]. This implies

C+c4+(A), C-c4+(A).

THEOREM 3. Let AdcH be a symmetric linear manifold. Then dimv(A-X) is con-stant on C+ (or C-), say dim v(A -X) = n, with the exception of at most k-n pointsin C+ (or C-). Hence also dimv(A+-X) is constant on C+(orC-) outside the sameexceptional set.

88

A proof of this theorem may be given using the remark following Theorem 1and the fact that

v(A -X)c nRA -)", X,y C, X .ntEN

In particular, if in Theorem 3 we have n = 0, then the defect indexdim(A+-X) is constant in C* (or C-), say dimv(A+-X) = m, with the exception ofat most k points in C (or FC-~), where

dimv(A+-X) = m + dimv(A-X);

cf. Iohvidou, Krein, and Langer [1982: Theorem 6.1].

If in Theorem 3 n > 0, then we can show a,,(A) = C, and hence we have thefollowing alternative. Let A cIe be a symmetric linear manifold. Then either|au(A)nC I sk andI Ia,(A)nC- |Isk, or ap(A) = C.

An eigenvalue XEa (A) is called critical if there exists an element SoEHk,op # 0, such that f o,XoEA and [po] s 0. Similarly co is called a critical eigen-

value if there exists an element IoEH, SP ? 0, such that J0,oyEA and [p,p] s 0.

THEOREM 4. Let A cIJi be a symmetric linear manifold with ap(A) C. Then A hasat most k critical eigenvalues in C uRuloi, and at most k critical eigenvalues inC-URUAeH.

We shall not give further properties concerning eigenvalues and root mani-folds here, but state a result concerning maximal symmetric linear manifolds.

THEOREM 5. Let A cH be a symmetric subspace. Then p(A) i 0 if andronly if A ismaximal and ap (A) C. In particular, if A is a selfadjoint subspace, thenp(A) 7 10if and only ifcai(A) C.

We close this section with the remark that in the case of symmetric rela-tions the situation with regard to the degeneracy of the spaces v(A-X) is morecomplicated than in the case of densely symmetric operators; cf. Krein andLanger [1971/72].

5. Compressed ResolventsThis section contains some results of our work with Langer [Dijksma,

Langer, and de Snoo '1984]. Let H be a Hilbert space, and let IL be a Pon-tryagin space with k negative squares. We use the notation Hc.,H to indicatethat H is a subspace of % and that the indefinite inner product of H restrictedto H coincides with the Hilbert space scalar product of H. Let A be a subspacein H with p(A) o 0. On p(A) we study the locally holomorphic [H]-valued func-tion R defined by

R(X)) = PRA(X) | IB.XEp(A),

where P denotes the orthogonal projection from k onto H. This function R iscalled the compressed resolvent of A in H. It is possible to characterize thecompressed resolvents of various subspaces in IH. We shall give the result forthe situation where A cA+, p(A)0 0. First we introduce the following notions.

If A cH is a symmetric subspace with p(A)nC4 0, and HCHk, where H is aa

Hilbert space, then A and H are said to be closely upper-connected, if the closedlinear span of H and all ranges (A-A)-1H, XEp(A)nC+ is the space H.

Let DcC+, D # and let K:DxD-[H] be a mapping. We say K has k negativesquares if

89

i) it is Hermitian, i.e., K(LA) = K(Xl)', L,A ED,ii) for all choices of n EN, LItED, f EH, i = 1....,n, the matrix

( [K(.li g)ft.F'f g ])t.f=1..n

has at most k and for at least one such choice it has exactly k negative squaresLet R be a function defined on some set DcC+, D 0 0, and with values in [H]. ByRR we denote the kernel

RR(l,X)= R(L)-R(X)_ - R(X)'R(L), I,XED.1-A

The set of all functions R, which are defined and meromorphic in C+ with valuesin [H], such that the kernel RR has k negative squares on the domain of holo-morphy of R, will be denoted by R.(H).

THEOREM 6.

(i) Let Hk be a Pontryagin space, and let A cH be a symmetric subspace withp(A)nC+ 0. If HcH 4 is a Hilbert space and R:p(A)-'[W ] is the compressedresolvent in H associated with A, then RE+R(H) for some k',05 k' ! k. If A andH are closely upper-connected, then k' = k.

(ii) If R ER+(H) is given, then there exists a Pontryagin space . , HCHk, and asymmetric subspace AcH with p(A)nC' $ 0, such that R is the compressedresolvent in H associated with A Here Uk and A can be chosen such that A andE6 are closely upper-connected, in which case A and 1 are uniquely deter-mined up to isomorphisms, which restricted to H are equal to the identity opera-tor on H.

Similar results are valid for the case where A is selfadjoint in H, withp(A) s 0. Also these results can be translated into results for the associatedfamily of .traus subspaces, defined by

T(l) = R(l)-'+L, LEp(A).

Assume we are given a symmetric subspace ScH2, where H is a Hilbertspace. Let A be a selfadjoint subspace in IJ with p(A) i , such that Hcdk and

SA extends S, i.e., ScA. Then we have

ScT(l)cS, LEp(A).

Now T(L)/S can be completely determined in the boundary space S/S, analo-gous to the above theorem. In this way it is possible to associate subspaces inPontryagin spaces with concrete boundary value problems. For details, we referthe reader to Dijksma, Langer, and de Snoo [1984].

References

Bognar. J. 1974. Indefinite Inner Product Spaces. Springer-Verlag, Berlin.

Coddington, E. A., an + Dijksma, A. 1978. "Adjoint subspaces in Banach spaces,with applications to ordinary differential subspaces." Ann. Mat. Pura Appl.118:1-118.

Coddington, E. A., and de Snoo, H. S. V. 1981. "Regular Boundary Value ProblemsAssociated with Pairs of Ordinary Differential Expressions." Lecture Notesin Mathematics 858, Springer-Verlag, New York.

90

Dijksma, A., Langer, H., and de Snoo, H. S. V. 1984. "Selfadjoint 11k- extensions ofsymmetric subspaces: an abstract approach to boundary problems withspectral parameter in the boundary conditions." Integral Equations andOperator Theory.

Dijksma, A., and de Snoo, H. S. V. 1984. "Symmetric and selfadjoint , elations inKrein spaces" (in preparation).

Gohberg, I. C., and Krein, M. G. 1960. "Fundamental theorems on deficiencynumbers, root numbers, and indices of linear operators." Amer. Math. Soc.Transl. (2) 13: 185-264.

Iohvidov, I.S., Krein, M. G., and Langer, H. 1982. "Introduction to the SpectralTheory of Operators in Spaces with an Indefinite Metric." MathematicalResearch 9, Akademie-Verlag, Berlin.

Kato, T. 1966. Perturbation Theory for Linear Operators. Springer-Verlag, Ber-lin.

Krein, M. G., and Langer, H. 1971/72. "Defect subspaces and generalized resol-vents of an Hermitian operator in the space Ilk." Functional Anal. Appl. 5:136-146, 217-228.

Langer, H. 1972. Zur Spektraltheorie verallgemeinerter gewbhnli herDifferentialoperatoren zweiter Ordnung mit einer nichtmonotonenGe'wchtsfurktion. Univ. Jyvaskyla Dept. Math., Report 14.

Langer, H. 1982. "Spectral Functions of Definitizable Operators in Krein Spaces."Lecture Notes in Mathematics, Vol. 948. Springer-Verlag, New York.

Langer, H. , and Textorius, B. 1982. "L-resolvent matrices of symmetric linearrelations with equal defect numbers; applications to canonical differentialrelations." Integral Equations and Operator Theory 5:208-243.

Ricner, V. S. 1982. "Linear relations and indefinite geometry." Candidate'sdissertation, Voronesh.

Sorjonen, P. 1978/79. "On linear relations in an indefinite inner product space."Am. Acad. Sci. Fenn. Al, 4:169-192.

Sorjonen, P. 1980. "Extensions of isometric and symmetric linear relations in aKrein space." Am. Acad. Sci. Fenn. Al, 5:355-376.

91

SPECTRUM OF SELFADJOINT AND POSITIVE OPERATORS WITH COMPACT INVERSE

J. Fleckinger

Abstract

This paper is a rough English translation of a seminar held in Orsay several yearsago. It surveys different methods for studying the spectrum of elliptic opera-tors. In particular, two methods for estimating N(X) are given: 1) the transformof the spectral function, and 2) the "max-min" formula. An appendix is devotedto the variational formulation of elliptic problems.

Notations and Main Results on Sobolov SpacesWe introduce some notations used throughout the paper:

Let m and n be integers greater than or equal to 1.

Let 0 be an open set in R" with boundary 610 For a = (al, ... , a. )EN", wedenote by Da the derivative of order I a I = a1 + - -- +a : Da = Di' -D," withD, =a/z.

Hm (0) denotes the usual Sobolev space of order m; i.e., Hm (fl)= Iu EL2 (Q)tDau EL2(fl), la L s m J. H"(0) equipped with the normIIu IIHm(f) = f( 2 |Dcu(z)I2)dz j is a Hilbert space. Recall that if 60 is

n ats"smooth enough,' Hm (0) is the restriction to 0 of elements of Hm (R").

H"o(0) is the completion of CE(0) with respect to the norm of Hm (0); here Ce(0)- or D(0) - is the space of infinitely differentiable functions with compact sup-port in 0.

REMARK. When 60 is "smooth enough,"

Ho"(0) = IuEH'(f)Isuppucflj.

The following results are well known:

* When 0 is bounded, the inbedding of Ho (0) into L2 (f) is compact.

* When c is bounded, there exists c (0) > 0 such that

II'u IIL(n) s c (ft) EIIDu IIe() for all uEH"'(0).

1. IntroductionLet H be a separable Hilbert space, and let A be a selfadjoint and

unbounded linear operator in H with domain D(A). We suppose that the imbed-ding of D(A), equipped with the graph norm, into H is compact.

Moreover, we suppose that A is positive; this is always possible by consider-ing A + tI or A2 in place oftA.

92

When these assumptions are satisfied, the operator B =A-1 from H into H iscompact, and it is possible to apply the classical results of the spectral theory ofselfadjoint operators with compact inverse; see, e.g., Akhiezer and Glazman[1967], Yosida [1968], and Goulaouic [1973].

PROPERTIES.

1.1. The spectrum of A, denoted by o(A), consists only of eigenvalues.

1.2. The eigenvalues of A are real and positive.

1.3. The set of eigenvalues is a countable sequence

0<A 1SX2 < k

and A tends to +cm as k tends to +cc.

1.4. The eigenvectors associated respectively with two distinct eigenvalues areorthogonal.

1.5. The eigenspaces L are of finite dimension.

1.6. The Hilbert space H is the (Hilbertian) direct sum of the eigenspaces 1.

2. Spectral Decomposition of an Operator

For further information on spectral decomposition, the reader is referredto Akhiezer and Glazman [1967] and Yosida [1968].

DEFINITION 2.1. Let L be a subspace in H, and Let M be such that L@M = H. Onesays that L "reduces" A if

(i) L and M are invariant under A, and

(ii) For aLL zcED(A), PL(z) is in D(A); here PL demotes the orthogonal projectionon L.

REMARK. It follows that P(z) is also in D(A).

We now have the following results.

PROPERTIES.

2.1. The subspace L reduces A if and only if PL and A commute.

2.2. Let E1,E2,...,Ek,... be a sequence of orthogonal subspaces such that eachEk reduces A; then E = eEk reduces A too.

2.3. If A is an eigenvalue of A, then LA, the associated eigenspace, reduces A.

Let us denote by 0 <X1 <X < (-X < XK < ..- the (strictly) increasingsequence of eigenvalues of A, by LK the associated eigenspaces, and by PK theprojections on LK. We then have the following property.

2.4. Each TED(A) can be written as

93

X = >XzK with ZKELKK=1

and

Ax = K = 2 XKZK.1

DEFENITION 2.2. A "resolution of the identity" is a family of projections Ex whichdepend on the real parameter X, such that

(i) If j S X, then EAEM = EE4= E,,

(ii) Ex is right continuous in X, and

(iii) E-+I as X-+c" and EA-+Q as X- min the strong topology.

Let us now introduce, like a Stieljes integral, the quantity

(Sx,y) = f Xd(Eaz,y), zEH, yEH,

which defines the operator S.

Coming back to the operator A, we consider

EA = E PK;

EA is a resolution of the identity which is called the "spectral family" of A.

THEOREM 2.1. It is possible to associate with each selfadjoint operator T in a Hil-bert space a unique spectral family EA such MLat

(i) Ex is a resolution of the identity, and

(ii) D(T) = xEHI f? 2 d(EAxx) <im:.-. o

Then Tx = fXdEAz (this is a semiconvergent integral). Conversely, every

operator defined by such an integral is selfadjoint.

REMARKS. When T is positive, all eigenvalues are positive and

T = fXdEAxZ with D(T) = xEH IfX2d(ExZ,x) < ci.0 0

If we consider the positive operator T_ = VT (such that 'IT = T), we have inthe same way

7x = v'XdEX with D(v7) = xEHI|f(v'X)2d(Ex,) <.

More generally, when f is a continuous function on R, it is possible to define

f T(x) = ff (X)dEx with D(f T) = xEH I|f[f (X)]2d(EAx,z) < mi.0 0

94

If A is a regular value for T, it is possible to define the resolvent RA = (T-AI)-by

Rxz = w .

THEOREM 2.2. The regular values X of T are such that there exists an interval(A-e, A+)in R on which Ea is constant.

THEOREM 2.3. The eigenvalues A of T are the real numbers where E has a jump;E,o - E- is the projection on the associated eigenspace.

3. Transforms of the SpectrumFor further information on spectral transforms, the reader is directed to

Boutet de Monvel [1970] and Hormander [1968].Let fl be a bounded open set in R"; let Abe a 0ifferential operator in D(fl).

We suppose that A is elliptic of order 2m, formally selfadjoint, and positive on 0.We consider a boundary value problem in f2 associated with A, and we denote byA its "realization in L2()."' We suppose that A is a positive selfadjoint andunbounded operator in L2(fl).

EA is defined by a kernel e (z,y,X) which is called the "spectral function" ofA:

Ef (z) = fe(x,yX)f(y)dy.a

If we denote by N(X) the number of eigenvalues of A less than or equal to A(each eigenvalue counted according to its multiplicity), N(X) is the trace of EA.Therefore, it is possible to use transforms of the spectral function in the studyof N(A). Estimates of N(A) can be deduced from estimates of the tra.form bymeans of (primarily) Tauberian theorems.

i) The Stieljes transform.: It leads to the resolvent; for z ER+, we consider

G1(z) = f(A-r)-1dEx = (A-zI)-1; an asymptotic estimate for G,(z) can be0

obtained outside an angle Iargz j ! t (see, e.g., Agmon-Kannai [1967]).

ii) The Laplace transform: It leads to the fundamental solution of the diffusion

equation associated with A: G2 (t) = fe"dE satisfies (at+ A)G2(t) = 0 with0

G2(0) = I, t z 0 (see, e.g., Hormander [1968] and Minakshisunderam [1953]).

iii) The Riemann transform: G3 (s) = f-'dEA when Res > n/ 2m ; G(s) is an

operator with a kernel gs(z,y,s). Let us define t(s) = TrG3(s); then

c(s) = fgs(z,z,s)dz = -',D k=1

where Ak denotes the k th eigenvalue (each eigenvalue counted according to itsmultiplicity); see, e.g., Minakshisundaram [1943] and Minakshisundaram andPleijel [1949].

'Throughout this paper we write "realization" instead of "realization in L 2(O)", i.e.,D(A) = tuEL(0)IAu = AuEL()I.

95

iv) The Fourier transform: G 6(t) = Ye bIdEA has been introduced by

Hormander [1988]. G satisfies (i -Al"2)G5(t) = 0 with G5(0) = I. The

spectral function can be deduced from G5 by the inverse Fourier transform,which is easier to use than a Tauberian theorem as in the previous cases. Butthis method requires the use of Fourier integral operators: Al/2n, which isdefined by its spectral function, is no longer a differential operator. In his paperHormander shows that there exists a Fourier integral operator L, (u) whichdescribes all the singularities of G5 :

Lgiq(u) - G(u)EC".

Then he proves the following theorem.

THEOREM 3.1. e (z,z,X) = Xn/2sc (z) + o (X( -')') with c (z) = (2r)J" f d;

A'(z,E) is the symbol of the leading part of A

4. Example: Elliptic Degenerate OperatorsFor further information, the reader is directed to Baouendi-Goulaouic

[1969].

Let 12 be a smooth bounded open set in W. We suppose, as above, that Ais adifferential operator in D'(0), which is properly elliptic of order 2m > n, withcoefficients in C"(O). Let us denote by A the realization of a boundary valueproblem which satisfies the hypothesis of Section 3 and such thatHre(f)C D(A)C H(f). A is an isomorphism from D(A) into L2(0).

4.1. Local EstimateLet us associate with (A+tI) the Green operator (A+tI)- 1 = G , which is an

operator with a kernel G (x,y) = G (y), such that

f (z) = QfG(xy)(A+tI)f(y)dy (4.1)

for all xEft and for all f ED(A). It follows that

||Gu,||(O) = 1L.A ( )1 -(4.2)

Therefore, there exists a positive constant C such that IIGzIIL ns C. A is anisomorphism from D(A) continuously imbedded in Hm (f2) into L (2); hence, theeigenvalues of A form a nondecreasing sequence (X)iEN, tending to + atinfinity:

0<X1 X2 , i

(each eigenvalue repeated according to its multiplicity). Let (pS)EN be an ortho-normal and Hilbertian basis in L2(0) of eigenfunctions associated respectivelywith the eigenvalues X. For z and y in 0 and for t z 0 we have

Ga = (X +t )-19p(2)% in L2 (0). (4.3)i=0

Let us now consider the operator (A+tI)X, which is selfadjoint also. If we denoteby gtz(y) its Green's kernel, we have for all ZEf and f ED((A+tI))):

96

f(z) = fgg,(z)(A+tI))f (z)dz. (4.4)

|I9t|IIL|(D) = SP ED ( tI) ) i(Ifr- fIIiaT(4.5)

(Obviously, these equalities are analogous to 4.1 and 4.2.)It follows from 4.1 and 4.4 that

G,(x) = fgt(z)gt3 (x)d7. (4.6)

Moreover, g:,(z) = gL,(z). Hence

Gt,(z) = |Itz II2and, by 4.5,

~If (z)12G (z) = supt.)12(4.7)

f ED((A +I)Y) ((A+t)f,'f )2(n)

Let us now write 4.4 for the eigenfunction pz:

(A+t)4 pg(z) = f(z)pt(z)dz.

By Parseval's equality we obtain

G:,(z) = gpe N~t . (4.8)

A local estimat, of the Green's kernel is obtained in Baouendi-Gaoulouic by prov-ing the following results:

a. Behavior of e Green's kernel outside the diagonal. For each compactKclxQ that does not intersect the diagonal, the function t - su G(z,y)I,

which is defined for t large enough, decreases rapidly when t tends to +00.

b. Comparison of Green's kernels of two operators having the same symbol.Let () be a smooth open set such that z ETCOc cQ. Let us denote by Gd theGreen's kernel in z of the realization in L2(01 ) of the Dirichet boundary valueproblem in 01 defined by A

In each open set 02 such that 02c01, the function t -4sup I G(y) - G(y) I

decreases rapidly when t tends to +m0.

c. Asymptotic estimate of the Green's kernel on the diagonal.

THEOREM 4.1. For all z Ef, lim t 1-n/ 2 'Gu,(z) = 1(x) 'with

1(z) = (27r)~"f(1+A'(z, ))-1d ,sn

where A'(z,() is the symbol of the leadingpart of AThis theorem is known for operators with constant coefficients. The general

result is established by comparison of A with A(zoD), which has constantcoefficients: the coefficients of A that are "frozen" in zo.

From 4.8 and Theorem 4.1, we deduce the following.

97

THEOREM 4.2. For all xEn,

limt I-(n/2m) (+t )-II|Pg(z)|2 = l(z).

Moreover, for z y, z ES), y E, the function t -+2 (X+t)-op (x)o(y) decreasesi=C

rapidly as t tends to +0.

We then apply to Theorem 4.2 the following Tauberian theorem.

THEOREM 4.3. Let a be an increasing function from R+ into R, a a real number in(0,1), and c a positive number. If

1

f(X+t)'da(X) = cta-1 + o (ta- 1), as t -*++t,

then

a(X) = c sinr A + (') a X+rc

If we choose a(X) = I |p (x) 12, we obtain the following theorem.

THEOREM 4.4. For all xE, yE, z # y.

limaX-(n/ 2 m) L foj(x) p7() = 0.

4.2. Global EstimatesWe now suppose that D(A) is continuously imbedded in H2 m(). It is possi-

ble to obtain the asymptotic behavior of fGi (x ,z)dz.

THEOREM 4.5. There exists c ; 0 such that for all x E, t z 0, f ED(A ),

If (z)I2 s ct(n/ 2 m)-1 [||A f II(fl) + t|lf I|Ifg)].It follows that

su i G (x ,x) s ctin/2,n)-1, 49

zcf)

Integrating and using Theorems 4.1 and 4.2, we have

lim tl-(n/ 2 m)f G (z,x)dx = limt1-(n/ 2 m)E (X+t)-1 = fl(z)dz.

Applying again the Tauberian theorem, we find

limX n'/'"N(X) = h(mn)fl(x)dz as Xf-+00,

with

h(m,n) = sinimn 2rn

This is equivalent to

98

X cjm'"" with c = [h(m,n) f(z)dx]-2'" as j-f++o.

5. The Max-Min Principle

5.1. The Classical FormulaeWe suppose that the hypotheses of Section 1 are satisfied. Let

Ali, AZ... ./p,... denote the nonincreasing sequence of the eigenvalues of B = A~(which is selfadjoint, positive, and compact). We suppose that each eigenvalue isrepeated according to its multiplicity. Let 91,92, .. .. , ,... be an orthonormalsystem of associated eigenfunctions. Let us denote by the subspace spannedby the first p eigenfunctions J 1,..., p , and by F the set of all p-dimensionalsubspaces in H.

THEOREM 5.1. The eigenvalues of B are characterized by

p = s (Bu,u) = in sus(Bu,u) = su inf(Bu,u).uE r E_1EF,-1 LEE _iE uE

Iul l Iui p UIh1=

This theorem implies an analogous theorem for A which is an unbounded opera-tor. The eigenvalues of A are A = p,-'.

THEOREM 5.2. The eigenvalues of A are characterized by

, =inf (Auu)= su inf (Auu) = sup(Au,u) = inf s (Au,u).uEE 1 E E-1 EJEF, uE

DUN-' p 1 -i u i i Hun1 1*11

PROOF. Let v ED(A). Then (Avv) = X (v, di)2 . When v E WL1 ,i=1

(Av,v) = Ei( ) v, (v,pi) 2 (because the (A,)pE are nondecreasing).i= w.. i=p

When IIvII = 1, i.e., E (v,po )2 = 1, we obtaini=1

inf (Avv) XAp.v E -1

If+l "i

By choosing v = p,, the infimum X, is reached and the first equality is proved.

In the same way, v E W implies (Avv) =X (v, pi2) 2SA, (v ,cp)2 and(=1 i=1

sup (Av,v) s A,. The third equality is then obtained by taking v = pp.V p

To prove the second equality, we notice that for all , EF-1, there exists

uoE Wp such that u0EE-..1 and Iuol| = 1. Let c, = (uo,opp). We have uo = cipifi i=1

and c 2 = 1. Thereforet=1

inf (Au 0,u) S (Au,uo) = c%.2XA,lull=1 i=1

And

99

su inf (Au ,us)

If we choose Ep_1 = W_1 , we obtain the equality.

The fourth equality is proved analogously. For all E, EF, there exists

u0 EW 1,.,IluoI=l;;henceu = cip andt=p

Therefore

inf sup (Au,u)Xp.EEF, utP

Itull-'

The equality is obtained by choosing E = W,.

5.2. Other Formulations of the Same Theorem

DEFINITION. Let K be a compact set in H. The "p-width of K in H," which isdenoted by d,(K,H), is defined by

i(K, H) = in! su fin lu -f I|.EEF, ts E E,

This definition has been introduced by Kolmogorov (see, e.g., Boutet de Monveland Grisvard [1971] and Singer [1970]).

If K is an ellipsoid with diameters 6 (we suppose that 6a- 6 - -'- ) and if

(cap)pEN is an orthonormal basis in H, with

K = ju = Zakax/ Eak 6k-2 < ig

then the following theorem holds.

THEOREM 5.3. d,(K,H) = 6p+1

PROOF. Observe that inf lu-f I|I= u -PE u.fE+ P

For all E EF,, there exists u1 EEpWYi1' 1 such that u1 = Eago andp+1 i

2 a26 72 = 1. Hence:=1

su 1inf |u -f "if in! lur-f II = II| 1 6pi.

As above, the theorem follows from

by taking E, = W, and u = 6, 17o4+.It is possible to characterize the eigenvalues of the operator B = A-1 by

means of this definition. Let

KB = Bf !llfl! 1j = fulllAul!s 1.

KB is an ellipsoid:

100

KB = iu= ap p,/ app-1.p=1 p=1

The diameters of KB are the eigenvalues , of B. The following theorem is theusual max-min formula written in a different way.

THEOREM 5.4. /p =Al = d,-1(KB,H) with KB = JBfI ILfI|slJ.

PROOF. If we write Theorem 5.1 for B2, we have

p1 = in f Nsup1(B 2u,u) = sup (Bu,u) = suptBu112.Ep, |16|C-"

Moreover, u is in W,. if and only if Bu E W. Hence

M,+1i= sup |BujjI= sup in IBu -f |||| 1||DUE Du| E I

\|QuI\-1 fE 1 1 fII vY E - 11

= in! sup in? |iv-f|1= dp(KB,H).EE1, V Exj J EER

The equality can be proved as above. Of course the eigenvalues of A can becharacterized in the same way: Ai = dC-1(LA,H) with LA = KA-i = u IIIAuII s 1;.

5.3. Consequences

COROLLARY 5.1. Suppose that the eigenvalues v of a positive and selfadjointoperator T in H are explicitly knoum and that (7lu ,u) s (Auxu) for all u ED(A).Then vs X.

COROLLARY 5.2. Let EcD(B) and . = B IE. Let & be the eigenvalues of ff. Then

6. Application to Variational ProblemsLet (VH,a) be a variational triple with V compactly imbedded in H. The

operator A associated with this triple is positive, selfadjoint, and invertible, withcompact inverse; V = D(AN). Therefore we have the following theorem.

THEOREM 6.1. (X+1)% = dp(LA%.H) with

LA = uEVIllA uII s 1 = uEVIa(uu) s 1j.

6.1. Boundary Value ProblemsLet us now copsider a variational boundary value problem: H = L2();

H '(0) C VC Hm (n). Therefore

S.JcLAnc~7M1,

where 7 and 6 are positive constants of continuity and coerciveness for a andwhereY1 denotes the unit ball of V. Hence

101

SH'() cO-CVcLAcy-*.Lcy-71f_.

An estimate of the widths of H'"(f) and ) gives a lower bound and an upperbound for the eigenvalues of A. These widths have been computed by El Kolli[1971].

THEOREM 6.2. Let 0 be a bounded open set in r, and let s be a positive realnumber. Then

0 < lim inf p'/" d(H (1), L2(Q)) s lim sup d (H'(0), L2 (0)).p- +- p-.+m

This theorem can be proved by means of the isomorphism between L2(T")and 12, where T" denotes the torus in 1", which, in turn, implies an isomorphismbetween Hm (T") and lfgemy/), where iz )= 1 We(a)EN"/ jenanag<1 Wehave an ellipsoid whose widths in 12 are known. Using the isomorphism, wededuce the result for the torus. Moreover, there exists a continuous linearextension P from H"(0) into H"(T") such that RP = IIHm()., where R is therestriction to 0. Therefore, App , ~p~"in.

THEOREM 6.3.

0 < lim inf p~( ")Ax s lim sup p-(2/n)A<c.

6.2. ConsequencesLet us consider a variational triple (V,H,a), and let us denote by N(A, Va)

the number of eigenvalues A (defined by Theorem 6.1) less than or equal to A.Under the same hypotheses as above, we have

N(A,V,a+t) = N(A-t,V,a) for all t ER

N(A, V,ta) = N(t -1A, V,a) for all t ER.

COROLLARY 6.1. If (V,H,a1 ) and (VH,a2 ) are two variational triples such thata1(uu) < a2 (u,u) for all uEV, then

N(A, V,a2) s N(A, V,a1 ).

COROLLARY 6.2. Let (V ,H,;a,)EI cN be a sequence of variational triples for whichthere exists c >0 such that a (u ,-) cIdu Ifor all iEI, u; E V. Let

H = = (ut)XEI IuiEH1; IluIIH = EIjuhIIj 4A< O)

and

Y= Iu = (v; XEIu.EV; a(u,u) = ~u~ <Doi.

The triple (V,H,a) is a "variational triple" and

N(A,v,a) = EN(A,v;,a;) for all A > 0.{

Let 0 be an open set in R" (0 is not necessarily bounded), and consider a"variational triple" (V(0), L2 (f),a) where V(0) is a "weighted Sobolev space." Letus denote by V0(0) the completion of Cc"(fl) with respect to the norm of V(D).When ocf, V 1(c) is the set of the restrictions to w of elements of VcD(0).

102

COROLLARY 6.3. Let C1 and 0D be two disjoint open sets in 0 such that Dru1v = T.If v(01)elA(0 2 )cVO(0) and V'(01) + Vl(1 2 )DVl(Q), then

N(X, 1(fl 1),a) + N(X, V(0 2),a) s N(kV 0 (0),a)

s N(X,V'(0),a) s N(,1V(0 1),a) + N(a,VI(02 ),a).

7. Example: Eigenvalues of Operators of Schrdinger TypeFor further information on Schrbdinger-type operators, see Reed and Simon

[1978], Rozenbljum [1975], and Fleckinger [1981].Let 0 = R"' (n z 1), and let a be an integrodifferential form on H1(R") which

is Hermitian and continuous:

a(u,v) = fb(z)(Vu VU + ui)dzn

with b uniformly continuous and positive and bounded away from zero. Let q bea positive and continuous function on R' tending to +oo at infinity and such thatfor all r > 0 there exists au> 0 such that q(z) - q (y) Is eq(z) wheneverIz-yI a< . When OcR", let V,(0) be the completion of C ' (0) with respect to thenorm

||Ua l =if[ I e z 2 + g( | Iz|]dz (.

The space VT (R") equipped with this norm is a Hilbert space, and the imbeddingof VQ (1") into L2 (f') is compact.

Moreover, when wcW", we can define Vg (w) as in Section 6, and Corollary 6.3can be applied.

Let A be the realization of the variational problem (V'(Ik), L 2 (I"). a+q).

REMARK. When b (z) = 1, Aq= -A + I + q= -A+ V is the usual Schrodingeroperator.

We deduce from the compactness of the imbedding of V (R") into L2 (1")that the spectrum of A, is discrete.

THEOREM 7.1. If fb(z)' 2 dx = +00, then

N(X,A,,r)~(2ir)~con f b (z)n/2 (X-q(z))n/ 2 dz as X-++oo,fzrEVL |q()<Aj

where wn is the volume of the unit ball in R" .

PROOF. We consider a partition of 14 into nonoverlapping cubes (Q)tE withcenters z. and sides r7. By Corollary 6.3 we have

FN(X, V, (Qt),a+q ) s N(X,AQ,R") S. E N(X, F4(Q),a+q ). .r (71 )tEIP CEP

Let =x ER"|g (z) < Xj, J = f EZ"IQtr 1# #s and I = EZ"|IQtcQxj. We sup-pose that q is such that 0A is a "Jordan contented set" with meas (a0X) = 0 and

b (z)n' 2dx y f b (z)"'dz for some 'y > 0.

It follows immediately from the max-min principle that N(X, V(Q), ag+q) = 0when cA J. Therefore 7.1 can be written as follows:

103

E N( X.V( QC)a+q ) s N(X, AqR" ) E N (X.V, ( QC),a+q ). (7.2)CEI CEJ

On each cube we compare a +q with an integral form with constant coefficients:

((aC+g q)uv) = f(b(zc)vuvii + (q (z.)+1)ui1)dz.

We deduce from the hypothesis on b and q that, for all e > 0, there exists 77 > 0such that I((a+q)u,u) - ((a.+g )u,u) e((a+q)u,u). Hence, by Corollaries6.1 and 6.2, we have

E N((1-e)X, VF(QC),a.+g e) i N(X,A ,R') s E N((1+E)A,, q( Q.), a.+t ). (7 3)CEl CE

N(X, ' (Q ),a +q) is the number of eigenvalues less than X of the operatorLC = b(zC)(-A+I) + q (zt) defined on the cube Q with Dirichet or Neumannboundary conditions, and it is easy to prove that

N(X, l'(Q ),ar+gq ) - (2rr)-"cwnb (z)n/ 2 n (X-q (zx))"/1 I s 7 (1+ "-W )(7.4)

By choosing 7 = -"114 " and by letting E tend to 0, we obtain Theorem 7.1.

References

Below is a (partial) list of references cited during the Orsay seminar.

Agmon and Kannai. 1967. "On the asymptotic behavior of spectral functions andresolvent kernels of elliptic operators." Israel J. of Math. 5:1-30.

Akhiezer and Glazman. 1967. Theory of Linear Operators in Hilbert Spaces.English transl. Ungar, New York.

Baouendi and Goulaouic. 1969. "Reguarite et theorie spectrale pour une classed'operateurs elliptiques degeneres." Arch. Rat. Mech. Anal. 34(5):361.

Boutet, de Monvel. 1970. Seminar on spectral theory. Genua, Italy.

Boutet, de Monvel, and Grisvard. 1971. "Comportement asymptotique desvaleurs propres d'un operateur." C.R.A.S. 272(1):23.

Courant, R., and Hilbert, D. 1953. Methods of Mathematical Physics, Vol. 1.Interscience.

El Kolli. 1971. "No. epaisseur dans les espaces de Sobolev." C.R.A.S. 272(8):537.

Fleckinger, J. 1981. "Estimate of the number of eigenvalues for an operator ofSchrbdinger type." Proc. Roy. Soc. Edinburgh 83A:355-361.

Goulaouic. 1973. Spectral Theory. Edizioni Cremonese.

Hbrmander. 1968. "The spectral function of an elliptic operator." Acta Math.3(4):193-218.

Minakshisundaram. 1949. "A generalization of Epstein Dzeta functions." Canad.J. Math. 1(4):320-327.

104

Minakshisundaram. 1953. "Eigenfunctions on Rieriannian manifolds." J. IndianMath. Soc. 17:159.

Minakshisundaram and Pleijel, A. 1949. "Some properties of the eigenfunctionsof the Laplace operator on Riemannian manifolds." Canad. J. Math. 1:242-286.

Reed, M., and Simon, B. 1978. Mathematical Physics. Academic Press, NewYork.

Rozenbljum, C. V. 1975. "Asymptotics of the eigenvalues of the Schrodingeroperator." Prob. Mat. Anal. 5:152-166.

Singer, I. 1970. Best Approzimation in Normed Linear Spaces by Elements ofLinear Subspaces. Springer-Verlag, New York.

Triebel. 1967. "Erzeugung nuklearer lokalkonvexer Raume durch singulareDifferentialoperatoren zweiter Ordnung." Math. Ann. 174:163-176.

Triebel. 1970. Nukleare Funktionenraume und singulare elliptischeDifferentialoperatoren. Colloquium on Nuclear Spaces and Ideals in Opera-tor Algebras.

Yosida. 1968. Functional Analysis. Springer-Verlag, New York.

105

Appendix Variational Boundary Value Problems

A. 1 Definition

A "variational triple" (V, H,a) is defined by

i) Two Hilbert spaces V and H such that

" V is continuously imbedded in H.

* V is dense in H.

ii) A sesquilinear form a( ,) which is continuous and coercive on V; i.e., thereexist two positive contants 7 and d such that

a(u,v) ; <alluy lvlv ly for all (u,v)EVxV

Ia (u,u) I z 7||a ll for all u EV.

We hav.: V C HC V'. By the Lax-Milgram Theorem, it is possible to define an iso-morphism A from V onto V by

<Au,V>Vyxv = a(u,v) for all (u,v)EVxV

Thus, A EL( V,V'). A can be considered as an unbounded operator in H withD(A) = JuEVIAuEHj.

A.2 Examples: Boundary Value Problems

Let 0 be an open set in R" (n z 1) and set H = L2(0). The following varia-tional problems will be studied.

1. V 1 = Ho (0); al(uv) = f(uv+L -- av)dz. By Green's formula (orn i-1 a z=1j

integration by parts), it is easy to see that

a(uv) = f(ui7 - (Au)i)dz = ((I-t)u,v) = (A u,v)n

for all v EHd (0) = V1 and for all u EH2 (0)nHd (f) = D(A1). Therefore we haveassociated the differential operator A 1 = I-A with Dirchet boundary conditionsand D(A1) = H2 (O)nH (C) with the "variational triple" (lid (0), L2(O), u).

2. n is bounded, and 7 is a function that is equivalent to the distance at theboundary. (For example, when 0 is the unit ball in M", co(z) = 1-Iz 12.)

V2 = luE H|V7Diu EH, i = 1,..,n .

a2(u,v) = f(ovvvi + ui)dz.

The associated differential operator is A2 = I-div(cp grad), which is an extensionof the Legendre operator.

3. 0 is unbounded, and q is a positive function tending to += at infinity.

V(0) is the completion of C '(f?) with respect to the norm

106

I|uIIV, = (ff(2 IDu 12 + q(z)Iu(z) 2 )dz)%;C) t=1

cls(u,u) = tIUIIl3.

A9 = -D + q is the Dirichlet Schrcdinger operator.

A.3 PropertiesIt is obvious that if the imbedding of V into H is compact, then the imbed-

ding of D(A) into H is compact too.

PROPOSITION 1. The operator A is selfadrjoint (resp., positive and selfadjoint) ifand only if a is Hermitian (resp., positive and Hermitian).

The proposition follows from the fact that D (A ) = Ju E Vlv -,(Av,u) is con-tinuous on D(A) for the norm of H3.

REMARK. When A is a linear operator in the Hilbert space H with D(A) continu-ously imbedded in H and dense range, we can define the variational triple(V 4,H,a4) where V4 = D(A), and a4 (uv) = (Au,Av). The operator associated withthis "triple" is A2.

107

ASYMPTOTICS OF EIGENVALUES OF VARIATIONAL ELLPTIC PROBLEMSWITH INDEFINITE WEIGHT FUNCTION

J. FleckingerH. El Fetvwssi

Abstract

This paper is concerned with spectral theory of "right non-definite" elliptic boun-dary value problems, i.e., Au = Agu on OcW, with Dirichlet homogeneous boun-dfry conditions, when g changes sign. Asymptotics are obtained for the eigen-values in two cases: 1) when 0 is bounded, and 2) when 0 is unbounded. In case 1,the usual Weyl-Courant estimate is generalized; in case 2, the de Wett-Mandl for-mula is extended for operators of "Schrodinger type."

1. IntroductionSince the beginning of the century, a vast amount of research has been car-

ried out on elliptic problems with indefinite weight function in the one-dimensional case. Until recently, however, the multidimensional case hasattracted less attention. The first paper we know of on the subject is byHolmg-em [1904]; it is concerned with the Dirichet problem on 0, a boundeddomain in IR : Au + kg (z,y)u = 0. In this case there are an infinite number ofpositive and negative eigenvalues. Holmgren proved that these eigenvalues canbe defined as the solutions of a problem in the calculus of variations, exactly likethe well-known Courant-Fisher result when g is positive.

In 1942, Pleijel [1942] gave an estimate for these eigenvalues; he improvedthe Weyl-Courant estimate on the asymptotics of the number of eigenvalues lessthan X [Courant and Hilbert, 1953; Weyl, 1911]. Two years later Pleijel studiedthe nature of the spectrum for a Schrbdinger problem: +Au - qu + ?gu = 0when q is positive and when 0 is not necessarily bounded [Pleijel, 1944].

More recently, Weinberger [1974] established the variational characteriza-tion of eigenvalues for abstract problems whose right member is not necessarilydefinite. Also, Manes and Micheletti [1973] applied Weinberger's results for thehomogeneous Dirichlet boundary value problem

(1.0) Au=Xgu onDO; u/au =0.

When g changes sign, 0 is bounded, and A is an elliptic operator of order 2,Manes and Micheletti prove the "max min" ("min max") principle for the positive(negative) eigenvalues. Similar results were given in the de Figueiredo [1982]survey paper. Recently, Lapidus [1984] improved Pleijel's results on the asymp-totics of the number of eigenvalues of the Laplacian when the weight is notnecessarily smooth.

In the last decade, the eigenvalue problem (1.0) with an indefinite weightfunction has been intensively studied, primarily as a result of its connection withthe theory of semilinear elliptic boundary value problems (see de Figueiredo[1982] and the references there). Nevertheless, most of this work has focused

" Permanent address: Laboratoire knalyse Nurn. VER MIG., Univ. P. Sabatier, 118 rte. de Nar-bonne, 31062 Tououse-Cedex, France. 'The author wishes to thank Argonne National Laboratory forits hospitality during the Workshop on Spectral Theory o Sturm-Liouville Differential Operators,May-June 1984.

108

on establishing the existence of a positive principal eigenvalue.

Our goal here is very different. This paper is devoted to the asymptotics ofthe number of eigenvalues of the variational problem associated with (1.0) in twocases:

1. When 0 is bounded and A is of order 2m (we generalize the earlierpapers of Pleijel and Lapidus).

2. When O is unbounded and when A = L + q is an operator of "Schrodingertype."

In the one-dimensional case, when A is degenerate, asymptotics are given inKaper, Lekkerkerker, Kwong, and Zettl [1984].

Notation. The following notation is used throughout the paper:

" H" (0) denotes the usual Sobolev space of order m.

" When a = (a1,...,a, )EN", Da is the derivative of order I a j = a+...+an:

Da = 6.

" The norm in Hm((0) will be

|uII|rn) =(f X IDau(x)12 dz).

* (, ) is the usual inner product in L2(0) .

We will now give more precise results.

(1.1). Let O be an open set in H" and A a formally selfadjoint elliptic operator oforder 2m, defined on 0: A= Da(aapDP), where aap = dpaEL((0) when

IIra+alI <2m and apEC(Q) when Ija+p I = 2m.

Let us denote by D(A) = u EL2()/ Au EL2(f);u/ an = 0 and by A, the posi-tive selfadjoint and unbounded realization in L2(C)) of the homogeneous Dirichetboundary value problem associated with A.

(1.2). We suppose that the embedding of D(A) (equipped with the norm graph)into L2(0) is compact.

(1.3). Let g be a continuous function on 0, which changes sign. Let us denote by0+ = z EO/g (x) > 0; and n_ = z E0/g (z) < 03. We suppose that 10+|> 0 and0_ I > 0 where |I- denotes the Lebesgue measure.

We study the spectrum of the boundary value problem

(P) JAu g=gu onf1, u ED(A)/an = 0

We first prove that this spectrum is discrete. It consists of two countablesequences (one positive and one negative) of eigenvalues, tending to infinity:

We write the "max min" and "min max" formulae and then estimate the

109

asymptotics of N*(X,Ag ,f) where N+(XA,g ,C)) [N-(X,A,g ,0)] denotes thenumber of positive (negative) eigenvalues less (greater) than X.

Under suitable assumptions on the regularity, we prove the following esti-mates:

1. When 0 is bounded and A is an elliptic operator of order 2m such thatH2"(Q) D(A) H2 m(), then

(1.4) N+(X,A,g ,O) ~ (X (z)) y ' (x)dx, X- +c0,

where 'A (z) is the usual "Browder-Garding" density:

I'LA(z) = (2rr)~" meas fCE1/A(x,) < 1;A'(x, E) denotes the symbol of the leading part of A

(1.5) N-(X,A,g,0) I (Xg(x))sm'(x)dz, X-+-c.

REMARKS.

1. We notice that in (1.5), X and g (x) are negative; hence Xg (x) is positive.

2. It is very easy to deduce (1.5) from (1.4) because Au = Xgu can be writtenas Au = (-X)(-g)u. This remark will be used throughout the paper. All resultsproved for positive eigenvalues imply analogous results for negative eigenvalues.

3. Both (1.4) and (1.5) can also be written

nnN*(X,A ,g ,) -~f(Xg,(x)) y '(x)dz, A-+0

where g.,(z) = max(g (z),0), and g _(x) = min(g (z),0).Of course, when g is positive (i.e., 10.. = 0), we have no negative eigen-

values, and we find the usual "Browder-Garding" estimate where N(A,g ,0) isusually denoted by N(a,A,g,1i). When g (x)=1, we write N(XA,0) instead ofN(X,A,1,0) or N+(X,A,1,0). Hence, (1.4) and (1.5) extend earlier estimates[Browder, 1953; Courant and Hilbert, 1953; Fleckinger and Metivier, 1973;Garding, 1953; Lapidus, 1984; Manes and Micheletti, 1973; Metivier, 1977; Pleijel,1942; Reed and Simon, 1978, and Weyl, 1911].

2. When 0 is unbounded, we assume that A is an operator of "Schrodinger type,"i.e., A = L +q, where L satisfies (1.1) and (1.2).

(1.8). q is a positive and continuous function defined on 0, tending to +c atinfinity.

Then, under suitable hypothesis, a) when fig m< , (1.4) holds; and whenfgn/2m is infinite:

(1.7) N+(X,L + q,g,f2) ~^'zEf/q(w)(<(z)I 'L(x)(Xg (x) - q (z)) 2 mdx, X-++0o

110

(1.8) N-(XL + q,gO) ~ -zD/g(=) ( ) ())

Again, we notice that (1.7) and (1.8) can be written as

N*(XL + q,g,O) ~ Ef/q(s) (s) (z)(g(z)-()) 2 mcix,

Both these formulae generalize de Wett-Mandl asymptotics of N(X,-A + q,R")[Courant and Hilbert, 1953; Reed and Simon, 1978; and Titchmarsh, 1958] and itslater extensions for N(X,-A + q ,0) [Rozenbljum, 1975] or N(XA = L + q .0)[Fleckinger, 1981]. (Of course, we write as above N(kA,0) instead of N(X,A,1,0).

2. An Abstract " Max Kin Principle"To prove the above estimates, we first write a variational characterization of

eigenvalues for an abstract problem. Our proofs in this part are almost thesame as in Pleijel [1942], Weinberger [1974], Manes and Micheletti [1973], and deFigueiredo [1982].

(2.1). Let H be a separable Hilbert space with inner product ( ,) and norm |Il.

(2.2). Let us now consider a "variational triple" (V,H,a) where V is dense in Hwith compact embedding and a is a Hermitian continuous and coercive formdefined on V:

Sa > 0 |a(u,v) I s a||uI| v |y1 Y,(u,v )EVxV

: fl> 0 a(u,u) a #|||$ Vu EV

We associate to (V,H,a) the operator A which is linear, positive, selfadjoint, andunbounded in H, with compact inverse: V = D(AM) and

VuED(A) YvEV a(uv) = (Auv ).

(2.3). C is a linear and self adjoint operator in H which is "A bounded," i.e.,

7 > O, |(0u,,u) I sya (u,u) Yu E V

In this part we are concerned with the abstract variational eigenvalue problemto find (X,u)ECxH st.

(2.4) Au =X Cu. uEV.

It follows immediately from the assumptions (2.3) that 0 cannot be an eigenvalueof (2.4) and that a defines an inner product on V; hence, using the Frechet-Rieszrepresentation, we prove the following proposition.

PROPOSITION 1. The operator T defined on V by

(2.5) (Cu,v) = a(7zsu,v) y(u,v)EVxV

is linear and continuous in V.

This is a simple consequence of (2.2) and (2.3):

lIIThIIu' a(Tt,Tu) = (CTu,u) s ya(u,7 u) ! ay|II7\i.IIluI|yand then

111

IIThIv -!59uIIvV uEV

(2.6). We suppose that T is compact.

Indeed, we note that formally T = A~1 C and hypothesis (2.6) will besatisfied, for example, when C is bounded. Moreover, T is "a-selfadjoint," i.e.,selfadjoint on V equipped with the inner product a( , ):

a(Tu,v) = a(u,Tv) V(u,v)EVxV

Now we introduce the eigenvalue problem.

(2.7) Tu = Au uE V We can apply to T the usual spectral theory of selfadjointcompact operators in Hilbert spaces.

PROPOSITION 2.

i) T has no continuous spectrum except, possibly, at 0.

ii) The eigenvalues of T are real.

iii) If p; and y1 are two distinct eigenvalues of (2.7) with Sp and Sp. the associ-ated eigenfunctioras, then a ( 9, p) = 0 ( mandSp are "a-orthogonal").

iv) The spectrum of T (except, possibly, 0) consists of two countable sequencesof eigenvalues (one positive and one negative) tending to 0:

A1 S Ali... A ,~ < -.- S.<4t0 - p.p ... p p

The a-selfadjointedness of T implies ii) and iii): If Tu = pu, a(Tu,u) = a(u, Tu)implies that is real and if Tv = vu, we obtain (-v)a (u ,v) = 0.

To prove i) and iv), we use the following lemma, which is a consequence ofthe T compactness:

LEMMA 1. There exists a non-zero function Sp E V for which

j = (Cu,u).

a(u.,u)-1

Then T cat= A191.Let us take the restriction of T to the orthogonal complement (for the

inner product defined by a) of the linear subspace generated by Sp. We obtain aselfadjoint and compact operator and hence

pu2= sup. (Cu,u)....u EV

a(uFrj)=Da(u~u)=1

(2.6) ,j = sup (Otu,u).

a(u. ?)=D,.<<11....3-11a(u~u)=1

In the same way we can prove that

(2.9) i = inf (Ctu,u).a(uuE)=+1

112

Let us denote by Sp the associated eigenfunctions such that a(pfr) = 1.

PROPOSITION 3. The eigenvalues of (2.7) are characterized by the variational

principle

(2.10) / 4L+1LCZ=

where A5 denotes the set of all j dimensional subspaces in V:

a(u,u)

j+1Let E, EA , and choose u = Cpo, such that u is orthogonal to E for the

k=1

inner product associated to a. We suppose that a(u,u) = C2 = 1.k=1

(Oau)Y= p+* C2 5 ++1 C2 +k=1 k=1

and

(ah,u) s

We obtain the equality by taking E, the linear subspace generated by Soj,...,cand u = p1.

We notice that the eigenfunctions of (2.4) are eigenfunctions of (2.7) with1X = -, the associated eigenvalues.

Let us denote by yu(V,a), in place of uj, the "min max" of (CU,u)/ a (uu)introduced in 2.10. We deduce from Proposition 2.3 the following corollarieswhich we will use throughout this paper.

COROLLARY 1. If (V1,H,a) and (V2,H,a) are two "variational triples" with Vc V2 ,then

pa (V,a) s p (V,a).

COROLLARY 2. If (V,H,a1 ) and (V,H,a2 ) are two "variational triples" such thata1 (u,u) S a2 (uu) VuEV, then

a (V,a1) :e p (V,a2).

3. Application to Boundary Value Problems Defined on Bounded Domains

(3.1). Let 0 be a bounded domain in R". We consider on 0 the variational eigen-value problem:

(P) Au = \gu on f;u/ en = 0,

where A and g are satisfying assumptions (1.1) to (1.3).

We will apply the previous results with H = L2 (0), C defined byCu(z) = g (z)u(z) and V = Ho (0), where Ho (0) denotes the completion ofC'(0) for the norm of H''. We will denote by a( , ) the integrodifferential formdefined by a(u,v) = (Au,v) VuED(A) Y EV = Ho (0) and problem (P) is

113

studied in its variational formulation:

(3.2) a (uv) = Af g (x)u(x) tz-)dx Vv E Hb (0)).

THEOREM 1. We suppose that hypotheses (1.1) to (1.3) and (3.1) are satisfied.Under these hypotheses, (P) has a purely point spectrum. It consists of twosequences of eigenvalues, one positive and one negative, tending to infinity:

Moreover,

+ (gu,u)1+i1= nJd, uEII() a(u u),

where (,) denotes the usual L2(O) inner product and A the set of j dimensonalsubspaces of Ho (0) .

(gu,u)

S1

and estimates (1.4) and (1.5) hold.

PROOF. The first part of Theorem 1 follows immediately from the former abstracttheory. Hence we have only to prove the estimate on N*(X,Ag ,f).

When g is positive, this estimate is well known [Weyl, 1911]. We deduce(exactly as in the positive case) from Corollaries 1 and 2

(3.3) X (g.,) -- X (g ,Q') if ('cO

(3.4) X 3(g ,) s X) (g 1,O) if g 1i -g

where X (g ,i2) denotes the j th positive eigenvalue of (3.2). These inequalitiesare equivalent to the following:

(3.5) N(kA,gS') s N(X,A,g,) when O'C0.

(3.6) N(kA,g 1 ,Q) < N(X,A,g ,0) when g 1 s g.Let us denote by g+ = max(g,0); g+ is non-negative on n, and it follows from

(3.5) and (3.6) that

Ye > 0 N(XA,gSf+) < N+(XA,g.0) s N(a,A,g++sQ).

Let us denote by

(g,0,X) = (X)=X 2 m f (g(x)) 2 m A'A(x)dz.

The function g being positive on f2+, we have N+(XA,g,4,) = N(X,A,g,fl), and

(3.7) lim c-1(X)N(kA,g,0+) = 1.

The same limit holds for '-1(g++,X,))N(A,A,g++t,11). Hence,

114

YE > 0, p-1( ,)N( ,A g ,n+) s q-1(X)N+(XA,g ,i)

s (pv-1(X)so(g+,+e,f,X)) -1(g ++E,12,X)N(X,A,g++e,O).

By letting a tend to 0 and using Lebesg ue's theorem, we prove thatfp~ 1()p(g++EfX) tends to 1. We notice that Mpg+,OX) = sp(gQ+,X). By letting Xtend to +cc, we have (3.7).

4. Application to Operators of Schrodinger TypeWe now suppose that 0 is unbounded in R".

(4.1). Let q be a positive and continuous function defined on R"', tending to +m'at infinity.

(4.2). Let L be a differential operator of order 2m defined on 0 satisfyinghypothesis (1.1).

(4.3). Let g be a continuous function defined on , such that g (z)q-1(z) tendsto 0 at infinity.

We consider the variational eigenvalue problem

(Q) Au = (L + q)u = Xgu on 0(Q) /an =0

To obtain estimates (1.7) and (1.8), we will apply again the results of the secondpart and work exactly as above. First, however, we write the estimateN(XL + q,g+,0). This study is divided in three parts. To begin, we discuss theright definite case and the right non-definite case when fg/2m = oo. Then wegive the estimate when fgn/2'" < co. We now suppose that the following assump-tions are satisfied:

(4.4). There exist two positive numbers to and A' such that for all 5 > 0,q.g ++8 andcazp (with |a + #| = 2m) can be extended ton~ = jz ER'/ dist(z,f+) < :oj and

VcE]oeo[. VA ', 7]aV77j5 7x|z-y ,I<'m?. Then

|q(z) -q(y)I segr(z)

g+(z) -g+(y)I eJ(g (x)+5)J

1 azp(z) - aap(y)I I ! I ap(z) I.

(4.5). For any positive number A, O = zrcO/g(z) < Ag (x) is a Lebesguemeasurable set and 2 A" Z A' y 'y > 0 such that [(j ] 71 [0A/] for all A e A"where

[C_] = [ g "(z) dz.

(4.6). We consider a partition of R" into non-overlapping cubes (Q)Er with sidei7 and centers xC. We suppose that

115

E g/ 2 m

tcJ

where gc = g (zX) , I = CEZ"/ Qcjxj and J = f EZ"/ QgnfQ l.

(4.7). We suppose that there exists 72 > 0 such that, for all

Wccf, (L'u,u) a 72fu|!fl(w) VuEHT (c),

where L' is the leading part of L We note that (because of 4.1) when X > 0,)cCf+A = z x E 12+ , =zE/ q (z) <kg (z)j. We first give two results when fgn/2m

is infinite.

A. The Right Defir i*e Case

(4.8). We suppose that g (x) z 6 > 0. Let us denote by W(2) the completion ofC6"(O) with respect to the norm

|lull( Pa=llulL,(n) + (qu,u)]3.

W(Q) is a Hilbert space, and we deduce from (4.1) the following proposition.

PROPOSITION 5. The embedding W (D) into L'(f) is compact.

This proposition is a consequence of the usual criterion of compactness forunbounded domains. We have

IluIIL( su p 1( f qu(xz)d se(R)tullI rR) ,

where 0'R = z E/ I x >R i and c(R) tends to 0 when R tends to +0.We now prove the following theorem.

THEOREM 2. When the hypotheses (4.1) to (4.8) are satisfied, and if fgn/2m is

infinite, then

N(X,L+q,g.0) f '(z) (Xg(z)- q(x))sm dx X-+co.CIA= j E n/ (z)<Ag (z)i

This estimate hold, for example, for

(-o + (1+ Ix 2)2k)u = X(1+ Ix 12)-k u on R

Theorem 2 will be proved in Section C.

B. The Right Non-definite CaseWe do not further assume that g is positive. We will obtain estimates (1.7)

and (1.8) as in the third part. We apply the results of the second part withH=L (0); V = W(0). The Hermitian form a is defined on W(D) bya(uu) = (Lu,u) + (qu,u). The operator C is defined as previously byCu (z) = g (x)u(z). We note that (2.3) follows from (4.3).

THEOREM 3. We suppose that hypotheses (4.1) to (4.7) are satisfied. WhenfgL/ 2 T&is infinite, estimates (1.7) and (1.8) hold:n

116

N*(X.L+q,g l) ~^ [i'(g - q)+/ 2 m ,a-++ .

PROOF OF THEOREM 3. The estimate (1.7) holds when g (x) z 6>0. We deduce fromCorollaries 1 and 2

N(X,L+q,g,Q) N(XL +q ,g.0f) N(X,L+q,g++6,fl)

and the estimate follows as in Section 3 by letting 6-0.

C. Proof of Theorem 2The proof of Theorem 2 will be almost the same as that given by Fleckinger

[1981]. We will use the two following results:

PROPOSITION 5. ?here exists two positive numbers C' and C" such that

C'A' 2n/2m[f] < <o(x) < C"Xan/m[QA]

where

o(X) = f' h (Ag-q)v/2m

PROOF OF PROPOSITION 5. The upper bound is obvious by use of (4.7). It followsfrom (1.1) that 'L(z) z C > 0 and

p(x) a C (A. f)n/ 2 ml:En+/ z (z)-q(Z)>o3(9 q

By use of (4.5) we have the result.When wch ( has been introduced in (4.4)), let us denote by W 1() the res-

triction to w of elements of W(?), and by N(X,A,g,w) [ Ni (X,A,g ,w)] the numberof positive eigenvalues less than X of the variational problem:

Au = (L+q)u = Agu inch; uEW(w) [uEW'(w)].

PW)PosMON 6. If c1 and w2 are two disjoint open sets in w such that r5, U5 2 = W,then

N+(XA,g ,w) + N+(X,A,g ,c2) s N+(X,A,g ,C)

s Nij(X,A,g ,c) s Ni (XA,gw 1 ) + Nij(X,A ,g , 2).This is a simple consequence of

W(wl)eW(2)c W(c) and W'(c)c W 1(w 1)9W'(w2).When (AJ, N1(A,Ag ,Q) = 0, because Xr-q < 0 on Q. It follows from thisremark and from Proposition 6 that

(4.8) E N+(X,A,g ,Qt) s N+(X,A,g .0) s E i (X,A,g , Q).CE! CEJ

On each cube we compare A with AC = + qC and g with g C, where rC = r (xC) and= a aap(zx)Da1+. It follows from interpolation inequalities and from

al= P I(4.4) that for all a small enough, there exist 7 > 0 and C > 0 such that, for alleEJ

(4.9) j(Au.,u)-(ACu,u) ! se(4 u,u) + C0la fg u2.Pt ,s,

117

We know that there exist a positive constant C, such that for all e

n-1

(4.10) jN(kA,gQt) - So(X4,) 1 C(Xg ) 277

with

(4.11) p(X4,) = I Q (Xg- - gt)2yn,

where p = .LL"

The same holds with N1 (- ). Choosing e = ~ 2(2m- 1), prove that

(4.2) upCel-n(41) sup p(a)E p((1-e)a - 22m, )-+1 A++ .

tEI 9

We have an analogous result for the upper bound.

D. The Casefg/ 2m< .a

THEOREM 4. We suppose that hypotheses (4.1) to (4.7) are satisfied and thatfg/ 2mn < o; then

N(X,L+ ,g ,O)~ f(ag+)n/2"m , -+mo.

The proof is almost the same as that of Theorem 3. The lower bound is obvious.

For example,

(--A + (1+z 2)u = Xx(1+x2)tu on 12 = ((x,y)ER 2 / y I < 1[.

If t = 0, we apply Theorem 3. If t < 0, we apply Theorem 4.

Note: We have recently been told that some estimates can be found in Birmanand Solomjak [1973].

5. References

Birman, M. S., and Solomjak, M. Z. 1973. "Asymptotic behaviour of the spectrumof differential equations." Jal Soviet Math. 12: 247.

Browder, F. 1953. "Le problem des vibrations pour un operateur aux deriveespartielles selfadjoint et du type elliptique a coefficients variables,"Comptes Rendus Acad. Sc. Paris t. 236:2140.

Courant, R., and Hilbert, D. 1953. Methods of Mathematical Physics. Vol. 1,Intersciences.

de Figueiredo, D. G. 1982. "Positive solutions of semilinear elliptic problems."Lecture Notes in Mathematics 957, Springer-Verlag, New York.

Fleckinger, J. 1981. "Estimate of the number of eigenvalues for an operator ofSchrodinger type." Proc. Roy. Soc. Edinburgh 89A:355.

118

Fleckinger, J. and Metivier, G. 1973. "Theorie spectrale des operateurs uni-formement elliptiques sur quelques ouverts irreguliers." Comptes RenduesAcad. Sc. Paris, 276:913.

GCrding, L. 1953. "The asymptotic distribution of the eigenvalues and eigen-functions of elliptic operators." Math. Scand., p. 237.

Holmgren, E. 1904. "Uber Randwertaufgaben bei einer linearerDifferentialgleichung zweiter Ordnung." Arkiv fur Mat. Astrofys. 1:401.

Kaper, H. G., Lekkerkerker, C. G., Kwong, M. K., and Zettl, A. 1984. "Full- andpartial-range eigenfunction expansions for Sturm-Liouville problems withindefinite weights," Proc. Roy. Soc. Edinburgh, 98A.

Lapidus, M. 1984. "Valeurs propres du Laplacien avec un poids piu change designe," Compte Rendus Acad. Sc., Paris, 298, p. 265.

Manes, A., and Micheletti, A. M. 1973. "Un'estensione della teoria variefinaleclasica degli autovalori per operator ellittici del secondo ordine." Bollet-tino 7:285.

M6tivier, G. 1977. "Valeurs propres de problems aux limits elliptiques irregu-Tiers." Bull. SMF mem 51-52:125.

Pleijel, A. 1942. "Sur le distribution des valeurs propres de problems regis parl'equation Au+Xku=0." Arkiv fur Mat. Astrofysik 29B (7):1.

Pleijel, A. 1944. Le probleme spectral de certaines equations aux derives par-tielles." Arkiv fur Mat. Astrofysik 30A (21):1.

Reed, M., and Simon, B. 1978. Mathematical Physics. Academic Press, NewYork.

Rozenbljum, G. V. 1975. "Asymptotics of the eigenvalues of the Schr6dingeroperator." Probl. Mat. Anal. 5:152.

Titchmarsh, E. 1958. Eigenfunction Ezpansions, Part 2. Oxford UniversityPress, London.

Veling, E. J. M. 1984. "Asymptotics analysis of a singular Sturm-Liouville boun-dary value problem." Integral Eq. and Op. Theory (to appear)

Weinberger, H. 1974. "Variational methods for eigenvalue approximation." Chap3, Regional Series in applied Math. 15, SIAM.

Weyl, H. 1911. "Das asymptotisch Verteilungsgesetz der Eigenwerte linearer par-zieller Differentialgleichungen." Math. Ann. 71:441.

119

A NONOSCILLATION THEOREM FOR SECOND-ORDER LINEAR EQUATIONS

S. G. Halvorsen*Man Kam Kwong '*

A. B. Mingarellit

Abstract

Let q: [O,-)-.R, q not a.e. zero be a locally Lebesgue integrable function. We areinterested in the existence of real constants c ui 0 such that the equationz" + cq (t )z =0 is nonoscilatory on [0,-). Applications to Bohr and Stepanoffalmost-periodic functions q are included.

1. IntroductionLet q: [0,w)-+R be a locally Lebesgue integrable function and c a real con-

stant. We consider the oscillatory behavior of the second-order lineardifferential equation

x"(t) + cq(t)z(t) = 0, t 0. (1.1)

Equation (1.1) is said to be oscillatory if any nontrivial solution vanishes aninfinite number of times. Otherwise (1.1) is said to be nonoscillatory.

For some potentials q (t) it may turn out that (1.1) is oscillatory for everyc # 0. For example, q(t) = sint is such a potential. In fact (see Markus andMoore [1956]) such potentials actually include the class of all nontrivial (Bohr)almost-periodic functions having mean-value equal to zero (for definitions see,for example, Besicovitch [1954]). An interesting problem is to discover classesof functions that either have (cf. Halvorsen and Mingarelli [1984]) or fail to havethis property. Below, we give an example of the latter.

2. A Nonoscillation Theorem

THEOREM. Let there exist positive constants M, T, e such thatt

|fq(s)dsl M (2.1)nT

for all t E[nT,(n+1)T] and n = 1,2,3,...,(n+1)T t

f (fqg(s)ds )2dt s M2 (2.2)nT nT

for each n z 1, and(n+1)T

f gq(s)ds !9 -r (2.3)

*Department of Mathematics, University of Trondheim, N.T.H., Trondheim, Norway.

"Department of Mathematics, Northern Illinois University, DeKab, Illinois.

tDepartment of Mathematics, University of Ottawa, Ottawa, Ontario, Canada KIN 934. The au-thor wishes to acknowledge, with thanks, funding from the Natural Sciences and EngineeringResearch Council of Canada under Grant U0167.

120

for all sufficiently Large integers n.

Then there exists a constant c > 0 such that (1.1) is nonosciLLatory on [0,me).

REMARK. It is well known that if (1.1) is nonoscillatory for c = c 1 > 0, then (1.1)remains nonoscillatory for all c E[0,c 1] (see, for example, Markus and Moore[1956]).

PROOF. It is well known that the nonoscillation of (1.1) is equivalent to theexistence, for some a z 0, on [a,oo) of a continuous solution of the Riccati equa-tion

r'(t) = -cq(t) - r2 (t), t E[a,co) (2.4)

which results from the change of variable r(t) = z'(t)/z(t) in (1.1). We willexhibit such an r (t) by choosing a suitable initial condition r (a) and then showthat the solution can be continued to one that is defined in [a,00). The continua-bility is established by showing that r (t) is always positive.

Without loss of generality we may assume that T = 1. We choose

c = min(0.01)M-1, (0.01)eM-2j. (2.5)

Let [a,p] denote the interval [n,n+1]. We first show that if

r(a) = 3cM, (2.6)

then r(t) can be continued up to P and, furthermore, r(t) remains positive andr(p) z 3cM. As usual, from (2.4), one easily derives the equation

ttr (t) = r (a) - fcq(s )ds - fr2(s)ds. (2.7)

a aUse of (2.1) and (2.6) readily yields the inequalities

t t2cM - fr(s)ds s r(t) s 3cM - fcq(s)ds. (2.8)

a a

Now suppose that r(t) = 0 somewhere in [a,p]. Let to be the first zero. Thenr(t) > 0 for t E[a,ta) and r (t0 ) = 0. Now the second inequality in (2.8) impliesthat

fr 2 (s)ds s f[3cM - fcq (s)ds ]2dta a a

to ts 2f[9c2 M5 + c(fq(s)ds)2]dt.

a a

It follows from (2.2) and the fact that to - a ! 1 thattofr2(s)ds S 20c 2M2. (2.9)a

The first inequality in (2.8) and (2.5) now imply that

r(t0 ) z 2cM-20c 2M2 > 0,

contradicting the definition of to. Thus r (t) > 0 in [a,]. The same argumentsas above easily lead to the following refinement of (2.9):

121

Pfr2(s)ds S 20c2M2. (2.10)a

Using (2.7), (2.5), and (2.3), we have

r (#) - 3cM + ce - 20c 2M2

S3cM

as claimed earlier.

If, instead of (2.6), we have

r (a) z 3cM, (2.11)

using Sturm's comparison theorem or results in the theory of differential ine-qualities (use of differential inequalities in the theory of oscillation can be found,for example, in Man Kam Kwong and A. Zettl [1982]), we see that the solution ofthe Riccati equation must lie above that which satisfies (2.6). In particular, if(2.11) is satisfied, then r (t) > 0 in [ac,#] and r (#) z 3cM. The possibility thatr(t) may blow up to infinity in [a,#] is excluded by the inequality

r(t) s r(a) -fcq(s)ds :gr(a) + cM.

We can now complete the proof of the theorem by choosing r (0) = 3cM so thatr (t) > 0 in [0,1] and r (t) 3cM. Induction shows that r (t) can be continuedindefinitely on [n,n+1] and r(t) remains positive.

COROLLARY 1. If q is uniformly bounded on R and (2.3) holds, then (1.1) is nonos-cillatory for somec > 0.

COROLLARY 2. If q is the restriction on [0,c) of a (real) Bohr almost-periodicfunction on (-oo,oo) 'with non-zero mean-vr'.ue, then (1.1) is nonoscillatory forsome c ; 0.

PROOF. Suppose that the mean-value of q (t) is < 0 (If a> 0, we choose a nega-tive c.) One of Bohr's fundamental theorems states that [Besicovitch, 1954]

a+T

rn T fmq(s)ds - y< 0 (2.12)

exists uniformly in a. Thus there exists a T large enough so thata+T

g-fq(s)ds s y<0

for each a. This yields (2.3) for c = (|yI T)/ 2. Since Bohr almost-periodic func-tions are uniformly bounded on (-oo,=o) [Besicovitch, 1954], the conditions (2.1)and (2.2) are certainly satisfied for suitable M.

REMARK. In fact, the nonoscillation of (1.1) with a Bohr almost-periodic potentialq implies that (1.1) is disconjugate on [0,o) (see Markus and Moore [1956]).

There is an extension of the class of Bohr almost-periodic functions to aclass that includes functions that are not necessarily continuous. The generali-zation we refer to is due to Stepanoff (see Besicovitch [1954] for definitions,etc.). It is known that functions in the Stepanoff class S1 are uniformly boundedin the Stepanoff metric, i.e., for some L > 0,

122

s+L

IIIs su g q(s)jds <o. (2.13)

As in the case of Bohr almost-periodic functions, these almost-periodic functionsof Stepanoff admit a mean-value, as in (2.12), that is uniform with respect to a.Since this is the case, it is clear from above considerations that (2.3) will besatisfied for such functions. In fact the following corollary holds.

COROLLARY 3. If q ESj is the restriction on [O,) of a (real) Stepanoff almost-periodic function on (-o, c) with a non-zero mean-value, then (1.1) is non-oscillatory (in fact, disconjugate - see Halvorsen and Mingarelli [ 1984] - on[Q,) forsomec #0.

PROOF. Let T = L > 0. It now follows immediately from (2.13) that there holds(2.1) and (2.2) with M = maxim |q|g|s T. |jq |s T3'2 1. The result follows.

Note: The techniques used abovve yield a different proof of Corollary 2 andCorollary 3, both of which are to be found in Halvorsen and Mingarelli [1984].Indeed, it is shown therein that the above corollaries remain valid even for cer-tain classes of Weyl and Besicovitch almost-periodic functions.

References

Besicovitch, A. 1954. Almost Periodic Functions. Dover, New York.

Halvorsen, S. G., and Mingarelli, A. B. 1984. "The large-scale structure of thedomains of non-oscillation of second order differential equations with twoparameters" (preprint).

Kwong, Man Kam, and Zettl, A. 1982. "Integral inequalities and second orderlinear oscillation." J. Differential Equations 45:16-33.

Markus, L., and Moore, R. A. 1956. "Oscillation and disconjugacy for lineardifferential equations with almost periodic coefficients." Acta Math. 96:99-123.

123

SOME PROBLEMS OF TRANSPORT THEORY

R. J. HangeLbroek*

AbstractIn this paper we discuss the linear transport equation and a rather abstractoperator V that occurs in the study of an associated eigenvalue problem. Theeigenvalue problem is or is comparable to an indefinite Sturm-Liouville problem.

1. IntroductionMany of the discussions during the Argonne workshop on Sturm-Liouville

operators considered the Sturm-Liouville eigenvalue problem

Av = X7Tu, (1)

where A is a selfadjoint Sturm-Liouville operator and T a real-valued function ofvarying sign. In linear transport theory a similar eigenvalue problem is con-sidered, and the questions studied are essentially the same. The operator A is,however, of considerably simpler structure in the latter case, and the physicalproblems that lead to the study of (1) are rather transparent. In discussingindefinite Sturm-Liouville problems, some acquaintance with the underlying phy-sical problems seems as desirable as in the case of definite Sturm-Liouville prob-lems, where one assumes some knowledge of the relation with standard physicalproblems like heat transfer. For these reasons we think it appropriate to spendsome time on the transport equation and its physical background.

The following are two examples of transport equations:

1. Neutron transport in a homogeneous slab:+1

S.'(z,,) = -afr(z, ) + ,ff(ps')(z,,')d', (2)

where E[-1,+1], zEJ (an open real interval), c E(O,1). The function f is nonne-gative, symmetric in the pair ( ,u'), and satisfies

+i

Sf (AA')dy'= 1.

II. Electron transport in a homogeneous metal plate:

y- -(xy)= -. (1iN,2) _-(z,), (3)

where and z range over the same intervals as in 1. The function 'i' is requiredto remain bounded for -+t1.

Both equations describe time-independent problems. For this reason weprefer the use of the variable z instead of t in the derivatives occurring in theleft-hand members of (2) and (3). Actually, the variable x stands for a positioncoordinate (see Section 2). Equations (2) and (3) are of the type Ti' = Af.

*Department of Mathematics, Western Illinois University, Macomb, Illinois 81455.

124

An application of the two-sided Laplace transform or separation of variables(or just plain experience) could lead one to study the associated eigenvalueproblems (1).

There do exist many more examples than the two given here. In particular,Beals and Protopopescu [1954] and Kaper [1979] discuss situations where Aranges over the whole real line instead of over [-1,+1].

2. Physical Meaning of the Transport EquationLet us give a brief description of the physical meaning of equation (2). We

consider a plane-parallel homogeneous slab of width a through which inert parti-cles (neutrons) move along straight lines. These particles may collide withnuclei of the medium within the slab. In principle, we need six independent vari-ables to represent the position and velocity of a particle. We use three Cartesiancoordinates for the position and three spherical coordinates for the velocity. Aline perpendicular to the slab is taken as the z-axis. The positive x-direction isalso taken as the direction of the polar axis for the spherical coordinates of thevelocity. We assume a situation in which the particle density 10 actually dependsonly on the position coordinate z and on the polar velocity angle gy, i.e., theangle between velocity direction and the positive z-direction. In other words, weassume translational invariance lateral to the slab, rotational invariance aboutthe x-direction, and all particles to have the same speed. In this way the phasespace is reduced to a two-dimensional space in which points are represented bycoordinate pairs (z, ), where zE[0,a] and = cospE[-1,+1].

The particle density 0 is then defined as a nonnegative function of the pair(z, ) such that '(z ,A)dxdM equals the number of particles in a region of volumedz dp at the point (z, ).

The derivation of equation (2) is similar to the derivation of the heat equa-tion with convection term. One sets up a balance equation for a region in phasespace by counting the number of particles that enter and leave the regionthrough its boundaries, taking into account the losses and gains resulting fromcollisions with nuclei of the medium. In this way we can interpret the termsoccurring in (2) as follows:

" First term on right-hand side: convection term due to the fact that the parti-cles move.

" First term on left-hand side: loss of particles due to collisions with nuclei of themedium. The coefficient equals -1 because of an appropriate choice of the unitin which z is measured.

" Second term on right-hand side: gain of particles because not all collisions withnuclei lead to a complete loss. The particle may be scattered into another direc-tion, or the collision may lead to the emission of one or more new particles. Theconstant c is the mean number of particles released per collision. The functionf ( , ') is a probability density giving the probability that a secondary particlehas velocity direction when it is the result of the collision of a primary particlewith velocity direction '.

The physical meaning of equation (3) is similar, with the different right-hand side resulting from a different behavior of electrons as compared to neu-trons.

Typical boundary conditions for each of the two equations are the following:

Finite slab (i.e., J = (O,a)):

125

1(0,u) = g+( ), 0 0< A: 1: incoming density at:x = 0;%(a, ) = g _( ), -1:9 y< 0: incoming density at:x = a.

Half space (i.e., J = (0,oo)):

Ji(Op) = g(p), 0 < u s 1: incoming density at:x = 0;limi(x,p) =0, -1 sys 1.

Anticipating the Hilbert space formalism which we introduce in the next section,we write these boundary conditions as follows. Let H = H®H-, whereH+ = L2[(0,1); d ] and H_ = L2 [(-1,0); d], and let P+,P_ denote the projectionsassociated with this direct sum. The above boundary conditions are then inter-preted as follows:

Finite slab: P+(0) = g +EH+, P_ (a) = 9 -EH_;

Half space: P+,'(0) = g EH, Ulim't(x) = 0 infH.

3. Hilbert Space Formalism

Let H = L 2[(-1,+1); d, ]. Then we may interpret equations (2) and (3) asordinary differential equations in H of the form

T (z) = -AO(x), :eJ. (4)

In both cases T and A are selfadjoint relative to the usual inner product inH. Of importance here are the spectral properties of A:

e For equation (2): E(A)c[1-c ,1+c]. There are three cases:

c < 1 (absorbing medium): J(A) is positive and bounded away from 0.

c = 1 (scattering medium): A has 0 as an eigenvalue. F(A)\0j is boundedaway from 0.

c > 1 (multiplying medium): F(A) has a negative part.

" For equation (3): E(A)cR+. A has 0 as an eigenvalue. E(Al0 is bounded awayfrom 0.

One should note the correspondence between the cases of equation (2),c = 1, and equation (3). The case of equation (2), c < 1, is by now well under-stood, up to the point that some people call it the trivial case. The case of equa-tion (2), c = 1, was (within the bounds of the Hilbert space formalism) firstattacked by C. G. Lekkerkerker [1976] for isotropic scattering (i.e., f = 1 in(2)). This same case was resolved by C. van der Mee [1981] for very generalfunctions f . The case of equation (3) (and also equation (2), c = 1, but in amore general setting than van der Mee's contribution) was solved by R. Beals[1984]. The case of equation (2), c > 1, remains quite hazardous and poorlyunderstood, notwithstanding some well-intended attempts.

In the remainder of this paper, we restrict ourselves to equation (2), c < 1.We do not include equation (2), c = 1, and equation (3) because these cases aremarred by (apparently unavoidable) technical complications.

In the case of equation (2), c < 1, the operator A has a bounded inverse. Werewrite (4) as

126

-d-(z) = -Slf(x), z E, (5)dxwhere S = A- 1 T and the existence of S~1 is implied by the existence of T-1. Theassociated eigenvalue problem (1) takes the form

S-1v = av. (6)

One may attempt to solve (5) subject to specific boundary conditions by expand-ing '(z) in a series of eigenvectors of S~1 (in this case, eigenfunctionals sinceS1 has not a discrete spectrum). Following this route, one has to prove, as afirst step, the completeness of the system of eigenfunctionals. Though he didnot use the Hilbert space formalism expounded here, it is exactly this route thatK. M. Case [1960] followed for isotropic scattering. Somewhat later [Hangel-broek, 1976] we avoided the hardships of having to prove this completeness pro-perty (also for f/= 1) by introducing a new inner product,

(v ,W )= (Av,w), v,wEH.

This inner product is equivalent to the original (, ). Since S is selfadjoint rela-tive to (, )A, we were able to invoke the Spectral Theorem. We do not have thetime here to discuss the details of Case's contribution, but we must mention hishalf-range completeness proof. Our endeavors to avoid this latter proof ulti-mately led to the introduction of the operator V [Hangelbroek and Lekker-kerker, 1977].

The spectrum of S-1 contains the intervals (-oo,-1] and [1,wo). Hence, S1does not generate a semigroup of operators by which (5) can be solved. Usingthe Spectral Theorem, however, we can decompose H in a direct sum

H = Hp@Hm (7)

of closed subspaces such that H, (Hm is the maximal invariant subspace inwhich S is positive (negative). Solving (5) separately in H, and Hm, we obtain asthe general solution in the case of a finite slab,

#(z) = (e-:S'P, + e(-:)S~-Pm)h, 0z<a., (8)where P and Pm denote the projections associated with (7) and h is some ele-ment of H to be determined from the boundary conditions.

In the case of a half space we obtain

1(z) = e-"s~'h1 , z>0, (9)where h1 must be in H, in order that lim,....'(z) = 0. Substitution of (9) in theboundary condition stated at the end of Section 2 yields that h, has to be a solu-tion of the equation (P',p + P-Pm)h i = 9 +, i.e.,

f =g9+, (10)where we defined V = P4,P + PPm. In the same way we find by substitution of(8) in the boundary conditions for the finite slab that h must be a solution of theequation

Vh = g++g_, (11)

where V = V+ (1-V)e IS-1 1, |S-1 = (P,-Pm)S~.

We conclude that in order to be able to solve the half space and finite slabproblems, we need to prove the existence of r' and V- 1.

127

4. Properties of the Operator VIn the following observations, we assume the existence of V~1 for the time

being.Since V = P4P, + P-.Pm, we have V: H,-.H+ and Hm 4-+H.. Then ~1 : H+-eH

and H_-+Hm. Actually, V~'P+ is a projection onto H along H_ and W-P. is aprojection onto H, along H+. The latter statements follow from the properties

P+V-1P+ = P+VV-1P+ = P+. (12)

and similarly,

P-V-1P- = P-.

One may call V an extension operator. It extends each element g+EH4 to anelement W1g +EH by adding an element in H_,

9+ = 9+ + PV-g . .

Let I = fdEA be the resolution of the identity for the operator S. IL g+EH+,then

V 19+_= d EA V-ig +,

since V'g+EH . Hence, because of (12),

9+ = P+ dExVg- = d(P+ExV-)9 +,

which proves the so-called half-range completeness of the system of eigenfunc-tionals associated with the operator S.

We have left to discuss the existence of V-1 and I'. There are two differentapproaches:

(i) We use the fact that the integral term in (2) defines a compact operator, Csay, in H. Thus, A = 1-C is a compact perturbation of the identity. Assumingthat C is a trace class operator [Hangelbroek, 1980] or, more generally, thatC = I T I D for some 0 < e < 1 with D compact [van der Mee, 1981], one canprove that 1-V is compact. It is not difficult to show that V is injective, eitherby a direct argument or by inference from the classical result that the halfspace problem has a unique solution. Using Fredholm's Alternative, we thenconclude that V 1 exists in H. The same argument yields that V~' exists since

V = V(1+(Vl-1)e--IIs1) (13)

with r 1 -1 compact.

(ii) We follow the approach initiated by R. Beals [1984], in which the compact-ness of C = 1-A is not used. We introduce two new inner products,

(v,w)7 = (I T |v,w) and (v,w), = (IS Iv,w)A,where I T I = (P+-P_) T and I S I = (P, -Pm)S. The equivalence of the inner pro-ducts (,) and (,)A implies the equivalence of (,)r and (, )s [Beals, 1984;Hangelbroek, 1984]. The completion of H relative to (,)T (or (, )s) is1? = L2[(-1,+1); j.II)d ]. We present a proof that V- 1 exists in H using an argu-ment that is simpler than the one used by Beals.

The relation between (, )T and ( , )s is

128

(vow)s =((2V-1)v,w)T.Hence, in terms of the partial ordering of selfadjoint operators in a Hilbertspace,

t < 2V-1 < t~1 relative to ( , )T

for some e > 0. Then

+t <W_ < -1+relative to (, )T.

We conclude that y-1 exists with ii|V~1IT < 2. Since

-1 < V- 1-i 1 relative to (, )7 and ( ,)s,

we obtain from (13) that VG~ 1 also exists as a bounded operator in R sinceI(VW1-1)e'IS-l1Is < 1.

References

Beals, R. 1984. "Indefinite Sturm-Liouville problems and half-range complete-ness." J. Diff. Eqs. (to appear).

Beals, R., and Protopopescu, V. 1984. "Half-range completeness for the Fokker-Planck equation." J. Stat. Phys. (to appear).

Case, K. M. 1960. "Elementary solutions of the transport equation and theirapplications." Ann. Phys. 9:1-23.

Hangelbroek, R. J. 1976. "Linear analysis and solution of neutron transportproblems." Transp. Theor. Stat. Phys. 5:1-85.

Hangelbroek, R. J. 1980. Timendependent One-speed Neuron Trwnsport Equa-tion with Anisotropic Scattering in Absorbing Media. Argonne NationalLaboratory Report ANL-80-60.

Hangelbroek, R. J. 1984. "On the stability of the transport equation." (preprint).

Hangelbroek, R. J., and Lekkerkerker, C. G. 1977. "Decompositions of a Hilbertspace and factorization of a W-A determinant." SIAM J. Math. Anal. 8:458-472.

Kaper, H. G. 1979. "Boundary value problems of mixed type arising in thekinetic theory of gases." SIAM J. Math. Anal. 10:161-178.

Lekkerkerker, C. G. 1976. '"The linear transport equation. The degenerate casec = 1. Proc. Roy. Soc. Edinburgh 75A:259-282, 283-295.

Van der Mee, C. V. M. 1981. Sernigroup and Factorization Methods in TransportTheory. Math. Centre Tract No. 146, Amsterdam.

Note: In the above list, we attempted to minimize the number of refer-ences, restricting ourselves to publications that, in our opinion, are the most

129\ k

important or informative within the context of this paper. Such a selection mayhurt many feelings. Especially, we feel that (unavoidable) injustice is done tothe many serious and very able scientists and mathematicians who are not men-tioned but whose studies have had considerable influence on our investigations.

Below, we give four additional references that are important to the under-standing of equations (2) and (3).

For background:

Bethe, H. A., Rose, M. E., and Smith, L. P. 1938. "The multiple scattering of elec-trons." Proc. Amer. Philos. Soc. 78:573-585.

Bothe, W. 1929. "Die Streuabsorption der Electronenstrahlen." Z. Physik 54:101-178.

Case, K. M., and Zweifel, P. F. 1967. Linear Transport Theory. Addison-Wesley,Reading, Mass.

Duderstadt, J. J., and Martin, W. R. 1979. Transport Theory. Wiley-Interscience,New York.

For a discussion of the Hilbert space approach

Kaper, H. G., Lekkerkerker, C. G., and Heijtmanek, J. 1982. Spectral Methods inLinear Transport Theory. Birkhauser, Basel.

131

ASYMPTOTIC BEHAVIOR OF SEMIGROUPS

J. Hejtmanek*

Abstract

A motivation is given for the study of the spectral mapping theorem for the ex-ponential function. Also, some results of this study are summarized.

1. SemigroupsLet X be a Banach space. Usually we have in mind an L2-Hilbert space or an

L1-Banach space. Both types of spaces hae an ordering structure'through thecone of positive functions X+. Both types of spaces are Banach lattices. Anoperator T: X-eX is called positive iff TX+ c X+.

Let W = [ W(t):t ERf] be a strongly continuous one-parameter semigroupwith generator A; W(t) = exp(At). It follows from the Feller-Mijadera-PhillipsTheorem that the spectrum a(A) of the generator is contained in some left half-plane. We define the spectral bound of the generator A

s: = supIRe X:XEa(A)j,

and the type of the semigroup W

wo: = infjcER: there exists an M z 1 such that IIW(t)IIsMexp(t) for all t z 0.

It follows from the Laplace transform that s s o, but in general we do not haveequality.

THEOREM. [ W (t )xo: t ER.] is the solution of the Cauchy problem

z(t) = Az(t), x(0) = zoEdom A.

A semigroup is called a semigroup of positive operators if W(t)X C X+ for allt 0.

We assume that the reader is familiar with the standard theory of linearsemigroups; references are the texts by Hille and Phillips [1957], Davies [1980],Goldstein [1983], and Pazy [1983].

2. The Time-Dependent Neutron Transport Equation

Let the physical domain of a neutron transport system be a convex subset Oof R, and let the velocity domain be a sphere or spherical shell

S = JEER9 : 029vo ICI v 1 cm.

Then f (x,E,t)dzdE is the expected number of neutrons in the volume elementdzd at (z,()E~xS at time t. We assume that the particle density functionf (-,-,t) is an element of L1 (OxS) or L2 (OxS). The linear transport equation is abalance equation which has the following form:

at aat'= - -Ef (z,t) - h(z,E)f (z,,t) + k(x,(-E')f(xE't)d'.

The operator defined by the right-hand side is a sum of a partial differential

Institute flfr Mathematik, Universit&t Wien, Strudlhofgasse 4, A-1090, Wien, Austria.

132

operator of order one, a multiplication operator, and a partial identity.

In addition, boundary conditions are prescribed: (i) in the reactor problem0 is a bounded subset of if, and the incoming flux on the boundary 80 vanishes;(ii) in the multiple scattering problem, we have 0 = P, and a bounded and con-vex target D is embedded in 0. In both cases, we can write the linear transportequation in functional form,

f (t) = (-T--A 1 +A2)f (t), f (0) = f 0 Edom T.

We assume that the linear operators A 1 and A2 are bounded.The solution of the differential equation f (t) = (-T-A 1 )f (t) has the expli-

cit formS

W 1(t)fo(z ,C) = exp(-fh(x -Es ,()ds)f o(x -t,), if: -t (E0,0

and zero otherwise.The solution of the differential equation f (t) = (-T-AI + A 2)f (t) can be

written as a Dyson-Phillips expansion of the corresponding perturbation prob-lem:

I

F(t)f 0 = W 1(t)f 0 +,1fWi(t-s)A2 Wi(s)fcds + ..0

We refer to the monograph by Kaper, Lekkerkerker, and Hejtmanek [1982] fordetails. There sufficient conditions on the functions h and k can be found suchthat the linear transport operator -T -A 1+A 2 has a strictly dominant eigenvalue.The existence of such an eigenvalue, which had been tacitly assumed sinceFermi's famous criticality experiment with the CP-1 reactor at the University ofChicago on December 12, 1942, implies that the solution of the linear transportequation, for large t, is determined by a decay constant A0 and a fundamentalmode o(z,C):

W(t)f o(x, )~ e 0"p(z, ).

The problem of predicting such asymptotic behavior from the knowledge of thespectrum of -T-A 1 +A 2 is, however, still unsolved. It requires the decompositionof the semigroup W into an asymptotic part e * Po and a transient part Zo(t)

W(t) = e * Po + Zo(t)(I-Po),

where io,(Zo) <co ( W) = Xo. We remark that the equality co( W) = Xo is not truefor general semigroups, but it is true for semigroups of positive operators in anL1-Banach lattice (Theorem of Derndinger).

3. Spectral Mapping TheoremThe validity of the spectral mapping theorem for the exponential function

would enable us to predict the desired asymptotic behavior for linear transportprocesses.

THEOREM. If [ W(t): t>to] is continuous in the uniform operator topology forsome t0 > 0, then exp(a(A)t) = o(exp(At)\ 0j for all t z 0. In partular, if thesemigroup is holomorphic, then the spectral mapping theorem is true.

THEOREM. If W is a semigroup of positive operators in an L1- or L2-Banach lat-tice, then s = co. Sre Nagel [1984].

133

Several examples have been given of semigroups for which s <cw . SeeHille-Phillips [1957], Fojas [1973], Greiner, Voigt, and Wolff [1981], Wolff [1981],and Nagel [1982].

We remark that the semigroup generated by the linear transport operatoris not holomorphic, but that it can happen that [ W(t): t > to] is continuous inthe uniform operator topology for some to > 0.

Furthermore, we remark that there are semigroups for whichexp(o(A)t) o u-(exp(At))\ 0; for all t > 0 (e.g., if s < c0), and that there aresemigroups for which equality holds for rational t, but inequality for irrational t.

4. Additional Examples

4.1. Schrodinger SemigroupsIn general, it is not true that the rotation of the generator of a semigroup,

especially by 900, results in the generator of another semigroup. If the genera-tor of the Schrodinger equation -iH is rotated by 900, then -H is generator of aholomorphic semigroup [exp(-Ht): t z 0], which is called the Schrodinger semi-group; see B. Simon [1982]. In this case the spectral mapping theorem is true,and a fortiori s = ca.

4.2. Stability TheoryWe refer to the articles of Pritchard and Zabczyk [1981], Wolff [1981], and

Nagel [1984]. Let us assume that s s w s0.

THEOREM. If s = wo, and if s < 0, then the semigroup is exponentially asymptoti-cally stable in the uniform, operator topology, i.e., there eist constants u > 0and M z 1 such that IIW (t) I I M exp(-ut) for all t o 0.

References

Davies, E. B. 1980. One-parameter SeTigroups. London Math. Soc. Monographs15, Academic Press.

Fojas, C. 1973. "Sur une question de M. Rhegis." Analele Universitatjii dir Tim-isoara 11, fasc. 2.

Goldstein, J. 1983. Semigroups of Linear Operators and Applications. TulaneUniversity.

Greiner, G., Voigt, J., and Wolff, M. 1981. "On the spectral bound of the generatorof semigroups of positive operators." J. of Operator Theory 5:245-256.

Hille, E., and Phillips, R. S. 1957. Functional Analysis and Semigroups. AMS Col-loquium Publ. 21.

Kaper, H. G., Lekkerkerker, C. G., and Hejtmanek, J. 1981. Spectral Methods inLinear Transport Theory. Birkhauser Verlag, Basel.

Nagel, R. 1982. "Zur Charakterisierung stabiler Operatorenhalbgruppen."Semesterberichte Funktionalanalysis Tubingen, Sommersemester 1982,pp. 99-120.

134

Nagel, R. 1984. "What Can Positivity Do for Stability?" in: Functional Analysis:Surveys and Recent Results III. Ed. K. D. Bierstedt and B. Fuchssteiner.Elsevier, North-Holland, pp. 145-154.

Pazy, A. 1983. SeTrigroups of Linear Operators and Applications to PartialDifferential Equations. Springer-Verlag, Berlin.

Pritchard, A. J., and Zabczyk, J. 1981. "Stability and stabilizability of infinitedimensional systems." SIAM Review 23:25-52.

Simon, B. 1982. "Schrbdinger sernigroups." Bulletin AMS 7(3):447-526.

Wolff, M. 1981. "A Remark on the Spectral Bound of the Generator of a Semi-group of Positive Operators with Applications to Stability Theory," in: Func-tional Analysis and Applications, ISNM 60, Birkhauser Verlag, Basel, pp.39-50.

135

SOME EXTENSIONS OF RESULTS OF TITCHMARSH ON DIRAC SYSTEMS

D. B. Hinton*J. K Shawf

Abstract

Spectral properties are considered for a one-dimensional Dirac system withsingularities at zero and infinity. Of primary concern is the asymptotic behaviorof solutions near zero. From the form of these solutions, criteria are given forthe singular point zero to be in the limit point or limit circle case. TheTitchmarsh-Weyl m-coefficient at zero is shown to be meromorphic. This result,when combined with known behavior at infinity, is used to establish the locationand class C(1) nature of the essential spectrum.

1. Introduction

The system considered is

Li( = -g1) c7'(z) - [ - z4) c +V(x) .(z)]

=Ag (x), O< z< e, p= 1(1.1)

where V is real and locally Lebesgue integrable on (0,mo), and k, X, and c areconstants with c > 0 and k,X possibly complex. Denote by H(I) the Iil ert

space of all equivalence classes of complex vector-valued functions = 2on

an interval I such that f 1 (I f 11 2 + If 2 j2 )dz < cc. The operator L determines(for k real) in H(0,oo) certain selfadjoint operators (cf. Weidmann [1971]) bysuitably restricting it to functions in H(0,o) satisfying certain boundary condi-tions. The number of such boundary conditions at each singular endpoint maybe determined by the method of Weyl (cf. Levitan and Sargsjan [1975]); thus, fol-lowing the terminology of this method, we say L is limit-point (LP) at 0 (lirmit-circle (LC) at 0) if the number of linearly independent solutions of Lg = Xg inH(0,1) for ImA 0 is exactly 1 (2). Similar definitions apply at z = co, althoughin the setting here L is always LP at 00 [Levitan and Sargsjan, 1975: p. 492].When L is LP at a singular point, no boundary conditions are required there; andwhen L is LC at a singular point, one boundary condition is required.

Another problem of interest in connection with L is the location of theessential spectrum of the selfadjoint operators determined by L. Under fairlygeneral conditions for Vi, V 2 "small" at 00, this essential spectrum is(-00,-c ]u[c ,ma); when V1, V2 are "large" at cc, this essential spectrum is (-oooo).Results of this nature may be found in Behncke [1980], Hinton and Shaw [1984a],Titchmarsh [1961 and 1962], and Weidmann [1971 and 1982].

For VI = V2, the system (2.2) was investigated by Titchmarsh [1961 and1962]. For the singular endpoint z = 0, we extend Titchmarsh's LP-LC criteria

'Mathematics Department, University of Tennessee, Knoxville, Tennessee 37996-1300.

tMathematics Department, Virginia Tech, Blacksburg, Virginia 24061-4097.

136

to allow for highly oscillatory potentials as well as having V1 V2 . In Section 3we show how these criteria, derived by asymptotic methods, also allow exten-sions of criteria for essential spectra to be -m,-c ]u[co) or (-ooao). Actually,we show the essential spectrum is of class 1) in its interior. Finally, in Section4 we prove that the essential spectrum is simple in its interior. Other exten-sions of Titchmarsh's LP-LC criteria, which are complementary to those givenhere, have been given by Kaf [1972] and Behncke [1980].

The operator (1.1) arises from the 3-dimensional Dirac operator by aseparation of variables, and its spectral properties can be related t, ;.hose of the3-dimensional Dirac operator (cf. Behnke [1980] and Rejto [1971a and 1971b]).Considerable literature exists on spectral properties of the 3-dimensional case;we refer to Barut and Kraus [1976]; Burnap, Brysk, and Zweifel [1981]; Kalf,Schmincke, Walter, and Wust [1975]; Klaus [1980]; Klaus and mist [1979];Schmincke [1972 and 1973]; and Wust [1977] for a partial listing of such proper-ties.

2. Asymptotics at 0We use here a theorem from Hinton [1984] for the system

7'(t )= [0'(t)O- 1 (t) + B(t) + C(t)]rj(t)), a. t < m, (2.1)

where 0 is a diagonal matrix each with 0Q nonvanishing and essentially increas-ing or essentially decreasing on [a,ao) (cf. Hinton [1984: p. 294]).

THEOREM 2.1 [Hinton, 1984]. Suppose in (2.1) that 0, B, and C are locally Lebes-gue integrable and that

i. Bo(t): = fB(s)ds exists.

ii. ||C(t )I|dt < Co.a

iii. fj|G(t)Jjdt < m, where G = -'[~1B 0 + B0 '1-1 + BOB.

Then there is a fundamental matrix 1 of (2.1) such that

limP(t)(- 1 (t) = I.

To apply Theorem 2.1 to (1.1) the singular point is transformed tom by

V(z) = [ -- m) 2(t), t = 1/z, 0 <z <1. (2.2)

Then I satisfies (- = d/ dt)

1(t -- k)/t _ (r+ ( (t)+c)t2 m-2-(t)=-A-V(t)+c)t - -(m-k)/ t (t,(2.3)

where $(t) = V (1/t). We state three theorems that distinguish the cases: i)k /z dominates V1,V 2; ii) k/z and V1. V2 are of comparable size; and iu) V1, V2dominate k /z.

THEOREM 2.2. Supposein (1.1) that forri = 1,2,

137

C

i. v(x): = fl(Y)dt exists,0

ii. f-'Ivi(x)Idz < Co,0

ai. f{vi(x)V2(x)I + IV2(x)Vi(x)Idz < m.0

Then there is a fundamental matrix Y(x ,X) of (1.1) such that

Urn Y(x,X) V zj = I. (2.4)

Moreover, fork real, Lis LC at 0 iff Ik I < 1/ 2.

PROOF. We write (2.3) for m = 0 as

- (t) = f 0k k + '() ZM1 + tt) M2 + C(t); 9(t), (2.5)

where

M1 = 0 , M2 = 1 o)C(t) = t-2(_-+O A~

To apply Theorem 2.1 to (2.4), we set

Et-k 0 t) 2t

l(t)= 0 . B(t)= -t M1+ MtZ 2.

By defining '(t) = V (1/t) = f s-2$I(s)ds, the matrix G in (iii) of Theorem 2.1is computed to be

G(t) = 2kt-1 Lt i(t ) - t-2 f(t)V(t) 't -1 2(t) 0 2J 1(O

It is readily computed that Theorem 2.1 applies and there is a fundamentalmatrix Z(t ,X) of (2.5) such that

limZ(tA,)F0 t 0 =f I(2.6)

The transformations (2.2) and (2.6) yield (2.4). The asymptotics (2.4) yieldimmediately the "moreover" part.

A prototype example for Theorem 2.1 is

V1(x) = V2(x) = a /x" + bxa sinxp, 0 <z <1,

where a,b,n,a,f are real constants with n < 1, < 0, and a > (g/2)-1.

THEOREM 2.3. Suppose in (1.1) that for i = 1,2,

i. 1' (x) = a / x + q(x ),;ai constant.

i. p0: = ,k/-aga 0.

iii. Q (x): = fqj()dt exists.0

138

,... f~x|I(x)|dz <mo.0

1

v. f{IQi(z)q 2(z)I + I Q2(x)g 1 (x)I);dx < cc.0

Then there is fundamental matrix Y(x,X) of (1.1) such that

limS-lY(x,X)X 0 0= k+, S = [ ( 'x-O J0 1 p k + L

Moreover, if a1, a2, and k are real, then Lis LC at 0 iff RepI <1/ 2.

The proof of this theorem is similar to that of Theorem 2.2. It is first neces-sary to diagonalize the leading matrix in (2.5) by the transformationa(t ) = S~1(t).

Results in case (iii), i.e., when V1, Va dominate k/x may be found in Hintonand Shaw [1984a]. For example, the following theorem is a corollary of theasymptotics in Hinton and Shaw [1984a].

THEOREM 2.4. Suppose in (1.1) for i = 1,2 and some a > 0,

a

00

asCi)sf Aiz(z)-+ as z-0, z VIA(z)Idz <. Thntee sC uxmna

(ii) [(V()-cY)/)(Vi(z)+c[)])Edz <m.],8

[(Vi(x)+c )/=(V2(z)-cp)]f(d< o,

0z

(iii) rl(x): = [(Vi(x)+c )/ (V(z)-c )]liz -k,

A(x): = rj'(x)/ g(z)[(V(z)+c )(V(z)-c )va

satisfy A(x)-*O as x -+0 and f' '(x)|Idx <--.Then there is a fundamental

matriY(x,X) of (1.1) s)s ch that0as.x -+,-z-* (z) 77 xza Y(x,) E( )1 E x) ~I

where for some b > 0,n

E(x): = exp fip0(x)x~2dz

0(x): = V1-D(x)2

T : = 11.

Moreover, (1.1) is LC at 0.

Note that for V(x) = V2(x) = k /x" , these hypotheses are satisfied for k > 0

and n >1.

139

3. Essential Spectrum Criteria

For the computation of essential spectra, it is sufficient to consider eachendpoint separately since: the essential spectrum for a two-singular endpointproblem is the union of two separate one-singular endpoint problems (cf. Weid-mann [1971]). However, it is also convenient to consider the two-singular end-point problem from the viewpoint of the Titchmarsh-Weyl m-coefficient whichmay be defined as follows.

Define solutions ', of (1.1) by the initial values

( (1,),(1,)) = , ) (1 ) = I.

Then the one-singular endpoint m-coefficients are given for ImA 0 by

M, (A) = lim 1(x , m-() = lim . (3.1)X 9 1(2,X) () z-.O 1(,)(3

Where x = 0 is in the limit circle case, the limit for m- in (3.1) is a sequentiallimit as m- is not unique; different sequential limits correspond to differentboundary conditions at 0. The m-coefficient (cf. Hinton and Shaw [1984b]) for aselfadjoint operator associated with (1.1) is given by

M(X) = [m-(X) - m+(X)]-4(m+(X)+m -(a))/ 2 rnm(X)r-(X)/ 2. (3.2)

For the Sturm-Liouville operator, fundamental relations between the singu-lar structure of the m-coefficient and the spectrum have been derived by Chau-dhuri and Everitt [1968]. These results were extended to systems by Hinton andShaw [1982 and 1984b]. We describe now the relations that we utilize here. letT be a selfadjoint operator associated with (1.1) whose rn-coefficient is M. Wesay M is analytic (has-a simple pole) at a real X0 if M has an extension that isanalytic (has a simple pole) at X0. Let p( T) be the resolvent set of T, a( T) be thespectrum of T, and P(T) be the isolated points of a(T). DefineE(T) = a(T) - P(I) as the essential spectrum of T. The set PC(T)cE(T) con-sisting of eigenvalues in E( T) is called the point-continuous spectrum andC(T) = E(T) - PC( T) is called the continuous spectrum. The following relationsare established in Hinton and Shaw [1984b]:

(i) XoEp(T)eM(X) is analytic at X0. (3.3)

(ii) XEP(T)aM(X) has a simple pole at X 0 .

(iii) X0EC(T)eM(X) is not analytic at X0 and limvM(X0 +iv) = 0.

(iv) k EPC(T)te limvM(Xo+iv) = S 0, and M(X)-iS(X-X)- 1 is not analytic atv.o+

xc.An examination of the proof of Theorem 2.1 shows that the matrix Y(z,X) in

Theorems 2.2 and 2.3 is entire in A for each fixed z. Since((x,X), O(z,X)) = Y(zX)Y(1,a)-1 we may then conclude that under thehypotheses of Theorem 2.2 or 2.3, m-(X) is meromorphic on C. Under thehypotheses of Theorem 2.4, m-(X) is meromorphic on C since (1.1) is LC at 0.We assume for the remainder of this section that m-(A) is meromorphic on C.

Under rather general conditions for V1, V2 "small" at c, m+(A) satisfies thefollowing:

140

(i) m+(X) is meromorphic on JX: -c <ReX <c ;. (3.4)

(ii) m+(X) has a continuous extension to X: ReX[-c ,c ], ImX 0 such thatImm+((X)> 0 for X real and X[-c ,c].

For example, if V% = V1 + Y 2 with V1(x)-'0 as x-'oo and Y 1', '2EL(1,0c), m+satisfies (3.4) [Hinton and Shaw, 1984a']. We now note some consequences of m+satisfying (3.4).

If we write mt (X) = at (X) + iyt (X) on JX:ImX > Qj with at, y* real, then wecompute that (with X suppressed),

mM = +--)2 + (a+-a-)2 +Q~~(3.5)7+(a-+~)2+7-(a++y+)2.(35

From (3.2), (3.4), and (3.5) we have the following:

(a) M is meromorphic on JX: -c < ReX < c j; hence the spectrum of T purelydiscrete on (-c ,c ).

(#) For AO in (-oa,-c)u(c,oo) a regular point of m-, there is an interval( -6,X+6) on which f- = 0 and

1+ +a

IITLM= (7+)2 + (a+-) 2 [a- )

(y) For No in (-00,-c)u(c ,ao) a pole of m-,

lilM(Xo + it) = i2 1/21

(6) If p is the spectral matrix of T, then by the Titchmarsh-Kodaira formula, i.e.,

p(X) - p( ) = n-lim fImLM(s+ie)ds,

we have on (-0o,-c)u(c ,co) that p is of class CM' and p'(X) has a rank 1.

For V1, V2 "large" at cc and of like sign, we have under general conditions,e.g., Hinton and Shaw [1984a: Theorem 2], that m+(a) has a continuous exten-sion to JX: ImA z 0 with Imm(X) > 0 on (-ceoo). Under these circumstances, thespectral matrix of p of T is of class CM' on ( ) with rank p'(X) = 1 on (-co,o).

Theorems of the above type for the one-dimensional and three-dimensionalDirac operator have suggested the following problems. However, results seem tobe scarce; one may be found in Glazman [1965: p. 207].

(i) Find conditions on the coefficients of (1.1) that ensure that a( T)n(-c ,c) isfinite.

(ii) Find conditins on the coefficients of (1.1) that ensure that c (-c) is a limitpoint of P(T).

For the three-dimensional Dirac operator, sufficient conditions for there toexist [-d,d] so that a(T)n[-dd] = 0 are given in Klaus and Wtist [1979],Schmincke [1973], and Wiist [1977]. Bounds on the eigenvalues in the one-dimensional case are given in Evans and Harris [1981].

141

4. Simple Spectrum

The expansion formula for (1.1) may be written for!f = EH(0,o) as (cf.

Levitan and Sargsjan [1979])

f(z) = fF-o(X)P(xX()dpuI() + 0(z,X)dP12 (X);

we

+fF,(X)f (z,X)dP2 1 (X) + 0(xX)dP22(X);, (4.1)

where

F,(a) = 7f T(x)4(x,)dz, F,(X) = ff T (x)(zX)dz.0 0

For simplicity we will say an interval IcC(T) is simple (see Naimark [1968] forthe definition of simple spectrum) provided the contribution to (4.1) over I canbe expressed in the form

fF(x)'i'(z,x)d(), F(A) = ff T (z)(zA)dz, (4.2)

where j is a solution of (1.1) and is a monotone nondecreasing scalar functionon I.

To see that these conditions hold for Ic(-oo,-c)i(c ,a) for V, V2 "small" orIc(-o,co) for V1, V2 "large," first suppose I contains no pole of m-(X). Then from(P) above, (P21'(X), P22'(X) = a~(X),(pu1'(X),P12'(X)) and since p is symmetric,P12' = P21' = aUPii'. Thus the contribution to (4.1) on I can be written as

fF 1,(x) + a~(X)F,(A)J$(z,XA) + a-(X)0(z,X)p11 '(X)dA, (4.3)

which is of the form of (4.2). At a pole X of m-(a), we have in a neighborhood ofX0, P12' = P21' = P22'/ o~ and Pu' = P22'/ (a-) 2 , and a formula similar to (4.3) can bederived.

References

Barut, A. 0., and Kraus, J. 1976. "Solution of the Dirac equation with Coulomband magnetic moment interaction." J. Math. Phys. 17:506-508.

Behncke, H. 1980. "The Dirac equation with an anomalous magnetic moment."Math. Z. 174:211-225.

Burnap, C., Brysk, H., and Zweifel, P. F. 1981. "Dirac Hamiltonian for strongCoulomb fields." I1 Nuovo Cimento 64B:407-419.

Chaudhuri, J., and Everitt, W. N. 1968. "On the specrum of ordinary secondorder differential operators." Proc. Roy. Soc. Edin. 68A:95-119.

Evans, W. D., and Harris, B. J. 1981. "Bounds for the point spectra of separatedDirac operators." Proc. Roy. Soc. Edinburgh 88A:1-15.

Glazman, 1. M. 1965. Direct Methods of Qualitative Spectral Analysis of Singular

142

Differential Operators. Israel program for scientific translation,Jerusalem.

Hinton, D. 1984. "Asymptotic Behavior of Solutions of Disconjugate DifferentialEquations." Diferential Equations. Ed. I. W. Knowles and R. T. Lewis. NorthHolland, Amsterdam.

Hinton, D., and Shaw, J. K. 1982. "On the spectrum of a singular Hamiltonian sys-tem." Quaest. Math. 5:29-81.

Hinton, D., and Shaw, J. K. 1984a. "Absolutely continuous spectra of Dirac sys-tems with long range, short range, and oscillating potentials" (to appear).

Hinton, D., and Shaw, J. K. 1984b. "On the spectrum of a singular Hamiltoniansystem II" (preprint).

Kalf, H. 1972. "A limit-point criterion for separated Dirac operators and a littleknown result on Riccati's equation." Math. Z. 129:75-82.

Kalf, H., Schmincke, U.-W., Walter, J., and Wust, R. 1975. "On the Spectral Theoryof Schroedinger and Dirac Operators with Strongly Singular Potentials."Lecture Notes in Mathematics, Vol. 448. Springer-Verlag, Berlin, pp. 182-226.

Klaus, M. 1980. "On the point spectrum of Dirac operators." Helv. Phy. A53:453-462.

Klaus, M., and Wust, R. 1979. "Spectral properties of Dirac operators with singu-lar potentials." J. Math. Appl. 72:206-214.

Levitan, B. M., and Sargsjan, I. S. 1975. Introduction to Spectral Theory: Self-adjoint Ordinary Differential Operators. Translations of MathematicalMonographs, Vol. 39. AMS, Providence.

Naimark, M. A. 1968. Linear Differential Operators, Part II. Ungar, New York.

Rejto, P. A. 1971a. "On reducing subspaces for one-electron Dirac operators."Israel J. Math. 9:111-143.

Rejto, P. A. 1971b. "Some essentially self-adjoint one-electron Dirac operators."Israel J. Math. 9:144-171.

Schmincke, U.-W. 1972. "Distinguished selfadjoint extensions of Dirac opera-tors." Math. Z. 129:335-349.

Schmincke, U.-W. 1973. "A spectral gap theorem for Dirac operators with cen-tral field." Math. Z. 131:351-356.

Titchmarsh, E. C. 1961. "On the nature of the spectrum in problems of relativis-tic quantum mechanics." Quart. J. Math. 2(12):227-240.

Titchmarsh, E. C. 1962. "On the nature of the spectrum in problems of relativis-tic quantum mechanics II, III." Quart. J. Math. 2(13):181-192, 255-263.

Weidmann, J. 1971. "Oszillationsmethoden fur Systeme gewbhnlicher

143(1

Difterentialgleichungen." Math. Z. 3 19:349-373.Weidmann, J. 1982. "Absolut stetiges Spektrum bei Sturm-Liouville-Operatoren

und Dirac-Systemen." Math. Z. 180:423-427.

Wlist, R. 1977. "Dirac operations with strongly singular potentials." Math. Z.152:259-271.

145

SEMIGROUPS GENERATED BY ORDINARY DIFFERENTIAL OPERATORS

Mark A. Kon*

Abstract

Certain regularity properties of semigroups generated by ordinary differentialoperators are considered. It is shown that semnigrcups generated by a generalclass of such operators (which allows for singular coefficients) are continuous,both pointwise and in LP. The semigroups are infinitely smoothing in the scale ofLP spaces, but only partially so in the scale of Sobolev spaces.

1. IntroductionThis paper will address some questions in the study of expansions in eigen-

functions of ordinary differential operators. Some of the results in Section 2 areessentially generalizations Lo higher order operators of results proved by Konand Raphael [1983] for Sturm-Liouville expansions on the half-line. Our initialmotivation arises from the heat equation -co < x < eo:

- 02U = -ka ; u(z,0) = uo(x); (0 s t < 00). (1)

This of course describes heat flow on an infinite rod. The equation on a finite ,,rsemi-infinite interval with homogeneous boundary conditions can be treateAusing similar techniques to those presented here. Our development will bedirected by the following questions.

QUESTION 1: In what sense does

u(z,t) -+ uO(x)? (2)

The answer will be the best possible in a much wider context; namely, (2)will be shown to hold for a large class of differential operators replacing(a2)/ (Ox 2), in all LP-spaces as well as almost everywhere. Related results forthe heat equation and eigenfunction expansions in general have been studied byBenzinger [1970, 1979]. This problem can, of course, be viewed as one of conver-gence to boundary values in partial differential equations, and our techniquescan in fact be adapted to study certain elliptic boundary value problems. Therehas been recent interest in this type of result, for example, in the study of theLaplace and heat equations on domains with C 1 boundary; see Fabes and Rivire[1979] and Jodeit [1979].

It is well known that forward time translation in (1) is infinitely smoothing,i.e., that if u0(x)ELP, then u(x,t)EL1 in z for t > 0 and any m > 0. Here, L isthe LP-Sobolev space of order m, i.e., functions with m derivatives in LP. How-ever, if (1) is replaced by, say,

02 a(-a 2 + q (z))u = - - (3)

or a more general even-order operator with singular coefficients is used, thensuch smoothing fails to occur. An equation such as (3), incidentally, describesheat flow, with a position-dependent rate q of heat loss; a physically interesting

*Departrnent of Mathematics, Boston University, Boston, Massachusetts 02215.

148

"drift term" a (z) _u relevant to heat diffusion in a fluid, can be added as well.

Thus, we will also address a general version of

QUESTION 2: How does the smoothness (in x) of solutions in (2) depend on that of4'?

The question of smoothing has been studied in Schrodinger semigrouptheory (see Simon [1982] and Carmona [1979]), as well as for hyperbolic equa-tions, where extremely precise results have been obtained (see Marshall,S&,rauss, and Wainger [1980] and Peral [1980]). It divides neatly into a simpleand a difficult problem, namely, that of smoothing in the scale of LP spaces, andtiat in the scale of Sobolev spaces.

The solution of (3) is, of course,u(z,t) = e-Uuo(z). (4)

Alternatively, if A is selfadjoint (for example),

u(z,t) = fe'v(x.x)iZo(x)dp(x). (5)

where v(xA) is an eigenfunction of A, 7o is a generalized Fourier transform ofu, and p(X) is the spectral function. Equation (5) shows that Question 1 is actu-ally one in summability theory. The integration is over the spectrum of A, whichin the case of the full line contains a twofold copy of R; it may equivalently beconsidered a sum of two integrals over the real line (see Levitan and Sargsjan[1975]).

For brevity we will concentrate hei e on results, and sketch some proofs.

2. Convergence to Initial ValuesConsider the semigroup generated by

A = (-1)m/2 + bm-,( ) 1 + ... + b o(x), (6a)

where m > 0 is even and

b(x)ELw + Lr (d = sup + < m). (6b)ism -1 Tj

The sum in (6b) means that b must be expressible as a sum of two functions inthe indicated LP-clAsses.

Let u(x,t) = e-tAuo(x), and define the kernel Kj(x ,y) by

e~4u = Kt (zy)uo(y)dy.

Our approach to studying K, (as well as other analytic functions of A) first leadsto analysis of the resolvent RC = (t-A)-1, for (EC.

THEOREM 1 (cf. Gurarie and Kon [1984]). The LP (1 s p s min r1 ) spectrum u(A)of A is contained in a complex domain

a

n = = pe": Cpn < | sin 2 12,

mIere d is given by (6b). Outside 0, the resolvent RC = (-A)- 1 has a kernelbounded by

147

| Rt(z,y)|I s h(z -y),

sm-212(7

where

h(X) = IZ - - , (8)

with t any number greater than 1.

We sketch the proof. Let

AD = (- 1) 2 d , B = bi(z) (-)i,

and R" = ( -A)-1. If appropriate cornergence occurs, we have the standardrepresentation

RC = R (BRa ). (9)R=0

Under Fourier transformation R is multiplication by ( +'n)-', where C is thedual variable of x. If we insert the definition of B into R (BR )k , and multiplyout, then a typical term in the product will be

RfbiD1RC b 2DieR ... bD'RC , (10)

where bi represents multiplication, Di = ( ), and i1 ,... ,ik are chosen from

(0,1,...,m-1). If K(z) is the convolution kernel of D!R P (i < m), then (4) haskernel

L.. y = R (z)bg(z)Kg1(z) * bP(z)K,(z) *--. * bik(z)K (z -y). (11)

The factors bg can be removed from the integral implied in (11) through itera-tion of Holder's inequality. The remaining convolution of functions, eachbounded by multiples of (8), can be shown again to be bounded by a multiple of(8). Summation of the resulting bound on (11) over all collections i1 , - - - ,i andfinally over all k leads to a geometric series whose sum bounds the kernel of (9)and is bounded by the right side of (7). This constitutes the proof of thetheorem, since the spectrum of A clearly lies in the complement of the domainof convergence of the above sum; this domain of convergence is precisely ~Q.Some consequences of Theorem 1 are discussed below.

THEOREM 2. If A is formally selfadjoint, and min r1 z 2, then it is essentially self-

adjoint on any essential domain of .

Theorem 2 is proved by first using the bounds of Theorem 1 to show that thelower order terms in A are a relatively bounded perturbation of the leadingterm, and then applying the Kato-Rellich theorem (see Kato [1980]).

We can also construct and find appropriate bounds on the semigroup gen-erated by A.

THEOREM 3 (cf. Gurarie and Kon [1984]). The operator A 'enerates an LP-continuous semigroup e~A l p min ra) analytic in the rjht half t-plane,and strongly continuous in L at t = 0.

148

To prove pointwise as well as strong continuity, we construct e-A explicitlyin terms of the resolvent; this will bound the semigroup kernel. Let Dr be thedisk of radius r centered at 0, and 00, = J : arg ,6t. Choose r > 0 and

0 < i< so that OcDrul 1. Thus the positively oriented contour

r = 8(Dr UR,1 ) contains the spectrum of A, as shown below:

We use the Cauchy representation

e-A = 1de.= (12)r--- (12)

By Theorem 1, the integral converges in the uniform operator topology on LP(1 : p min ri) and the kernel K of ea-A is bounded by

| K (z~y) = re-4RC(z,y)d( ,

s c le-CI II m h(I I|(zj-y)) Id(I, (13)

where h is given by (8). Essentially a change of variables shows that1 1

IK(z,y)I s cit I m h(It "(z-y)),uniformly in t for I|argt I s i <tr. This indicates that K behaves like anapproximate identity as t -'0, and a modification of some standard results in har-monic analysis (see Stein and Weiss [1971: Theorem 1.25]) gives Theorem 4.

THEOREM 4. iff A is given by (6), and u (z ,t) solves

149

-Au(x,t) = u(z,0) = uo(x)ELP, (14)

then

u(z,t) = e-Auo(x) -. u0 (z) (15)

in LP (1 s p < min rj) and almost everywhere inz.

If c is an analytic function defined on 1 and its interior, with p(0) = 1, then

So(tA) = 2-i (t )d,2ri r((-A)

if the integration converges absolutely in the uniform operator topology. Thequestion of the convergence sp(tA)f .f is one of summability of eigenfunction

expansions, since, in the notation of (5),

po(tA)f ~ f p(tA)T (X)v (x,X)dp(A).

A statement exactly parallel to (15) holds here, with some minor algebraic decayconditions on o.

3. Smoothing PropertiesWe first briefly consider smoothing of the semigroup in the scale of LP

spaces on R To this end, note that by a simple application of the Dunford opera-tor calculus,

e-tA = 14 e- t (C+)* d (k = 0,1,2,...). (16)2ni f( '-A) (C+A)*

That is, the semigroup is ex pressible in terms of high power of the resolvent.Some investigation of the L -smoothing properties of the convolution kernel in(7), and iteration of these through (16), shows that e-tA is infinitely smoothing,i.e., that e"LC c if LP (1 q s ).

The answer to the corresponding question for the scale of Sobolev spacesdepends strongly on the smoothness of A. This is indicated by (16), since a highpower of the resolvent is itself the inverse of a differential operator only if thecoefficients are sufficiently smooth. We have only a sufficient condition forsmoothing. Let Lm be the Sobolev space of order m, i.e., the functions with mderivatives in LP.

THEOREM 5. If A satisfies (6) and has sufficiently smooth coefficients, i.e., ifb (z)EI, then

e~tA: LP(R)-'+l+1m(R) (17)

is bounded for all t > 0.

We note that one cannot hope that a statement much stronger than (17) willhold, since, for example, the operator

A=- - |z|~ (0<E<1)

has eigenfunctions v (z) ~-c1 |z 2-6 + ca2 + c3 (z -0), so that e-tAu, in general,is not in L2, if u EL2, for e > M. A precise statement of necessary and sufficientconditions for Theorem 5 would be very interesting; it would presumably be inti-mately connected with smoothness of the eigenfunctions of A.

150

-References

Benzinger, H. E. 1970. "Green's function for ordinary differential operators." J.Differential Equations 7:478-499.

Benzinger, H. E. 1979. "Perturbation of the heat equation." J. Differential Equa-tions 32:398-422.

Carmona, R. 1979. "Regularity properties of Schrodinger and Dirichlet semi-groups." J. Functional Analysis 33:259-296.

Fabes, E. B., and Riviere, N. M. 1979. "Dirichlet and Neumann problems for theHeat Equation in C-Cylinders." Harmonic Analysis in Euclidean Spaces.AMS, Providence.

Gurarie, D., and Kon, M. 1984. "Radial bounds for perturbations of elliptic opera-tors." J. Functional Analysis 50:99-123.

Jodeit, M. 1979. "Applications of Calderon's Cauchy integral theorem." Har-monic Analysis in Euclidean Spaces. AMS, Providence.

Kato, T. 1980. Perturbation Theory for Linear Operators. Springer-Verlag, NewYork.

Kon, M., and Raphael, L. 1983. "New multiplier methods for summing classicaleigenfunction expansions." J. Differential Equations 50:391-406. -

Levitan, B. M., and Sargsjan, I. S. 1975. Introduction to Spectral Theory: Self-Adjoint Ordinary Differential Operators. AMS, Providence.

Marshall, B., Strauss, W., and Wainger, S. 1980. "L-L4 estimates for the Klein-Gordon equation." J. Math. Pures et Appl. 59:417-440.

Peral, J. 1980. "LP-Estimates for the Wave Equation." Harmonic Analysis ir.Euclidean Spaces, AMS, Providence.

Simon, B. 1982. "Schrbdinger semrigroups." Bull. AMS 7:445-526.

Stein, E., and Weiss, G. 1971. Introduction to Fourier Analysis on EuclideanSpaces. Princeton University Press, Princeton.

151

PROBLEMS CONCERNING ORTHOGONAL POLYNOMIALS ANDSINGULAR STURM-LIOUVILLE SYSTEMS

Allan MLKraLL*

Abstract

As singular Sturm-Liouville theory progresses, its application to the boundaryvalue problems describing various classical orthogonal polynomial sets is also be-ginning to be known. This article outlines what has been done and what remains.

1. IntroductionAnyone who has taught a course in Fourier series or intermediate

differential equations has encountered such classic problems as the calculationof the Legendre polynomials, eigenvalues, and eigenfunctions. Most such people,however, fail to recognize that many of these problems (such as the Legendrepolynomial problem) are singular and hence do not fit into the context of theregular Sturm-Liouville problem, which is usually discussed at the intermediatelevel. The reason for the failure is easy to understand. Quite simply, until fairlyrecently, the singular problem was not fully recognized, and indeed, even today,there ar e many aspects of the singular problem that remain to be solved.

The purpose of this article is to describe briefly what has been done and tolist some still unsolved problems. We shall begin with the classical orthogonalpolynomials of mathematical physics, which satisfy second-order ordinarydifferential equations and for which the theory is well developed. We also listsome polynomial problems that satisfy higher order differential equations forwhich the theory is less well understood. We conclude with some problems thatare wide open.

2. Second Order ProblemsThe Legendre polynomials satisfy the differential equation

((1-t 2)u')' + X "u = 0.

Traditionally defined on the interval [-1,1], they form a complete orthogonal setfor L2 [-1,1. By so doing, they become part of the blocks forming the solutionsof a number of problems in mathematical physics. On L 2[-1,1], the minimaloperator associated with the expression Lu = ((1-t 2 )u')' is symmetric, withequal deficiency indices (2,2), and so possesses a self-adjoint extension. Thisextension has a domain that is in part characterized by boundary conditions at

1. It is these that have caused considerable confusion over the years. We shallshow how they are derived.

Likewise, the Liguerre polynomials satisfy the differential equation--et (e ~tu')' + A, u = 0.

Traditionally defined on the interval [0,wx], they form a complete orthogonal setfor L2 ([0,oo); e~). The minimal operator associated with Lu = (-e~'u')' hasequal deficiency indices (1,1), and so again it possesses a self-adjoint extension,

VThe Pennsylvania State University, University Park, Pennsy]vania 16802.

152

whose domain satisfies a boundary condition at 0. It also is not well known. Weshall exhibit it as well. There is also a boundary condition at 0o, but since it isautomatically satisfied, we shall not need it.

Finally, the Hermite polynomials satisfy the differential equation

-et(et1"u') + ),u = 0.

Here the interval is (-o,o). On L2((-xce); e 2-) the minimal operator associatedwith Lu = (etu') has deficiency indices (0,0), and is therefore already self-adjoint. Elements in the domain of the operator satisfy automatically a boun-dary condition at both 00, much as in the Laguerre case at 00. Since these con-ditions are automatic, we need not concern ourselves with their exact nature.

How are these facts derived? Let us outline the method. Any symmetricsecond-order differential equation

(pu')' +pou + XrU = 0

can be put in a symmetric system format:

-0 1 r 0 + (0 0 ] .

Writing J,A,B for the matrix coefficients, and y for the vector, we see that J isskew Hermitian b' = -J = J-1 , and A and B are real and Hermitian. Over aninterval [a,b], therefore, we may consider the Hilbert space L)[a,b] generatedby the inner product

b

<yz> = fzAAydt,

and on LJ[a,b], we may consider the differential expression defined byLy = Jy' - By. The minimal operator is symmetric and has equal deficiencyindices. To extend the domain, we need to define appropriate boundary condi-tions in the sense of Dunford and Schwartz [1964].

Let c be an arbitrary but fixed point in (c,b), and let Y = (p be a fun-

damental matrix for the differential equation above. Assume without loss of gen-erality that Y(c) = I, the identity. The Weyl limit-point limit-circle theory hasestablished that for complex X, there exists a solution

in L)[c ,b ], and a solution

= [ I ~

in L2 [a,c ]. The coefficients M and m are the much-studied Weyl coefficients.We can use 0 and 0 to derive new fundamental matrices that are more suitablefor our calculations. Let

and

153

Z= = ] = O L

Then

Y = [ 1(M -[)()

If we compute Green's formula, we find after setting Jy' - By = Af ,Jz' - Bz = Ag ,that

b b<Lyz> - <y,Lz> = fz(Jy' - By)dt -f(Jz' - Bz)ydt = (z Jy)).

a a

In order for the operator generated by the expression L to be self-adjoint, theterm (z'Jy)a must be made to vanish in such a way that constraints on y and zare th'. same and at the same time minimal.

LEMMA 2.1. Let Y, Z ,Zb have conjugate transpose matrices, with X replaced byX,denoted by Y, 2, , respectively. Then

TJY = J, 1JZ5 a J. JZb = J.

The proof is quite simple. The derivatives of these expressions are 0; hence theyare constant. At t = c, they all equal J.

LEMMA 2.2.

-JYJJ = J, -JZJ2 J = J, -JZb J J = J.

PROOF. Note -(JV)(JY) = I. So (JY)-1 is -Jr. Hence these may be permuted.Multiplication by J completes the proof. The proofs for ZZb are the same.

Note that

where Sp = t(t,), o = o(t,a), the same A as in Zb, and not its conjugate. Let usnow consider the term (z'Jy)(b). Replacing J by (-JZb )J (Z4J), we find itbecomes

z (-JZb) J( J)y = (ZJz)*J( Jy).

Let us examine the expressions in parentheses. Since JZb' = (XA+B)Zb, Zsatisfies

-'J = '(XA+B).

Recalling Jy' = By + Af , we find

(4Jy)' = (a'J)y + 4(Jy'))

=- (XA+B)y + $(By+Af)

=f"A (f - Ay).

In component form this becomes

154

yli '-y2'P , _ (f 1-y1)1c'-y29 ~ pr (f 2-Xy i)'

Certainly the top component on the right is integrable. If Sp is likewise inL 2([ab];r), then the second component on the right is integrable as well.

THEOREM 2.3. Let bea solutionof (u')'+ p ou + Xru = 0, ImX 00, which is inL 2 ([a,b];r); let y and [(ply')' + poy]/r = f be in L2 ([a,b];r). Then limW[y,,t]

([ ]1Ylyt-+b

exists.

Conversely, if y and [(ply'))' + poy]/r are in L2 ([a,b];r) and limW[y,p]

exists for all such y 's, the Sp is in L2 ([a, b ];r).We have essentially proved the first part. The proof of the second may be

found in Krall [1984].The conclusion is that the expressions in

(z'Jy)(b) = (ZJ,)*J( 'Jy)

are Wronskians and have limits at b if and only if the solutions 't,p generatingthem are square integrable. These Wronskians are the generalized boundaryconditions. In any case, regardless of what happens to the components, theexpression (z Jy)(b) exists.

Likewise at t = a,

(z*Jy)(a) = (ZJz)*J(2Jy).

Again the various components have individual limits if and only if the solutions,Sp are square integrable.

THEOREM 2.4 (Green's Formula).

<Ly,z> - <y,Lz> = (Z'Jz)*J( aJy)(b)

- (ZZJz)*T((Jy)(a).

The situations at a and b depend on how many solutions ,p and ', ,respectively, are square integrable. We cite Littlejohn [1984] and Krall [1984].At t = b , if both ,1,cp are square integrable, then the simplest requirement is toset W[y,'i](b) = 0. Since the Green's formula term at b is

W[z,fc]W[y,9&] - W[Z,']W[y,so],

if Wronskians of both y and z with ,t vanish, then the entire term disappears.Actually, this can be generalized considerably [Krall 1984]. In particular, wenote that the equation W[y ,' '] = 0 is independent of the eigenvalue parameter X.

At t = b, if only 'p is square integrable, then W[z,t](b) and WJy, ](b) van-ish so rapidly that the entire term vanishes, even though W[z,o] and W[y,p]may become infinite. Again see Krall [1984].

The same occurs at t = a. So, for simplicity, we require thatW[y,1i])b)= 0, W[y,'7](a) = 0 in either case. The actual proof then that theextension of the operators L with domain so constrained is self-adjoint is stillquite complicated. The statements of the previous paragraph as well as self-adjointedness require the use of a Green's function. Again we cite Krall [1984].Nonetheless, at least the nature of the boundary conditions has been exhibited.

The boundary conditions for the orthogonal polynomials are as follows. Forthe Legendre polynomials, require

155

lim(1-t2 )u'(t) = 0, lirn(1-t2 )u'(t) = 0.t-.it+-1

For the Laguerre polynomials, require

limt(2u'(t)-u(t)) = 0.t -.0

At o, the constraint is automatic. For the Hermite polynomials, the constraintsat ' are both automatic.

We invite the reader to find the constraints for the Bessel operator at t = 0.

3. Fourth Order ProblemsEach of the operators associated with the polynomials of the previous sec-

tion can be squared to generate a fourth order problem. Only the Legendresquared operator has actually been considered [Krall and Fulton, 1982], and itneeds to be revised. In addition to these, three other sets of orthogonal polyno-mials satisfy fourth order differential equations. Found by Krall [1940], theyhave been named the Legendre type, the Laguerre type, and the Jacobi typepolynomials. They are of special interest since their boundary constraints areA-dependent.

The proper way to discuss these is also in a system format. Since everysymmetric fourth-order differential equation

(P2y ")~ + (ply')' + (Poy) + Xry = 0

is equivalent to

000 0-1 0 0 Wi0 0 1 0 2, 0000+0 -P1 0 1 20 -1 0 0 us [~ 0 0 0 + 0 0 -1/P2 0] 3 'S00u 0 0 1 0 Q

the theory outlined in the previous section holds fairly well, with only a few com-plications. However, none of these complications have been explicitly workedout. As a consequence, our description of the boundary value problems associ-ated with these polynomials is less than satisfactory.

The Legendre type polynomials satisfy the differential equation

((t2 -1)2u")" + 4((a(t 2-1)-2)y')' ++ Xy = 0.

The boundary conditions can be expressed as

Bau'(1) = Xu(i)

-Bau'(-1) = au(-1).

The Hilbert space setting is L 2 (-1,1)®R®R, required because of the A-dependentboundary conditions. These conditions really should be rewritten as indicatedearlier.

The Laguerre type polynomials satisfy the differential equation

(t ze-tu"s)"- (([2R+2]t+2)e-ty')' + e-tX4y = 0.

The boundary condition required is

-2Ru'(0) = Au (0).

This should be recharacterized as well. Since A again appears in the boundarycondition, the setting required is L 2[[0,o);e t ]®R.

156

Finally, the Jacobi type polynomials satisfy the differential equation

([(1-t)0+4 - 2(1-t)a+3 + (1-t)a+2]U~)~

+([(2a+2+2M)(1-t)a+2 - (2a+4+2M)(1-t)a+1]y')'

+(1-t )aA y = 0.

The boundary constraints are more complicated.

1. -2Mu'(0) = Xu (0).

2. If [ - , -] is the Laguerre bilinear concomitant associated with thedifferential equation, then

(a) lim[u,1] = 0, lim[u,t] = 0, -1<a<1.t-+1 t-~1

(b) lim[u,1] = 0,1 s a< (3.(c) no requirement, 3 : a <a.

For those who are familiar with the language, we say that 0 is in the limit-3 case.The point 1 is in the limit-4 case in (a), in the limit-3 case in (b), in the limit-2case in (c).

All of these need to be recharacterized in a singular format. We cite Krall[1981] for further information concerning these polynomials.

4. Shxth Order ProblemsLittlejohn [1982, 1984] and Littlejohn and Krall [1982] have come up with

two sets of orthogonal polynomials that satisfy sixth-order differential equations.The first of these, the H. L. Krall polynomials, satisfies the differential equation

((t2 -1)3u"')"' + 3(AC+BC)((t2 -1)u")"

+6((t2 -1)(t 2 -3)u)" + 12ABC2((t 2-1)u')'

+8AC((t2-1)(t 2 -3)u)' + 6BC((t2 -1)(t2 +3)u')'

+24u" + Antu = 0.

The boundary constraints are

24BOu"(1) + (24ABC2+24BC)u'(1) = Xu(1),

24ACu"(-1) + (-24ABC2-24AC)u'(-1) = Xu(-1).

The setting for these is L2[[-1,1];c ]®R®R.

The second set satisfies the differential equation

(t4e~ u"',)'"-6((t2+t )e-t u")"

+ ((6t 2+12t+12)e~tu')' + 6R(t 2e-tu')' + X u = 0.

One boundary condition is required:

12Ru'(0) = Au (0).The setting is L2 [[0,oo);e- t ]®R.The boundary conditions here need further workin a system context. The sixth-order differential equations can be written in theJy' = (XA+B)y system format. We cite Walker [1974].

157

5. Remarks

There are many problems suggested by the outline presented here.

1. What are the singular boundary conditions in the system format for theproblems described in Sections 3 and 4?

2. Can one develop a simple theory of singular Sturm-Liouville boundary valueproblems? Niessen [1971, 1972] has paved the way, but his papers are for-midable.

3. How far can one go in classifying such polynomial boundary value problems?In particular, are there other sixth order problems unrelated to the secondand fourth order problems?

4. Kaper, Kwong, and Zettl i[1984] have come up with a way of regularizingsingular points. How does this apply to these problems?

These questions shold keep us busy for some time to come!

References

Dunford, N., and Schwartz, J. T. 1964. Linear Operators, Part II. lnterscience,New York.

Kaper, H. G., Kwong, M. N., and Zetti, A. 1984. "Singular Sturm-Liouville prob-lems with nonnegative and indefinite weights." Monat. Math. (to appear).

Krall, A. M. 1981. "Orthogonal polynomials satisfying fourth order differentialequations." Proc. Roy. Soc. Edinburgh 87:271-288.

Krall, A. M. 1984. "Self-adjointedness in the Weyl limit cases: the role of boun-dary conditions." Quaes. Math. (to appear)

Krall, A. M., Shaw, J. K.. and Hinton, D. 1983. "Boundary conditions fordifferential systems in intermediate limit situations." Proc. 1983 Confer-ence, University of Alabama, Birmingham, pp. 301-305.

Krall, A. M, and Fulton, C. 1982. "Self-adjoint 4th order boundary value problemsin the limit-4 case" (Symposium on Ordinary Differential Equation Ops.1982). Lecture Notes in Mathematics, Vol. 1032. Ed. W. N. Everitt and R. T.Lewis. Springer-Verlag, New York, pp. 240-256.

Krall, H. L. 1940. "On orthogonal polynomials satisfying a certain fourth orderdifferential equation." Penn. State Coll. Studies 6:1-24.

Littlejohn, L. L. 1982. "The Krall polynomials: a new class of orthogonal polyno-mials." Quaes. Math. 5:255-265.

Littlejohn, L. L. 1984. "An application of a new theorem on orthogonal polynomi-als and differential equations" (preprint).

Littlejohn, L. L., and Krall, A. M. 1982. "A singular sixth order differential equa-tion with orthogonal polynomial eigenfunctions" (Proc. Conf. on OrdinaryPartial Differential Equations, Dundee 1982). Lecture Notes in Mathemat-ics, Vol. 964. Ed. W. N. Everitt and R. D. Sleeman. Springer-Verlag, NewYork, pp. 435-444.

158

Niessen, H. D. 1971. "Zum verallgemeinerten zweiten Weylschen Satz." Arch.Math. 22:648-656.

Niessen, H. D. 1972. "Greensche Matrix und die Formel von Titchmarsh-Kodairafur singulare S-herrnitesche Eigenwertprobleme." J. reine. ang. Math.261:164-193.

Walker, P. W. 1974. "A vector-matrix formulation for formally symmetric ordi-nary differential equations with applications to solutions of integrablesquare." J. London Math. Soc. 2:151-159.

159

SPECTRAL THEORY OF ELLIPTIC PROBLEMS WITH INDEFINITE WEIGHTS

Michel L. Lapidus *

Abstract

Consider the homogeneous Dirichet boundary value problem -Au = p(z)u on abounded open set 0 in I (k z 1), where the weight function p changes sign in 0.Let >, (k.) be the nth positive (negative) eigenvalue, and let N+(A) (N.(A)) be thenumber of positive (negative) eigenvalues larger than A (smaller than -A).

In this note we give the leading terms of the asymptotic expansions of Ak as n+mand N (a) as A-.o, as well as estimates for their remainders. We also give lowerbounds for I A I and Nt(X) which hold for all n.

1. IntroductionLet p be a real-valued function defined on a bounded open set 0 of R'

(k z 1). We consider the linear eigenvalue problem

(P) -Au =Ap(X)u, XED,

with (homogeneous) Dirichlet boundary conditions: u = 0 on 80, in the varia-

tional sense, where 80 denotes the boundary of 0 and A = E=2/ az 2. We shallassume that p changes sign in 0, in a sense to be made precise below; for thisreason, p is often called an indefinite weight function in the literature. Such aneigenvalue problem is typically obtained by linearization of a semilinear ellipticproblem; for instance, once linearized about the origin, the nonlinear eigenvalueproblem

au = Xf (x,u), xEO,

with f (x,0) = 0 and Dirichlet boundary conditions, yields the above linear prob-lem with p(x) = Of /Ox (x ,0); and, clearly, p(x) need not keep a constant sign inthis case.

These linear and nonlinear problems have recently been the object of muchattention. (See, for example, the review article of de Figueiredo [1982] and thereferences therein, or the paper by Hess and Kato [1980]; a survey of earlyresults on related linear Sturm-Liouville problems (k = 1) can be found inBocher [1913]. They are of current interest in applied mathematics, physics,and engineering (see, for example, Dee, Grube, and Harper [1972], Kaper,Kwong, Lekkerkerker, and Zettl [1984], Ludford and Robertson [1974], and Lud-ford and Wilson [1974].)** From a mathematical point of view, they lead to somerather interesting questions in the theory of partial differential equations andoperator theory.

*University of Southern California, Department of Mathematics, DRB 306, Los Angeles, California90089-1113, and Mathematical Sciences Research Institute, 2223 Fulton St., Room 603, Berkeley, Cali-fornia 94720. This work was partially supported by a grant from the USC Faculty Research and Inno-vation Fund and by Argonne National Laboratory; it was completed while the author was a member ofthe Mathematical Sciences Research Institute at Berkeley.

"I am grateful to Hans G. Kaper for providing me with these references on the applied litera-ture.

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We shall now focus our attention on the linear problem (P). Partly becauseof the potential applications to nonlinear problems, we want to consider weightsthat may be discontinuous. Precisely, unless otherwise specified, we assumethat pEL(f) with p z k /2 if k 3 and p = 1 if k = 1. Moreover, we supposethat n4 = JzEQ: p(x) > 0J and 0_ = xE0: p(x) < QJ have positive Lebesgue meas-ure in J . The scalar A is said to be an eigenvalue of the Dirichet problem (P) ifthere exists a nonzero u in HJ (0) satisfying the equation -Au = )ypu, in the dis-tributional sense;* accordingly, u is then called an eigenfunction of (P) belong-ing to A. Under the above assumptions, it is known [Manes and Micheletti, 1973;de Figueiredo, 1982: Chap. I] that (P) has a countable set of positive and nega-tive eigenvalues, written in increasing order according to multiplicity:

In addition X;1 is given by the "max-min principle":

1-= max minif pu2: Igrad u|IIz(f) = 1, ( 1)

where F~ runs through the family of all n-dimensional subspaces of HJ (0). Notethat, by the obvious symmetry of the problem,

X-.(p) = -A(-p). (1.2)

Many authors have been studying almost exclusively the positive eigenfunc-tions of (P); this is entirely justified in the case when an eigenfunction mustrepresent a population density, for example. (See, e.g., Brown [1983].) Recallthat a positive eigenfunction of the Dirichet problem must necessarily belong tothe first eigenvalues Al or A- 1 , as can be seen by use of the Krein-Rutmantheorem (see Manes and Micheletti [1973] and Hess and Kato [1980]). For cer-tain applications, however, the eigenfunctions of interest do change sign - sincethey represent, for instance, some component of a velocity field; hence, it seemsworthwhile to pursue the investigation of higher eigenvalues and eigenfunctionsof (P).

In this paper, we determine the asymptotic behavior of the eigenvalues Xtnof (P) and, under more stringent hypotheses, obtain corresponding remainderestimates. We also give lower bounds for I AXa I valid for all n. In addition, wetry to indicate some of the many problems that remain to be solved. For exposi-tory purposes, we present here the case of the Laplacian with Dirichlet boundaryconditions; under appropriate assumptions, however, most results hold forsecond-order elliptic operators with mixed homc eneous Dirichlet-Neumannboundary conditions. Because of space limitations, the preliminary findingsabout the eigenfunctions of (P) which were reported :t the Argonne workshopwill be presented elsewhere. Mathematically, our long-term objective is toextend to the case of indefinite weights some of the main results of the classicalspectral theory of linear second-order elliptic operators with weight p = 1 inbounded as well as unbounded domains of Re. For the "classical theory," theinterested reader might consult Agmon [1965], Gilbarg and Trudinger [1977],Courant and Hilbert [1953: Chaps. IV-VI], Reed and Simon [1978], Osserman andWeinstein [1980], and Yau [1982].

The notation H1(0) stands for the Sobolev space of functions u EL2 (0) with distributionalderivatives grad uELf(0); roughly speaking, the subspace H (0) is the set of functions in H1 (O) thatvanish on 00. (See Lions and Magenes [1968] and Adams [1975].) If one wants to consider Neumanninstead of Dirichet homogeneous boundary conditions, one must replace Ha (0) by H1(0).

161

2. Asymptotic Behavior of the EgenvaluesFor X > 0, let N+(X) be the number of positive eigenvalues of (P) less than or

equal to X; similarly, N_(X) denotes the number of negative eigenvalues X.larger than or equal to -X. In the following, IBI (resp., IntB) stands for theLebesgue measure (resp., the interior) of BcR. Here and thereafter, we adoptthe convention according to which at = bt means th.t. a+ = b+ and a... = b _;also, n will always be a positive integer.

Our first theorem -which extends a famous result of Hermann Weyl [Weyl,1911; Courant and Hilbert, 1953: Theorem 14, p. 435; Reed and Simon, 1978:Theorem XIII.78, p. 271] - gives the leading term of the asymptotics of X asn-+o, or, equivalently, of Nt(X) as X-+ .

THEOREM 2.1. Assume that I0+\Int (Q+) > 0 and I0-\Int (0..)I > 0. Then we;ave

()k X2 p /2], as X-+c (2.1)

and

|X .n~Ckn2/k11p.:1i -1e), as n -+, (2.2)

wuere Bk is the volume of the unit ball in RH and C denoies Weyl's constant;recall that Bk = ITk/ 2/1F(1+k/ 2) and Ck = (2nr)2(B )- .

REMARK. Note that when p is continuous and changes sign in 0, Theorem 1.1implies that (2.1) and (2.2) hold for any nonempty open subset 0 of RI; for, inthis case, 0+ and f2_ are open and the conditions I 11 > 0 and I 0t\Int(0 ) j = 0are automatically satisfied.

Before giving the proof of Theorem 1.1, we need to recall two simple com-parison principles; when necessary, we use the notation Xn(p;0) or N(X;p,0) toemphasize the dependence on the weight function p and the open set 0.

LEMMA 2.1 (Monotonicity with respect to the weight). If Pi P2 p..e. inf 2and ifX (p2 ;0) exists, then X(p 1 ;1) exists and X 2(p1 ;0) s X1(p2;:f).

LEMMA 2.2 (Monotonicity with respect to the open set). If 01cC)2 and if X,(p;QJ)exist for j = 1,2, Then ,,(p;C1) z X(p;02)"

These principles are "physically obvious" and well known in the case of posi-tive weights [Courant and Hilbert, 1953: Theorem 3, p. 403 and Theorem 7, p.411; Reed and Simon, 1978, Proposition 4, p. 270]. Mathematically, they followimmediately from the max-min formula (1.1); since the latter still holds, theirproof remains unchanged in this context. (See also Weyl [1911, pp. 58-62].)

PROOF OF THEOREM 2.1. The idea of the proof is rather simple and can be put touse in similar situations. Assume that we know the result for positive weights;we then reduce the problem to this case by means of the above monotonicityprinciples. Indeed, we "trap" N+(X) between two expressions having the sameasymptotic behavior.

In view of (1.2), it suffices to establish (2.1) for N+(X) since (-p)+ = p.; theestimate (2.2) will then follow since N+(X) = n.

Fix e >0. Since p p + e, we have by Lemma 2.1

N+(X;p,0) < N+(X;p+ + Ef). (z.3)

182

Set D = Int(f+). Since Dc() and I D+ I > 0, it follows from Lemma 2.2 that

N+(X;p,D)cN+(X;p,Q). (2.4)Combining (2.3) and (2.4), we obtain

N+(X;p,D) s N,(a) s V+(a;p+ + e,D). (2.5)Note that p (resp., p+ + e) is positive in D (resp., 0). By the "classical result" forpositive weights, essentially due to Weyl (see Lemma 2.3 below), we have

. N4 (A;p,D) = B (k /2hm = 27fkpk(2.6)

and

N+(X ;p+ +,0) Bk f(p+ k/2lim=Xk/2 (2r)k +e)(2.7)

Consequently, the equations (2.5) through (2.7) imply

Bkkf p/2 slim inf!N+(X)(27r)k D+/2

N+(k) Bks lim sup j (p+ + E)k/ 2 . (2.8)a+/a (2r)k

It follows from Lebesgue's dominated convergence theorem thatfco+ + e)k/2.f (p+)k/2 as e-+0; recall that, in particular, (IpI + 1)PEL'(0) with

p>k/2ifk 2andp =1ifk = 1.Now observe that

fpk/2 = f Pk/ 2 = [ (p )k/2.

for 110+ D I = 0 and the zero set of p does not contribute to the latter integral.We thus obtain (2.1) by letting e-+0 in (2.8). / / /In the course of the proof of Theorem 2.1, we have used the following

lemma.

LEMMA 2.3. Theorem 2.1 holds for positive weights.

PROOF. Since ihis is essentially known, we only outline the main steps. One mayargue as follows:

1. Theorem 2.1 holds for any positive continuous function p on 0. Indeed,when p = 1, this follows from Metivier [1977: Theorem 5.12, p. 188]* and, for posi-tive p in C(0), this is obtained by substituting the operator -A/p for -0.

2. Theorem 2.1 holds for any positive pELP (0). One proceeds much as inReed and Simon [1978: Proof of Theorem XIII.80, p. 274]; one approximates p inLk/2(O) [and possibly L 1(0)], by continuous functions with compact support in 0.To do just this, one makes use of an a priori estimate of the type

N+( a) s C Ak/2 ||p| AI,

*In his memoir Metivier [1977], by refining the method of Courant-Weyl, extended Weyl'stheorem to more general open sets (and operators).

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where c is a constant depending only on k, as well as of the following inequality,which results easily from (1.1) and Lemma 2.1:

N+(X;p 1 + P2)< N+(X;jp 1 + p2 I) s N+(X; ;p1 I) + N+(X;Ip2 D).Actually, Lemma 2.3 could be obtained directly by combining Theorem

X[II.80 of Reed and Simon [1977: p. 274] and Step (iv) of Li and Yau [1983: pp.317-318]. ///

Since, in particular, Xn = O(nr/k), we derive from Theorem 2.1 the follow-ing.

CoROLLARY 2.1. Under the assumptions of Theorem 2.1, the "zeta function"

A(a) = (X)-+ (X,)-a and the "eta function" 77(a) = E(X)-an=1 n=1 n=

- (Xn)~ are well defined for all complex numbers a with Rea sufficientlyn=1

large; here z~a is defined by cutting the complex plane along the negative ima-ginary axis.

REMARKS.

2.1. Theorem 2.1 was announced by Lapidus [1984]. From that paper werecover Theorem 1 (p. 266) by recalling that the boundary of a (Jordan) con-tented set (see, for example, Loomis and Sternberg [1968: Chap. 8, 6-7]; andReed and Simon [1978: p. 271]) has Lebesgue measure zero [Loomis and Stern-berg, 1968: Proposition 6.1, p. 332].

2.2. Note that no assumption has been made about the zero set of p in f. More-over, 0,, +, and 0_ need not be connected.

2.3. We deduce from Theorem 2.1 that the positive (resp., negative) eigenvaluesof (P) have the same asymptotic behavior as the eigenvalues of the ellipticoperator -A/p in Int(0+) [resp., -A/p in lnt(C_)].

2.4. The physical intuition that led us to Theorem 2.1 is the following: as is wellknown, for positive p the eigenvalues of (P) represent the fundamental frequen-cies of a vibrating membrane with mass density p; in the present case, the mem-brane has both positive and negative "frequencies," the large values of which aredetermined by its positive and negative "mass distributions."

After this work was completed, the author learned of several referenceswhere results related to Theorem 2.1 could be found: the article of Pleijel 1942](treating by variational methods the case of a continuous weight on (cI) andthe paper of Birman and Solomyak [1979] (studying by techniques pertaining tothe theory of pseudodifferential operators and differential geometry a moreabstract problem which, when specialized to the present setting, corresponds toa C weight in a smooth domain OcRk). The work of Birman and Solomyak, likethat of the author, must have been conducted independently of Pleijel's since itdoes not cite it. Indeed, its situation at the confluent of many mathematicalareas is what makes this subject very interesting.

We now give two instances of application of Theorem 2.1.

Example 2.1. Let 0 be a bounded open subset of R'. Let 0, f_ be two disjointmeasurable subsets of 0 of positive measure such that I D Int(0,) I = 0. Setp(z) = +1 if z E[), -1 if xE(L, and 0 otherwise. We then obtain a natural exten-sion of Weyl's formula:

164

stn~C (Inl)2/k as L-n+m . (2.9)

Equation (2.9) shows that the spectrum [i.e., the set of eigenvalues of (P)]determines the volume of 0+ and 0_. How much more information can beobtained from the spectrum? (For the classical isospectral problem, see, forexample, Kac [1966] and Yau [1982: pp. 23-24].) More generally, this suggeststhe following question.

Question. To what extent does the spectrum determine the sign of the weight?

Example 2.2. Let 0 = Jz ER: Ix < <1; and p(x) = I j2-2sign(lx I-1), where Ix Idenotes the Euclidean length of x and P < 1; note that p may be quite singular.It then follows from Theorem 2.1 that, as X-a+ao,

(Bk) 2 1(2r) 1-'

(Bk) 2 Ak(1-a)-1(27)k 1-

3. Remainder EstimatAsWe now indicate how, by the method developed in the proof of Theorem 2.1,

one can obtain further information about the asymptotic behavior of N+(X).Depending on the assumption made on p and 0, one can arrive in this manner atvarious kinds of error estimates. We give such an example below.

We make the following hypotheses:

(i) I 0 Int(0+)ulnt(0.))I = 0 and Int(0+), Int(0+)vint(0-), have "finite (k -1)-dimensional boundary."*

(ii) The restriction of p to Int(0+)uInt(0Q) is Holder continuous of order h,hE(0,1], and is bounded away from zero. Moreover,

|p(z) s Cst.[dist(z,Int(C_))]h , x EInt(0+).

Note that p need not be continuous in 0. We now state the following theorem.

THEOREM 3.1. If (i) and (ii) are satisfied, we have the remainder estimate

N+(X) = (27)-}BkXk/2[f(p+)k/2] + O(X(k-)/2), as \-*++0, (3.1)

Idwere r = h/ (h+1); with the obvious changes, a similar estimate holds forN4(X).

PROOF. In view of (i), we may assume that 0 = Int(0+)uInt(0). Let 0 < r s 1. SetD = Int(0+) and po(A) = (2r)-BkAk/ 2. We deduce from (2.5) that

N+(a;p,D) - p(X)IL|II|| A(D) N+(a) - c(X)IIP+II|L.A(fn)

"A subset u of 18 is said to have "finite (k -1)-dimensional boundary' if e o sup-1c6<+-, where

c.6=(zt Ec: dist(z,8&)<6j. (See ILEtivier [1977: p. 194].)

165

SN+(X,;p+ + E,Q) - oAI,)!p+ + iILk/2e(f) (3.2)

+ p(X)[I|p+ + eLII /2(f -) Ip,|| (a)]-

Note that if a function is bounded away from zero and Holder continuous oforder h, so is its inverse. It then follows from (ii) that 1/p [resp., 1/ (p.++)] isH6lder continuous of order h on D (resp., 0). Hence Theorem 6.1 of Metivier[1977: p.1 9 5] yields

IN+(a;p,D) - p(X)IIpLoteP|(f) I S),

IN+(X;p+ + c,Q) - (X)|jp+ + IIL(/2(f) I<'(X), (3.3)

for all sufficiently large X; here, 1'(X) = Cst.X(k-)/ 2, for some constant indepen-dent of a since 1/ (p, + s)i|L. is uniformly bounded in e. We now obtain (3.1) byfixing X large enough, inserting (3.3) into (3.2), and letting e-0. //

REMARK. Under a different set of assumptions, one could deduce error estimatesof the form O((- 1 )/2 ) or O(X(k-1)/2logx).

4. Lower Bounds for the EigenvaluesWhereas the last two sections gave results concerning the large eigenvalues

of (P), the following theorem - announced by Lapidus [1984] - provides lowerestimates for X~ which hold for all n.

THEOREM 4.1. Assume that 0 is a bounded domain of R 'with C2 boundary andthat k - 3. Set 7x = (k (k -2)/ 4e)(kBk) 2/k with e = exp(1). Then, we have foraL n z 1 and X>0:

Ix tnI a 7kn21 IV) IIk/ E),(4.1)and

kBk (k (k -2)/ 4e )k/2N (X ) xk2f (p )k/2. (4.2)

PROOF. As before, it is enough to establish (3.1) for Xn. Fix n z 1 and e > 0. ByLemma 2.1, Xn(p) e Xn(p.. + ).

Moreover, by Li and Yau [1983: Theorem 2, p. 314] applied to the positiveweight p+ + E,

Xn(p+ + e) 7e2/kp+ + II|kV2/cQ. (4.3)

To obtain (4.1) we now let s-'0 in (4.3) and apply the dominated conver-gence theorem. //

REMARK 4.1. An immediate consequence of Theorem 4.1 is that if I 0Q1 > 0, then|X Tn I exists for all n and tends to +c as n-.+o; for k z 3, we then obtain analternative proof of this fact recalled in the introduction. (Compare, for exam-ple, the proof of Manes and Micheletti [1973: Proposition 3, p. 290] or deFigueiredo [1982: Proposition 1.11, p. 43].)

The result quoted in the proof of Theorem 4.1 is a beautiful estimate of Liand Yau established for positive weights; in the same paper [Li and Yau, 1983], itwas used by these authors to improve the so-called "Cwikel-Lieb-Rozenbljumbound" for the number of "bound states" (that is, negative eigenvalues) of theSchrcdinger operator in R (k 3). (See also Lieb [1980] and Reed and Simon

166

[1978: pp. 101-106].)In the first part of their paper, Li and Yau attempted to solve Polya's con-

jecture. In the present context, and for Dirichet boundary conditions, thelatter would state that

nz CnukhIIP IILb/(n), for all n. (4.4)

From Theorem 2.1 we know that the ratio of the left- and right-hand sides of(4.4) tends to 1 as n gets larger and larger; of course, it can be checked that theconstant y7 occurring in the statement of Theorem 4.1 is smaller that Ck. Evenin one dimension, (4.4) does not seem to be known, whereas Polya's conjecture(p = 1) is easily verified in this case by explicit computation. Needless to say,whoever wants to attempt solving this problem in higher dimensions must beready to face considerable difficulties. For literature on the classical problem,see Polya [1961] where the analogue of (4.4) was established for "tiltingdomains" and the recent works of Lieb [1980] and Li and Yau [1983], where par-tial results were obtained on arbitrary domains.

Finally, we would like to point out a problem about which very little seemsto be known. Assume that 0 is unbounded, say, 0 = RI; moreover, in thedefinition of (P), replace -A by the "Schrodinger operator" -A + V, where V is areal-valued function defined on 0 and satisfying appropriate regularity condi-tions; suppose that V is unbounded from below. Consequently, (P) is now both"left" and "right" nondefinite.

Problem. Obtain, in terms of the properties of the "potential" V and the"weight" p, a partial classification of the types of spectra that can occur in thissituation.

Of course, one must first define precisely what is meant by terms like"essential spectrum" in this setting. This can be done, for instance, by using theconcept of approximate eigenfunctions or the language of indefinite inner pro-duct spaces (Krein spaces). Naturally, the "spectrum" need not be discrete ingeneral. Actually, a try at a few examples will quickly convince the reader thatthe "spectrum" could be quite complicated in this case.

References

Adams, R. A. 1975. Sobolev Spaces. Academic Press, New York.

Agmon, S. 1965. Lectures on Elliptic Boundary Value Problems. Van Nostrand,Princeton, New Jersey.

Birman, M. Sh., and Solomyak, M. Z. 1979. "Asyrnptotics of the spectrum of vari-ational problems on solutions of elliptic equations." Siberian Math. J. (AMSTrans. of Sibirsk. Mat. Zh.) 20:1-15.

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169

J-SYMMETRIC DIFFERENTIAL SYSTEMS

Heinz-Dieter Niessen*

Abstract

For differential systems, the notion of formal J-symmetry is defined in such away that systems arising from formally J-symmetric differential equations areformally J-symmetric in this sense. It is indicated how J-symmetric differentialsystems give rise to a J-symmetric operator in some suitable Hilbert space. Thegeneral theory of J-symmetric operators in Hilbert spaces can then be appliedto this operator to obtain analogous results for J-symmetric systems as they areknown for J-symmetric differential equations.

1. Introduction

Let T be a linear operator in a Hilbert space H with inner product (, andlet J be a conjugation operator on H. (For the problems considered here, Hcan be assumed to be a function-space and I the operator of complex conjuga-tion.) Then T is called J-symmetric if for all y,z in the domain of T

(Ty,Jz) = (y,JTz),

i.e., if

TcJT*J, (1.1)

where T* denotes the ordinary adjoint of T. If equality holds in (1.1), then T issaid to be J-selfadjoint.

The theory of J-symmetric operators in Hilbert space has been developedmainly by Glazman [1957], Zhikhar [1959], Galindo [1963], and Knowles [1980and 1981]. Two of the main results are given below.

THEOREM 1.2 (Extension Theorem). To every J-symmetric operator T there existsa J-selfadjoint extension 7. If A belongs to the regularity field of T, then T canbe chosen in such a way that X belongs to the resolvent set of 7.

Here the regularity field -denoted by 1](T) -is defined to be the set of allcomplex X, for which T-X is boundedly invertible (but not necessarily surjec-tive). The regularity field of a J-symmetric operator may be empty.

THEOREM 1.3 (Modified von Neumann Formula). Let T be a J-symmetric operatorwith non-empty regularity fielc and - according to (1.2) - let r be a J-selfadjoint extension of T with non-empty resolvent set. Then for all X in theresolvent set of 7

DJT.J = Dr + (?-X)--1N((T -X)') + JN((T-X) *).

Here DA and N(A) denote the domain and the null space of A, respectively.

Now let -r be the (in general) quasi-differential operator defined by

*UniversitAt Essen, Gesamhochschule, Postfach 103764, 4300 Essen 1, West Germany.

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177 = Z (-1){(pm-1n ) , (1.4){=0

where the coefficients pt are complex-valued functions defined on some realinterval I. For sufficiently smooth ,x' with compact support in the interior of Ithe equation

('trr,JO) = (,JTr) (1.5)

holds true, where J denotes complex conjugation. Therefore, the minimaloperator To in L2(I) induced by r is J-symmetric. Thus, the theory of J-symmetric operators can be applied to To. For example, by applying the exten-sion theorem and the modified von Neumann formula, all J-selfadjoint exten-sions can be described by boundary conditions, and their resolvents can be con-structed. This application of the general theory of J-symmetric operators hasbeen achieved, for instance, by Glazman [1957], Zhikh&r [1959], Knowles [1981],and Race [1980].

The aim of this paper is to generalize the theory of "formally J-symmetric"differential operators to differential systems. As an example we consider thesystem arising from (1.4) or, more generally, from the equation

T = Awr, (1.6)

where w denotes a (for the moment) complex-valued function. Lety = (,'jEOJ, . . . 7[2ln-']), where the fl1 are the quasi-derivatives of 17. Then (1.6)is equivalent to the system

-y' + Ay = XBy (1.7)

with (2m,2m)-matrices A and B defined by

0 Ii

p1 _ '%0 PO

pm 0

Thus the aim is to define -in a most general way -tne concept of "formallyJ-symmetric" differential systems such that (1.7) in particular becomes a for-mally J-symmetric system and such that there exists a suitable Hilbert spaceand a J-symmetric operator in it describing the differential system. Then thegeneral theory of J-symmetric operators can be applied to this operator inorder to get results analagous to those for differential equations.

For simplicity we assume all coefficient functions to be continuous, e.g., thep{ and w above.

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2. Definition and Algebraic Structure

Let I be an arbitrary real interval with endpoints a and b (-rsa <b soo), andlet n be a positive integer. Then the set of all locally absolutely continuousfunctions from I into C" will be denoted by A, and A denotes the set of allpiecewise locally absolutely continuous functions defined on I with values in C.Here a function y is called piecewise locally absolutely continuous if there existsa finite set 4 cI such that y is locally absolutely continuous on \4y. Further-more, the set of all measurable functions defined almost everywhere on I withvalues in C" will be denoted by M. Obviously, A,, A, and M are linear spaces, thelast one with equality almost everywhere.

Now for i = 1,2 let F, Gs, S be continuous mappings from I into the set ofall complex (n,n)-matrices. Then by

F'y: = Fly' + F2 y, y: = Gy' + G2 y, Sy: = Sy' + S2y,

we define differential operators F, G, and S mapping A, linearly into M. Furth-ermore, for XEC, denote by C the mapping from I into the set of all (2n,2n)-matrices defined by

=F 1-X G1 F2-XG]Ca: = S1 S2

We consider the system FPy = XGy, i.e.,

(F1 -XG1 )y' + (F2 -XG 2 )y = 0 (2.1)

and define it or -more correctly -the pair (F,G) to be formally J-symmetric inthe following sense.

DEFINITIoN 2.2. (F, G)is "formally J-symmetric with respect to S" ifF-AG1: I-+G,,,(C) (2.3)

for all XEC, and there exists a continuously differentiable function H :I-+GiC(C)such that for aLl XEC

C4 E C1 =inH). (2.4)

Then H is called "concomitant" of (FG) with respect to S.

Here 0 denotes transposition and E is the (n,n)-unit matrix. Condition(2.3) ensures that the Cauchy problem for (2.1) is uniquely solvable; equation(2.4) reflects a commutation property (see Eq. 2.6 below).

Equation 2.4 shows that H is uniquely determined by F, G, and S -but notby F and G alone - and that H(x) is skew-symmetric for each z EI. Since H(z)is supposed to be nonsingular, and since each odd-order skew-symmetric matrixis singular, the order n of the system (2.1) must be even.

The system (1.7) can be shown to be formally J-symmetric with respect to asuitably defined S:

Let n = 2m. Then F1 = -E.,,F2 = A, G 1 = 0, G2 = B. Defining

S1: = 0, SZ:=0 j

where E, denotes the (m,m)-matrix with 1's in the second diagonal and O'selsewhere, the pair (F,G) is formally J-symmetric with respect to this S, andthe concomitant equals S2. Thus, the concept of formally f-symmetric systemsgeneralizes that of formally J-symmetric differential operators.

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We use the notation

[y,z ]: = fz(z)y(x)dz

for all y, z E for which the integral exists -not only for square-integrable y andz.

Then the following characterization of formally J-symmetric systems canbe proved.

THEOREM 2.5. (F,G) is formally J-symmetric with respect to S if the followingconditions are fulfilled:

1. For all xEl

a. F1 (x) is nonsingular.b. F- 1 (x)G1 (x) is nilpotent.c. S (z)G2() = G (x)S2(x).d. S1 (x)F2(T)-Ft (x)S2(x) is nonsingular.

2. For all AEC and for all continuously differentiable y ,z EA with compact sup-port in the interior of I the equation

[(F-AG)y, JSz] = [Sy, J(F-AG)z] (2.6)

holds true.(Then H = SIF2 -FjS2 ).

Conditions (a) and (b) are a restatement of (2.3); equation (2.6) iseq uivalent to (2.4) with some H, possibly X-depending and singular valued. Then(c) guarantees that H is independent of X, and condition (d) ensures that H(x)is nonsingular everywhere.

Comparing the commutation property (2.6) for X = 0 with (1.5) indicatesonce more that we have defined a generalization of the concept of formally J-symmetric differential operators. In (2.6) the differential operator S occursinstead of the identity in (1.5).

Another characterization of formally J-symmetric systems, which giveseven more insight into the algebraic structure of the operators F, G, and S isthe following.

THEOREM 2.7. (F, G) is formally J-symmetric with respect to S and with concomi-tant H if the following conditions hold:

1. F1 (x) is nonsingular for all x El.

2. H is a continuously differentiable solution of H' = HT + TH withT: = Fif1 F2 such that H(x) is nonsingular and skew-symmetric for all x EI.

3. There exist continuous functions W. L from I into the set of all (complex)symmetric (n,n)-matrices such that WL is nilpotent everypwhere and such that

C,= 1YS (=1,2) and (2.8)

S1 = LF1, S2 = LF2 - (Fif1 )H. (2.9)

Equation 2.9 and Condition 1 show that L is uniquely determined; Equation2.8 implies the following corollary.

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COROLLARY 2.10. G = WS.

We shall use the same symbol for a matrix-valued function and for theoperator (in M) of multiplication by this function. So in Corollary 2.10, Wdenotes this operator of multiplication.

Let zo be a fixed point of I and let Y be the fundamental matrix of solu-tions of (2.1) such that YA(zo) = E . Then we have the following theorem.

THEOREM 2.11. If (F, G) is formally J-symmetric with respect to S and with con-comitantH, then for all XEC

1. YjHYx = H(zo).

2. (SY,)t (F1 -.XG1 ) YA = H(zo).

The first equation follows by multiplication of (2.4) from the left by Y'

and from the right byY ; the second equation can be proved by multiplying

(2.4) from the left by ,} from the right by and by using Eq. 1.

The second equation especially implies that (SYx)(z) is nonsingular for allz El. In connection with (2.10) this shows that

W = (GY)(SY)-1

is uniquely determined by (2.8). For example, in the problem arising from thesystem (1.7),

W = diag(0,...,0,w), L = 0.

Another consequence of Theorem 2.7 is that (2.1), solved for y', is a polynomialin X of degree n at most. This follows from (2.8), (2.9), and the fact that WL isnilpotent:

(F1 -XG 1)~1 = (F1 -XWS 1)'1 = (F1 -X WLF1)-1

= F 1-1 (E -X WL)- 1 = Ff1'(X WL) .t=o

Suppose from now on that (F,G) is formally J-symmetric with respect to Swith concomitant H and W,L as in Theorem 2.7.

In order to obtain a suitable Hilbert space and a J-symmetric operator in itwhich rejects the properties of the system (2.1), some further restrictions haveto be imposed.

3. Right-definite Systems

DEFINITION 3.1. (F,G) is called "right-semidefinite (with respect to S)" if W(x) ispositive semidefinite for all z EI.

From now on (F, G) will be assumed to be right-semidefinite. Then thereexists a uniquely defined continuous function K on I with values in the set ofHermitian (n,n )-matrices such that for all z EI

K(z) z 0 and K2(z) = W(x). (3.2)

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COROLLARY 3.3. For all x EJ W() and K(z) are real.matrices.

For A' this follows, since W(z) is Hermitian and symmetric by Definition 3.1and Theorem 2.7, respectively. By the symmetry of W(z), Kt is a solution of(3.2), too. Since this solution is unique, K(z) is symmetric. Being Hermitian,K(z) is real.

In the case of Example (1.7), right-semidefiniteness means that the "weight"function w is real and nonnegative. Then K = diag(0....,0,+.w ).

In the general case we define a linear operator U from A into M by

U: = KS. (3.4)

Then (2.10), (3.2), and (3.4) imply the following corollary.

COROLLARY 3.5. G = KU.

We use the following notations.

DEFINITION 3.6.

L2 : = fy Eli/ Iy (' )I 2 is integrable on I; ,

E' = U-1L2 = jyEA, / Uy EL2;,

D: = AnEnF- 2 ,

Fa: = jy EAnE / (F-XG)y = 03 for all XEC,

(y,z): = [UyUz] for ally,zEE.

REMARKS.

a. The space 4a of continuous solutions of (F-XG)y = 0 belonging to E iscontained in YACnD; since F'j = X G = K(X Uy) and Uy EL2 .

b. (y,z) = [Gy,Sz]= [Sy,Gz] for all y,zEE by (3.4) and (3.5), since K(z)is Hermitian. This shows that the positive-semidefinite inner product (,) on E isnothing artificial but that it is given by the problem itself. In the case of (1.7),for example, (y,z) = f 1rntwdz, where 7' and are the first components of y andz, respectively.

c. Obviously, U is a linear isometry from the semi-prebilbert space (E, (, ))into the Hilbert space (L2,[ , ]).

d. The space D plays a role similar to that of the domain of the maximaloperator induced by -r in L2 (f).

GREEN'S FORMULA 3.7. Let y,z ED. Then

1. [F y,JSz] and [Sy,JF'z] exist;

2. <y,z>: = (z'Thy)(b-0) - (z t Hy)(a+0) exists; and

3. [FPy,JSz] - [SyJFz]= <y,z>.

The integrals in (1) exist by the definition D. Then (2) and (3) fol ow by(1) on multiplying (2.4) from the left by [Z) , from the right by YJ, and

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integrating over I.

We now assume that (F,G) is right definite:

DEFINITION 3.8. (F, G) is "right-definite (with respect to S)" if for some XEC thepositive-semidefinite inner product ( ,) is definite on 4.

It can be shown that jyEE/ (yy) = 0j is independent of A. Therefore, if thecondition of Definition 3.8 holds for one A, it holds for all A. Especially one mayassume (, ) to be definite on FO. Then a compactness argument gives the follow-ing lemma.

LEMMA 3.9. (F, G) K right definite if for some compact interval M in the interiorof I the integral f( UYo)* UYo is positive definite.

This lemma can be used to show that for every y ED there exists a u EDwhich coincides with y to the left of M and vanishes to the right of M. This givesa decomposition y = u+v of y into elements of D which vanish in a neighbor-hood of the right and left endpoint of I, respectively. This decomposition is fre-quently used in the proofs of the following results.

In the case of Example 1.7 it can be shown that the problem is right definiteif, apart from being nonnegative, the weight function w is not identically zero.

4. An Appropriate Hilbert SpaceSince (E,( , )) is a semi-prehilbert space, there exists a completion (G,( , ))

of (E,(, )) and a unique continuous extension of U 3 from all of G into L . Thisextension will be denoted by the same symbol U. Then U is a linear isometryfrom (G,( , )) into (L2 ,[ ,1). In view of (3.5), G/s can be extended to all of G by

G: = KU, (4.1)

where again the same symbol G is used for this continuation.

Then the following theorem can be proved.

THEOREM 4.2. L2 = UG ® N(K).

Here N(K) denotes the null space (in L2 ) of the operator of multiplicationby K.

COROLLARY 4.3.

1. UG = N(K)-,

2. KL2 = GG,

3. D = AnEnF-1GG,

4. J(UG) = UG.The first - obvious - part of the corollary shows that one knows UG if K is

known; in particular, no completion process is needed in calculating UG. Thesecond equation follows by applying K to the decomposition of Theorem 4.2 andby (4.1). This equation then implies (3). L2 and -since K is real valued -N(K)are invariant with respect to J. Therefore, (4) follows from Theorem 4.2.

DEFINITION 4.4. H: = UG, k: = K/.

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Then (H.[,]) is a sub-Hilbert space of (L2 ,[ ]); by Corollary 4.3 (4), J is aconjugation on H; U is a linear isometry from (G,(, )) onto (H,[ , ]); and k mapsH bijectively onto GG since HnN( K) = J0 by Theorem 4.2. Furthermore,G = KU (compare the right-hand part of the commutative diagram 5.9).

5. The J-Symmetric OperatorBy Corollary 4.3 (3), F maps D into GG. Therefore, the following definition

makes sense (compare the upper left-hand part of Diagram 5.9):

DEFINITION 5.1. 1: = k-1Fy,

Then F,D = R' = KT, and since J and K commute, Green's Formula 3.7implies the following theorem.

THEOREM 5.2. [Py,JUz] - [ Uy,JIz] = <y,z> for a.l y,z ED.Let I denote the orthogonal complement of D in itself with respect to the

bilinear form < , >:

D: = yED/<y,z> = 0Qfor all z EDj, (5.3)

and denote by I' the set of all y ED with compact support contained in the inte-rior of I. Then Ik'cIkcD, and (5.3) and Theorem 5.2 imply the following corol-lary.

COROLLARY 5.3. [Py,JUz ] = [ Uy, J'z ]for all y Eb andaLLz ED.A converse of this corollary is the following.

THEOREM 5.4. Let f ,g EH and assume that

[ly,Jf ] = [ Uy,Jg ] for all y EIb'.

Then there exists a uniquely defined z ED such that f = Uz and g = l'z.

As a consequence of Corollary 5.3 and Theorem 5.4, we get the following.

COROLLARY5.5. (UIk'= JP(DnN( U)).Here I..denotes the orthogonal complement in H and the null space N( U)

contains all elements of G with norm zero.We now make an additional assumption.

ASSUMPTION 5.6. DnN(U) = 0 J.In terms of U, right-definiteness means that FbnN(U) = Oj. Since by

Remark (a) following Definition 3.6 Fe is contained in D, Assumption 5.6 slightlystrengthens the assumption of right-definiteness. In most cases this additionalhypothesis is fulfilled, especially in the case of the system (1.7) arising from thedifferential equation (1.6) or in the case of systems arising from differentialequations 777 = Xar, where a is another formally J-symmetric differential opera-tor.

Assumption 5.8 and Corollary 5.5 immediately imply the following corollary.

COROLLARY 5.7.

1. UaD UD is bjective.

2. D' is dense in G.

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Now it is possible to define an appropriate linear operator A from DA: = UDinto H by the following definition.

DEFINITION 5.8. A: = '(U,D )-1.

The following commutative diagram elucidates the situation:

FDGD G surj

D G 6

U/D bij r K bijU surj

H > UD-DA A H

From this diagram the following remark arises.

REMARK 5.10.

A = U(G~1F,D) (UD)-1;

that is, apart from the isometries U and U,, A is the relation G~1FD. Thisshows that the operator A reflects the properties of the differential system (2.1).Now let

A0': = A , . , AD = A, D.

Then A0'cAocA and by Corollary 5.7, (2), these operators are densely defined.The bilinear form <, > can be transferred to DA by the definition

<f',g>: = <(U,D)-1f , (UD)~1g > for fg EDA.

Then Theorem 5.2 and Corollary 5.3 can be formulated as follows.

THEOREM 5.11.

1. [AfJg ] - [f ,JAg ] = < f ,g > fr all f .g EDA .

2. [Aof ,Jg ] = [f ,JAg ] for all f EDAQ and all g EDA.

The last assertion shows that AD is J-symmetric and that

JAJcAo. (5.12)

In the same way Theorem 5.4 can be reformulated as follows.

THEOREM 5.13. Letf ,g EH, and assume that

178

[Ao'h,Jf] = [h,Jg] for all hEDAd.

Then f EDA and g = Af . In particular, this implies

(A') *cJAJ. (5.14)

Since Aoc(A0 ')', (5.12) and (5.14) imply the following theorem.

THEOREM 5.15. JAJ = Ao = (A').Consequently, A is closed and A0 = A0'. Since by Theorem 5.11 (1), < , > is

continuous with respect to the graph norm on DA, A can easily be shown to beclosed, too. Therefore, we have the following corollary.

COROLLARY 5.16. A is closed, and AD = A0 '.

Theorem 5.15 also implies that every J-symmetric extension B of A0 is arestriction of A:

BcJBJ CJA;J = A.

Therefore the domain of B equals UC for some linear space C with DbcCcD.Define for such a space the space C0 by the following definition.

DEFINITIoN5.17. C: = jyED/ <y,z> = 0 for all z EC.(D in the sense of this definition is the same as the old Db.)

Then Theorem 5.11 gives the following characterization of J-symmetric andJ-selfadjoint extensions of A0, respectively.

THEOREM 5.18. Let B be a linear operator with A0 cBcA, and let C: = (UD )-1 DB.Then B is J-nsymmetric if CcC0, and B is J-selfa &joint if C = Co.

Since AD is J-symmetric, the general theory of J-symmetric operators cannow be applied to AD. Together with the special structure of the operators A0and A this application yields a lot of new results for our problem. We will sketchonly a few of these results.

First of all, the extension theorem (1.2) shows that there exist J-selfadjointextensions of A0. Another result of the general theory gives that the "deficiencyindex in A", i.e., m(X): = dimR._A is constant in the regularity field P(A0 ). Nowan easy calculation shows that

RoA = N((A0 -X)') = JN(A-X) = JUFK. (5.19)

Therefore, m(X) = dimEA is constant in 11(Ao). In particular, 0 s m(A) s n. Itcan be proved that m(A) , if a or b belongs to I.

For XE 1(A0) the modified von Neumann formula gives in connection with(5.19) and Theorem 5.15

DA = DA 0 + (BA-X)~'JUE, + UEO,

where BA is a J-selfadjoint extension of A0 with A in its resolvent set. Applying(UD)~1 to this equation gives

D = Db + RA JUr + 4. (5.20)

Here RA = (U,D)~1(BA-X)~1: H-D.

If 1(A0 ) # 0, then (5.20) holds for some AEC. Therefore,dimD/ Do = 2m (a) = : 2m, and for every linear space C with DbcCcD there exist

179

WIA....,W2mED linearly independent modD0 such that for some I 2m

C = Db + [w 1, . . . , w], D = C + [wL+1,...,W2m].

Then C = C is equivalent to

I = m, <wi,w1 > = 0 (i,jsrn),C= jyED/<yw{> = 0 (ism){.

This is the description of J-selfadjoint extensions of A0 by boundary conditions:

THEOREM 5.21. Let 11(Ao) 0, and let m: = X dimD/Db. Then DB = U/D C is thedomain of a J-selfadjoint extension B of A0 if there exist w1, .. . ,wm EDlinearly independent modD0 such that

<1kw> = 0 (i,jsm)

and

C = jyED/<y,w 1 > = 0 (ism)l.

In contrast to the situation here, in the case of formally symmetric systemsthere do not always exist selfadjoint extensions; and even if there exist such self-adjoint extensions, it may happen that none of these extensions is generated byseparated boundary conditions. In the case of formally J-symmetric systems,the situation is quite different. If 11(A0 ) s 0, then there always exist J-selfadjoint extensions which can be described by separated boundary condi-tions. We cannot go into details here.

Let us finally consider the "resolvent" RA: = (U,)-1(B-X)- 1 of the J-selfadjoint extension B with X in its resolvent set. It can be shown that RA is anintegral operator from H into D, although (B-X)-1 in general is not an integraloperator. For simplicity, let a EI and define m: = m(X),

= [- 2: = [ E )

Then there exists a constant regular (nn)-matrix P such that the first m solu-tions of ZA: = YAP span F. Let Ho: = Zj(xo)H(xo)Z(xo). With these assump-tions and definitions, the following theorem holds true.

THEOREM 5.22. Q2 H0Q2 = 0 and there exists an (n,n)-matrix T such that for allf EH and all x El

(RfJ)(x) = fG(z,s)f(s)ds,

where

G(z,s): = ZQ(z)- 1H +Q 1 TQ 1 (UZ(s)forssxs1 Z xjQ2H..1Q 1+ Q1 TQ 1} U .()jfor s>x

The matrix T can be determined by the boundary conditions describing B.Theorem 5.22 gives the following necessary condition for X to be in the regularityfield of A0 .

COROLLARY 5.23. If XEI(Ao), then Q2 HOQ = 0.

This also shows that in the case a EI, m(X) = m z -2

180

References

Galindo, A. 1963. "On the existence of J-selfadjoint extensions of J-symmetricoperators with adjoint." Comm. Pure Appl. Math. 15:423-425.

Glazman, I. M. 1957. "An analysis of the extension theory of Hermitian operatorsand non-symmetric one-dimensional boundary-value problems on thesemi-axis." Dokl. Akad. Nauk. SSR 115:214-216.

Knowles, 1. 1980. "On J-selfadjoint extensions of J-symmetric operators." Proc.Amer. Math. Soc. 79:42-44.

Knowles, 1. 1981. "On the boundary conditions characterizing J-selfadjointextensions of J-symmetric operators." J. Diff. Eq. 40:193-216.

Race, D. 1950. "The spectral theory of complex Sturm-Liouville operators." Ph.D.Thesis, Univ. of the Witwatersand, Johannesburg.

Zhikhar, N. A. 1959. "On the theory of extensions of J-symmetric operators."Ukrain. Mat. Z. 11:352-364.

181

POINTWISE EQUISUMMABILITY OF ELLIPTIC OPERATORS

Louise A. Raphael*

Abstract

We present criteria for determining pointwise equisummability of expansions ineigenfunctions of certain pairs of elliptic operators on general domains of r".Applications are given for Sturm-Liouville systems and the heat equation.

In this paper we give criteria for L -equisummability of eigenfunctionexpansions for certain pairs of elliptic operators on W. Namely, L"-equisummability of two elliptic operators, or the pointwise convergence of thedifference of two summability means, is reduced to showing that the differenceof the modified resolvent operators is uniformly bounded. The class of ellipticoperators is a generalization of the class of Sturm-Liouville operators. Our ellip-tic operators have leading terms that are positive and lower-order terms whosecoefficients are singular on nowhere dense sets. Our work is motivated by thefact that it is easier to study expansions in eigenfunctions of unperturbed opera-tors, say, Laplacian = -A on R", than of perturbed ones, -A + q (x).

The prototypical case fur equisurnmability is found in the study of equicon-vergence for differential operators; see Haar [1910], Walsh [1922-23], Birkhoff[1908], and Tamarkin [1912]. That is, the difference between expansions withrespect to eigenfunctions of a Sturm-Liouville operator and the ordinary Fourierseries tends to zero uniformly in every finite interval. In some one-dimensionalcases where equiconvergence fails, one can show (see Stone [1926]; Levitan andSargsjan [1975]; and Benzinger [1970]) equisummability for Riesz typical meansof eigenfunction expansions of differential operators. For equisummability inhigher dimensions, Gurarie and Kon [1983] give conditions under which theexpansion of an IP function in eigenfunctions of an elliptic operator isequisummable with the corresponding expansion obtained from its leading term.

We wil' illustrate our theory with two examples. The first shows that forf EL2 (R), the generalized Fourier transform of f associated with certain classesof Sturm-Liouville problems is analytic summable to f (z) pointwise if and only ifthe Fourier transform of f is analytic summable to f (z) pointwise. The secondapplication is to convergence of solutions of the heat and perturbed heat equa-tions to their common initial value.

1. Equisummability for Differential Operators in IP(RW), 1 s p cWe begin with basic definitions. Let A be a differential operator on

LP(R"f), 1 < p < . Let u(z,X)j be the set of generalized eigenfunctions associ-ated with the eigenvalue X belonging to the spectrum a(A). Let f ELP(W"), andassume that the eigenfunction expansion

f (z)~ )F(X)u(z,X)dp(X) (1)aA)

'Howard University, Department of Mathematics, Washington, D.C. 20059. Research partialysupported by the Army Research Office under grant number DAAG-29-84-G-0004.

182

and its generalized Fourier coefficient

F(X)~ff (z)u(z,X)dz

exist. Here ~ denotes LP convergence as the limits of integration becomeinfinite, and p is a combination of spectral functions. Let o(X be continuous andp(0) = 1. We say that the eigenfuwction expansion (1) is p-summable in anappropriate topology if the summability means

V (sA)f = iF(X) (eX)u(r,X)dp(X)fAA)

as I I -'0, e again belonging to some domain D in C. When the summator func-

tion p(eX) = 1 ---=i, fore=- - andz f-+f as |z |oo,z in1l-tX 1-- ,x zAffsz~rz

D'= I- z eDJ, the eigenfunction expansion is resolvent summ.able to f .Two differential operators A and B on LP(R'), 1 s p <00c, are p-

equisummable from LP (R") into L'(R") if

IIP(tA)f -so(eB)f IILu= |pII(tA)f (z)-p(eB)f (z)| -e0 asIe 1eQ+0

for all f inLP(R").

If summability does not occur with respect to the LP(R') norm, 1 s p < cc,then equisummability from LP (R") to L(R") becomes interesting. Our firsttheorem states conditions under which equisummability from LP (R') to L(R")holds for 1 sp < .

THEOREM 1. Let A, B be closed deferential operators on LP(R') into L(R'"),1 sp <o. Assume z (z -A)~ 1 - z (z--B)~1 is uniformly bounded from LP (R") toL(R") for z in DCC. If z(z -A)- and z(z -B)-1 are unifnrvraly bounded fromL'(R) to L'(Rf'), then A and B are resolvent equisummable fromLP(R')-L-(R') in D.-

Our proof depends on the fact that the space of infinitely differentiablefunctions with compact support, C (W"), is dense in LP(R"), p ? fc; the alge-braic identity

z (z -A)-1f -z = z (z -A)-1[z-1! ]

and a Banach space theorem.The setting for our principal theroems follows. Let K be a set in C. Here 8K

denotes the boundary of K, while K' denotes the complement of K in C. Theregions ., (i=1,2,) defined in Theorem 2 can be informally thought of as con-centric keyholes, the intersection of one of which with the exterior of the othercontains the contour I. These regions will be used also in Theorem 3.

THEOREM 2 (Kon). Let A, B be closed dii erential operators LP(R') for 1 sp < 00.Assume R, f = z (z -A)~'f -z (z -B) f is uniformly bounded fromLP(R")-'L(R"). Let D, =7 izECI |I|z |Isrd, and g= zECI |Iarg z I 3s ,and K = D,.u4for i =1,2 besuch that K2cK1.

Let A and B be resolvent equisummable on the complement of K2 . If rp isanalytic on K1 such that o(0) = 1, c(z) = 0(z -6), (6>0) z in KInK , then A andBare rp-e quisummable in s. ein D = J zIIarg z | <1-2i from LP (R")-L(iR')for 1 p <0.

183

The proof uses the Dunford operator calculus and the Dominated Conver-gence Theorem.

2. Equiuummability of Elliptic Operators on R"

We now present some notation. Let a = (a1, ... , a") be a multi-index, and

Da= -L "81ta |= a.iI z i 8,"

Consider the differential operator

A = E ba(z)Da=Ao+C,

where A0 contains the leading terms and C is the remainder. We assume that A0is constant coefficient positive elliptic [i.e., A0 F baDe and F baza > 0 for

Ial=m Ial=mz = (zi, ... , z,) 0]. We assume that the coefficients of C can be expressed assums of functions in certain LP(I") spaces, i.e., ba(z)eLra + L, Ia I<m, whered = s T' + Ia| <m. We choose for the domain the L Sobolev space L .

(If p is outside 1spsmin ra, then A may not be densely defined.) The nexttheorem is an application of the theory in Section 1.

THEOREM 3 (Gurarie-Kon). Let A = Ao+C and B = Ao+D be closed elliptic opera-tors defined as above on the Sobolev space L f, 1sprminra, where A0 is constantcoefficient positive elliptic containing the leading teams of order m. Assumethe coefficients ba(x) of AB are singular on a nowhere dense set. Let K, andK2 be as in 7heorem 2 and ro be analytic onK1, 9(0) = 1, cp(z) = 0(z-), 6>0, z in KfnK . Then A and B are lp-equisummablefrom LP(R") into L'(R")for p > n/(m-d),mand r in D = jz Iarg z | <'62-61;.

The key technique of the proof is to analyze kernels of the resolvents enduse LI-radial bounds of the resolvents developed in Gurarie and Kon [1984 and1983].

We observe that when A is a differential operator, (z -A)- 1 is LP-smoothing,so we expect equisummability from LY(R") into L"(W). For pointwise equisum-mability when A,B are sp-equisummable, then p(cA)f -f at exactly the set ofpoints where p(eB)f -'f , since (op(eA)-(eB))f converges uniformly to 0. Thuswe state the following corollary.

COROLLARY. If A,B are as in Theorem 3, then 9(eA)f -f at zER" if and only iffo(cB)f -+f at x.

3. ApplicationsIt follows immediately that under the conditions of Theorem 3, the general-

ized Fourier expansion associated with certain classes of Sturm-Liouville prob-lems and the ordinary Fourier transform are analytic equisurmmable.

For example (see Levitan and Sargsjan [1975]), consider the equation-d 2u~z) +q(z)u(z)=Xu(x) xEc(-cooo)

dzand q (x)E(L'(r") + L"(R") is real and continuous. Let u1 (zX) denote the solu-tion of this equation which satisfies the initial conditions

184

uI(0,X) = 0, u(0,A) = -1,

and by u2 (x,A) the solution under the initial conditions

u2(0,A) = 1, a-u2(0,A) = 0.

Let f EL2 (R) and F(A) be the generalized Fourier coefficients of f with respectto uh(z,A), (=1,2. Let p(z), i=1,2 and 77(z) be the standard limits of step func-tions where the jumps occur at the eigenvalues of the boundary value problemson a fSnite interval [a,b]. Here, pt are monotone and bounded, while ri is ofbounded variation in every finite interval. Then we can write

f (z)~fFi(A)ui(z,A)dp1 (X) + fFi(A)u2 (z,X)d17 (X)

+ fF2 (A)u 1(z,A7?(X) + fF(X) 2 (zX)dp 2(X), (2)

where ~ is L2-convergence as the limits of integration become infinite.It is well known that in the case q (z) = 0, (2) is the Fourier transform,

namely,

f (z)~J f i(-)sin /z(f(V) - f (-V))dd

+ f cosv'Xz(f (vX) +f(-v'X))dv, XA 0,

where denotes the Fourier coefficient of f .Now when the summat or function p(A) is analytic and satisfies the condi-

tions of Theorem 3, and the summability means with respect to p of each of theintegrals is absoluteFy convergent, then the jp-equisummability of the general-ized Fourier expansion and Fourier transform follows, that is,

w w

fF1 (x)p(tx)u 1 (z,X)dp1 (X) + fFi(X)o(x)u2 (x,A)d17(A)- w -

+ f F2 (A)P(EA)u(z,A)dr7(A) + fF2 (X)co(eX)u 2(x,A)dp2(A)-+f (z)

pointwise if and only ifw

(ek)f(k)e-dk -f(z), k2=A, asE-+0

pointwise.

For our final example, consider the heat equation Aou = -Au = at-u with

the initial condition u(z,0) = f (z)ELP (I), p z n, and an associated perturbedheat equation

Au* =(-b~ (z)u' = (u'), u'(z,0) = f (z)ELP (R"), p z n.

Here q(z) is the sum of L"(I) and Li(R") functions. So by Theorem 3, when= e-A ,|e4f - e - 0 f j..-+0 as t -+0. In other words, the solutions u(z,t )

18511&,

and u.(z,t) converge to f (z) as t -+0 at the same set of points z in R'.

Proofs of the theorems will appear elsewhere.

Acknowledgments

The author wishes to thank Drs. Hans Kaper, Man Kam Kwong, and Tony Zettl forthe opportunity to participate in the Argonne National Laboratory Workshop onSpectral Theory of Sturm-Liouville Differential Operators, and Professor MarkKon of Boston University for mathematical conversations and valuable sugges-tions.

References

Benzinger, H. 1970. "Green's function for ordinary differential operators," J. Diff.Eq. 7:478.

Birkhoff, G. D. 1908. "Boundary value and expansion problems of ordinary lineardifferential equations." Trans. Amer. Math. Soc. 9:373.

Gurarie, D., and Kon, M. 1983. "Resolvents and Regularity Properties of EllipticOperators." Operator Theory: Advances and Applications. Birkhaiiser Ver-lag, Basel, pp. 151-202.

Gurarie, D., and Kon, M. 1984. "Radial bounds for perturbations of elliptic opera-tors." J. Functional Analysis 56:99.

Haar, A. 1910. Zur Theorie der orthogonalen Funktionensysteme.." Math.Annalen 69:331.

Levitan, B. M., and Sargsjan, I. S. 1975. Introduction to Spectral Theory: Self-Adjoint Ordinary Differential Operators. AMS.

Ramm, A. G. 1981. "Spectral properties of some nonselfadjoint operators." Bul-letin of AMS 5:313.

Ramm, A. G. 1982. "Perturbations preserving asymptotics of spectrum withremainder." PAMS 85:209.

Stone, M. H. 1926. "A comparison of the series of Fourier and Birkhoff." 'Ilns.Amer. Math. Soc. 28:695.

Tamarkin, J. D. 1912. "Sur quelques points de la theorie des equationsdifferentielles lineaires ordinaires et sur la generalisation de la serie deFourier." Rendiconti del Circolo Matematico di Palermo 34:345.

Walsh, J. L. 1922/23. "On the convergence of the Sturm-Liouville series." AnnalsMath. 24:109.

187

THE ESSENTIAL SPECTRUM OF A ClASS OF ORDINARY DIFFERENTIAL OPERATORS

Bernd Schultze

Abstract

The essential spectrum of ordinary differential operators is investigated usingthe theory of relatively compact perturbations. For several classes of ordinarydifferential expressions, the essential spectrum can be determined exactly,results that seem to be new even for the even-order symmetric case.

In this paper, we will consider the (in general) non-symmetric differentialexpression

My = ipyll(1)t-o

on I = [1,m) and determine the shape of the essential spectrum of closeddifferential operators T(M) generated by M in L2(I), i.e., of the set

a,(T(M)) = IEC | R(T(M)-X) is not closed;.

There are a large number of important results concerning the essential spec-trum of differential operators generated by symmetric expressions M. However,there are only few examples where this set is not empty and entirely known: if nis even, in the case that M is some perturbation of the Euler differential expres-sion or of an expression with constant coefficients, as in Mfiller-Pfeiffer [1977]and Evans, Kwong, and Zettl [1983], and if n is odd, the limit-point criteria [Hin-ton, 1978; Schultze, 1984] imply that the essential spectrum must be the wholereal line.

In the more general case of expressions of the form (1), the basic theory forthe essential spectrum has been developed by Evans, Lewis, and Zettl [1984] andby Rota [1958]. It turns out that as in the symmetric case, the decompositionprinciple holds and that the essential spectrum is invariant under finite dimen-sional extensions of the operator. For this reason, we can confine our considera-tions to a, (T0 (M)), where To(M) is the minimal operator generated by M, anddefine a,(M) = a,(To(M)). In Rota [1958], the following spectral mappingtheorem is proven: If p is a polynomial with constant complex coefficients andp(a.(M)) 9e C, thenp(a,(M)) = a.(p(M)).

The most frequently used method to get information on the essential spec-trum consists in determining the numerical range, since it contains the essen-tial spectrum. Sometimes this numerical range coincides with the essentialspectrum, but it can also happen that there is a large gap between these twosets. Thus if, for instance, M is a symmetric odd-order expression, then itsnumerical range is R, but a subclass of such expressions is known having emptyessential spectrum.

The method applied here is more direct. Up till now, it has been restrictedto have real powers of the variable t as dominating functions for some sub-scripts i. But with this method the essential spectrum of several classes ofexpressions can be exactly determined. These results seem to be new, even for

* UniversitAt Essen, Fachbereich, 8 Mathematik, D-43 Essen 1, Germany.

188

the even-order symmetric case.We shall use the theory of relatively compact perturbations developed by

Kauffman [1977] for differential operators (slightly generalized in Schutze[1984] and also use the information on the structure of MM g vn by a1982].

To be more precise, we first consider special real expressions

M0 y = APcy("), (2),=0

where s ENO, 0,...,C, integers with 0s o <...<(, = n, p ,(t) = a.t rithaER (a = 0,...,s) satisfying for s z 1 the condition

a c< 1 (3)

(1-fand for s > 1

+1a .<a,-a for a=1,...,s-1. (4)

For the coefficients aoER we make the following restrictions:

sgn(-1) 'a, = const for , even

sgn(-1)*C +lla,= const for , odd()

We want to mention that the conditions (4) and (5) may be weakened withoutaffecting the results. Thus we can admit the " " relation in (4), and (5) can bereplaced by a condition of the following form: The "sign" of these even-order(odd order) coefficients must be constant as long as they lie between two con-secutive odd (even) kink points. (In this context O and , are defined to be evenand odd.) But from now on we always assume that M0 satisfies conditions (3), (4),and (5). Conditions (3) and (5) mean that the points (,,o,) in R2(a=0,...,s) forma polynomial path with decreasing slopes smaller than 1.

For such expressions M0 we are able to determine M M0 , which is a realsymmetric .xpression of the form

M0 M 0 y = (-1)k(gy(k))(k).b =o

The form of the coefficients gq is given in the next lemma.

LEMMA 1. For , s k s C,+1 (a = 0,...,s -1), we have

g (t ) = (ck + 0 (1))t C.+1~Co

with ck z 0 (k = ,o,...,n) and ct, = aQ for a = 0,...,s.

This representation will be used to give lower estimates of

i|Mof |II=(M1 Moff)= fgI ( (k) I2k Co

for test functions f ECo (I). Since not all ck are positive, the idea is to blur thegrowth rates between the kink-points , and to the left of o. To do this, we usethe following inequalities.

189

0

LEHMA 2. Lt n EO, 0 s i <n, aER. Thenwehave for aUf ECo():

Pam. f (n) I2 Z [2~ n(2a-2 +1)]2 ft 2(0-n +t) If()12.I [ 1 I

Besides this known inequality, we need the following lemma (cf. Kwong and Zettl[1981: Theorem 9].

LEMMA 3. Let n d' n z 2,a,PER. Then for every e > 0 therE ezist K(e) > 0 andtsE I such that for all f EC ((rj*,a)) we have

{a( |fj()I 2 s!ef tea f(n) I2 + K(e)ft2$ If 12.t:ll I I

With these tools we are able to prove the following crucial lower estimation.

LEMMA 4. here are tJc, b > 0 (k = 0,...,n), K a0 such that for alfEECo ((1,co))we have

C +i 1(k -C,)a,+i+(C,i-)awl(k

|iof| IIIa iYfb t3|f ()12eaO0k=C,+1 I

+2 fb,t f(k) 12--K I|f|I|Ik=0 I

with(ca0-o+k) for a0 > 0 or = 0

Ck-L2aokt(g- for ao s 0 and to > 0

In addition, we can choose K = 0 if ao Z to.This result enables us to apply perturbation theory in order to get informa-

tion for more general expressions. As admissible perturbations of Mo we cantake expressions

My = rk1y(k) (6)k o

with complex-valued rkEC*(I) (k =0,...,n) satisfying

o (t ) if Ef0,...,s j exists with ,<k 5t,1(rk(t)= (t ) for k=0,....(o

0Then we have for f EC;(])

|| Mf |III' (t I ||f ||)2s (n+1)2 ||ryf(k) ||2

or 2 + ( + i ,

f t(1) If I) + f f *g(1) If ( )1 2 .

9=0 kwCi,+1 I k O I

First we consider the case a0 > to. Here we can take K = 0 in Lemma 4, andwe have the following lemma.

LEIMA 5. If M is given as in (6) with coefficients satisfying (7), then there is

190

0 <a < 1 andEIsuchthatfora fECo ((,u)) we harve Mf 12 s ca lIMof 112.

REMARK. Condition (7) implies that M is even relatively compact with respect toM0 in this case a0 > to if r~ = 0. But using information that the proof of Lemma4 gives for the constants bk, the assertion of Lemma 5 can be obtained underweaker conditions than (11).

The main result in this case holds for every bounded perturbation of M0with bound smaller than 1.

THEORElM 1. Let a > to and M be given as in (6) such that there is 0 < a < 1 andvjEI with l|Mf|lie s al|Mof lie for allf ECo ((i,co)). Then a,(Mo+M) = 0.

Next we consider the case a0 < Co. Here we have to make the furtherassumptions that all a(a=O,...,a) are odd or all are even. Let us first assumethat all t, are odd. Then M0 has the form

M0y = a czta,y(~) =,ty( Q+)quO qoo

with W, = ( a-), ~, = (-1)%*a. By reason of (5) we have sgn 2, = const, so wecan assume a,%> 0. Defining now the symmetric odd-order expression

Noy = 2 (-1) a~, (ty ), + (t y, (2 ,

it can be shown that WM0 - No is relatively compact with respect to N0 -i. There-fore No and ifMo have the same essential spectrum. But it is shown by Schultze[1984] that No is in the limit point case for ao 5o; therefore a, (N0 ) = R sinceNo is of odd order. So we have the following theorem.

THEOREM 2. Let ao < to and all t,(a=0,...,s) be odd. If M is given as in (6) withr~ = 0 and the other coefficients satisfy (7), then a(MO+M) = iR

-Similar arguments hold in the case that all (a=...,s) are even. But herewe have only the information a,(Mo)c[0,o) if (-1) 'a, >0 (=0....,s); and in thisgenerality, more precise results cannot be obtained, as examples will show.

THEOREM 3. Let a0 < o. all t, be even, and (-1)'a,,> 0 (=0,...,n). If Mo isgiven as in (6) with r = 0 and the other coefficients satisfy (7), theno, (Mo+M)c[0,r).

If we again specialize this case to s = 0, to > 0, then we can write Mo in theform

M0 y = (-1)mtay(2n), (8)

where m = )Wo, a = a0 . Denoting the corresponding symmetric expression by

Noy = (-1)1 (ty(m))(m1) (9)the spectral mapping theorem and perturbation theory give for perturbations

i-1 .kMy = ry(k) with rk(t) = o(t2 ") (k=0,...,2m-1) (10)

the following result (if M is symmetric, given in the symmetric form withcoeficientsrqj, differentiation carries it over in the form (10) with analagous con-ditions on rk and some of their derivatives).

THEOREM 4. Let M0 be given as in (8), go as in (9). M as in (10) witha < 2m (m EN). Then

191

a,(Mo+M) = a,(11 0+M) = [O,m).

For more involved expressions M0 the essential spectrum may be differentfrom the whole non-negative real line. If we take for M0 (9 0) the two-termexpressions with c z 0:

Moy = tfty(4") + (-1)mctay(mI), (11)

resp.,

Eoy = (tSy(2 m))(sm) + (-1)"c(tym))(m), (12)

then again the spectral mapping theorem (for the polynomial p (z) = z2+c) andperturbation theory give for perturbations

47n.-1 .Ayl= rky(k) with rk(t)=o (t 2 ") (k=0,...,4mv-1), (13)

k m0

the following generalization of Theorem 4.

THEOREM 5. If M0 is given as in (11), Do as in (i2), and M as in (13) witha < 2m (m. EN), c mo, then

Q.(Mo+M) = or,(Mo+M) = [c,oo).

Taking other polynomials, we can generate expressions with more and morecomplicated essential spectrum in the complex plane. This also seems to hap-pen in the case singled out till now: a0 = Co.

If N is "some" perturbation of the Euler-expression Noy= = b tky(k),k .0

thew its essential spectrum is shown to be

a, (N) = be + fbkfII(z-(j+$)) IRez=01;

see Goldberg [1966:M.7 and VI.8]. For one-term expressions Noy = iCtCyt) thisgives pictures for the essential spectrum, which are only qualitative (see Fig. 1).

For expressions M0 given as in (2) satisfying (3), (4), and (5) with ao = Co andperturbations M given as in (6) with coefficients satisfying (7), similar methodsof proof as for Theorem 1 show the following result:

Co-1If AEC with IA <I|o 12~ II (21+1), then ASEa.(Mo+M).

i=0

192

in

-0 mm --

I

AU

iR

-10395

64

945 .32L

S=5

rnrnA15

3

Ai -4R

R

Fig. 1. Essential Spectrum for One-term ExpressionsNoy = iftC (C)

R

-i i:t4

iR

I1f5 R

S=4

I- ,w

193hIT

References

Evans, W. D., Kwong, M. K., and Zettl, A. 1983. "Lower bounds for the spectrum ofordinary differential operators." J. Diff. Equations 48:123-155.

Evans, W. D., Lewis, R. T., and Zettl, A. 1984. "Non-selfadjoint operators and theiressential spectra." Proc. Roy. Soc. Edinburgh (to appear).

Goldberg, S. 1966. Unbounded Linear Operators. McGraw-Hill, New York.

Hinton, D. 1978. "Deficiency indices of odd-order differential operators." RockyMountain J. Math. 8 (4):627-64?. -

Kaufmann, R. M. 1977. "On the limit-n classification of ordinary differentialoperators with positive coefficients." Proc. London Math. Soc. 35:496-526.

Kwong, M. K., and Zettl, A. 1981. "Norm inequalities of product form in weightedL'-spaces." Proc. Roy. Soc. Edinburgh 89A: 293-307.

M " ler-Pfeiffer, E. 1977. SpektraLeigenschaften singuLtrer gewhnlwcherDifferentiJaloperatoren. Teubner Verlag, Leipzig.

Read, T. T. 1982. "Positivity and discrete spectra.for differential operators." J.Diff. Equations 43:1-27.

Rota, G. C. 1958. "Extension theory of differential operators." Comm. Pure andApple. Math. 11:23-65.

Schultze, B. 1984. "Ordinary differential expressions with positive supportingcoefficients" (preprint).

195

DIRAC SYSTEMS WITH OSCILLATING POTENTLLS ANDABSOLUTELY CONTINUOUS SPECTRA

J. K Shaw*D. B. Hintont

Abstract

A Dirac system y' = [XA(z) + P(z)]y is considered on an interval [ a,b) of reg-ular points, the equation being singular at z = b. Theorems that locate the con-tinuous spectrum are proved, and sufficient conditions that the continuous spec-trum be absolutely continuous are given. Then externsions are given for twosingular endpoint problems.

1. IntroductionA Dirac system is a first-order system of the type

= s_ Xxa 2(z)+p2(z)]] , asz<bso, (1.1)

where the real-vaied coefficients p, a, and p are locally integrable over [a,b),cxt(z) > 0, and X is a complex parameter. Initially, we regard (1.1) as regular atx = a and singular at z = b, but see Section 4 for extensions of our results totwo singular endpoint problems.

It is well known that the Weyl limit point-limit circle classification holds forsystems (1.1) (see Kogan and Rofe-Beketov [1974] and Levitan and Sargsjan[1975]); i.e., the dimension of the space of solutions of (1.1) belonging to

La[a,b) = i )If(cilIyi)2 +a2IyI2 )dz<c , = g1J is either exactly 1 for all non-

real X or exactly 2 for all nonreal A. In this paper we will consider only the limitpoint case (dimension = 1 ).

For systems of limit point type, we affiliate an operator on L [a,b) with(1.1) by introducing a boundary condition B()=sinPy 1 (a)+cosfy 2 (a)=0 andthen letting T:Da.p-LQ[ab) be the operator defined by

T(?)= [~gla2f1] i]1 1 .[piP]l; (1.2)

the domain Dap consists of all locally absolutely continuous 7 EL [ a ,b) withB(g) = 0 and T(g)EL [a,b). Then T is selfadjoint and thus has a real spectruma((T) (see Levitan and Sargsjan [1975]).

One of the central problems in differential operator theory is to determinethe spectrum and, in particular, to determine where it is discrete and where itis continuous. The present paper takes up this question, together with the prob-lem of finding sufficient conditions that the continuous spectrum be absolutelyconinuous (defined below). The absolutely continuous spectrum is important

'Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-4097.tDepartment of Mathematics, University of Tennessee, Knoville, Tennessee 37996-1300.

196

for its applications to scattering theory.

One of our objectives will be to extend the work of Titchmarsh [1961 and1962], Erdelyi [1963], and Weidmann [1971 and 1982] on Dirac systems withCoulomb potential,

[-k/z A+V(z)+c <= I-A-V(z)+c -k/x O<X<ao, (1.3)

where k and c are constants and V(x) is locally integrable on (0,0). IfV(z) = Z/z, Z constant, then (1.3) is the radial wave equation in relativisticquantum mechanics for a particle in a field of potential Z/x. In Titchmarsh1961 and 1962] and Erdelyi [1963], the spectrum of the two singular endpoint

problem (1.3) is considered under assumptions that V(z) - (7'/x) be absolutelyintegrablk toward z = 0, for some constant y, and that V(z) should be, in a cer-tain technical sense, either "small" at z = = or "large" at z = w; included in thetechnical assumptions are V(z)-+O or V(z)+fo, respectively, as z-'. In thesmall potential case, the spectrum is continuous in (-o,-c ]u[c ,oa) and discretein (-c ,c); in the large potential case the spectrum continuously covers thewhole real line. It is implicit in Titchmarsh and Erdelyi that the continuousspectrum is absolutely continuous, in fact continuously differentiable. Weid-mann [1971 and 1982] discusses various subjects concerning oscillation proper-ties and the spectrum (1.1) or 6 = co. In the case where alx(z) = a2(z) = 1,

the potential term P = I 1 is broken up into long- and short-range terms

P = P1 +Ps, where P1 is of bounded variation, P1 -j _0+ (z -'os j_,+), and

PEL1 (z0 ,.o), z0 )a. Under these assumptions, the spectrum is absolutely con-tinuous in

In the sections that follow, we give generalizations of the results of Titch-marsh and Weidmann by applying a general asymptotic technique and a changeof variable developed by Hinton and Shaw [1984]. We allow weight functions, onthe one hand, and we permit a decomposition of the potential into long-range,short-range, and oscillatory parts. Specific hypotheses and statements aregiven in the next section. In Section 3 we outline proofs, and we close the paperwith some remarks in Section 4.

2. eAtmenb of .itu

The resolvent set of the operator T in (1.2) consists of all complex A forwhich (T-XI)- 1 is a bounded operator from La[a,b) into itself [Levitan andSargsjan, 1975; Hinton and Shaw, 1982]. The spectrum of T is the complementof the resolvent set.

Let 4(zA) and O(zA) be the unique solutions of (1.1) which satisfy for allcomplex A the initial conditions

['a(a), (a)]_= [sin1cosl .(2.1)

Then the limit (in the limit point case)

m(A)_= -lim 'dl(- )(2.2)z-b 9(xA)

exists and defines an analytic function for Im(X) 0 0 [Levtan and Sargsjan, 1975;Hinton and Shaw, 1981]; m (A) is known as the Titchmarsh-Weyl m-coefficient.The spectral fraction for T is a nondecreasing and right-continuous functionp(X) uniquely determined by p(0) = 0 [Levtan and Sargsjan, 1975; Hinton andShaw, 1982]. One has the Ttchmarsh-Kodaira formula [Hinton and Shaw, 1982]

197

N2

P(/2)-P( 1) = lim n-1f Im m ( +ie)dM (2.3)

at points of continuity , and 2 of p. The spectrum of T consists of the pointsof increase of p [Levitan and Sargsjan, 1975]: jump discontinuities occur ateigenvalues (both isolated and nonisolated), the continuous spectrum consists ofpoints where p is continuous and nonconstant, and an open interval belongs tothe absolutely continuous (resp., C1) spectrum if and only if p is absolutely con-tinuous (resp., continuously differentiable) there.

We now state the two theorems to be discussed here. More general resultsare possible; see Hinton and Shaw [1984a]. The following ones illustrate thetechnique of mixing asymptotics, change of variable, and the Titchmarsh-Weylcoefficient to describe the spectrum of (1.1).

THEOREM 1, In (1.1), let

b = to, a1(z) = a.,(z) = 1,

p(z) = A,(z)+A(z), p1(z) = p11(z)+p 12(z)+p13 (z),

P2(Z) = p21(x)+p2(x)+p 8 (z),where A1, p i, and P21 are differentiable and t 1 -+0, p 11 -+-X1 , p21-- X2 (z- o)

Here X<42; let phi'. A', p9, A EL'[zo0cx), and suppose the integralsPh2(z) = f phS(s )ds exist (conditionally), and define functions that belong toL1[zo,o). suppose further that all the functions

fIPs(s)pJ,'(s) Ids,

ftPh(s)A'(s)ds,

Ph2(z)Pjg(z),pm2(z)f I Ph2(s )p 1 '(s) I ds,and

Pm2(z)f |IPk2(s)o'(s)Idslie in L1[z0 ,oa). Then (1.1) Lies in the limit point case (and thus T is selfadjoint)and the spectrum of T is continuously differentiable in (-o,X1 )u(X 2,a) anddiscrete in (X,,X).

The potentials ph2(z) may be viewed as "oscillatory" terms which must cou-ple with the long-range terms ph,(x)+Ah and A,(z) in the way prescribed. Anexample of functions satisfying the hypotheses are A 1 = x-1 = P11, P21 = -1+z1,and ph2(Z) = zxsin(z 4 ), a < 2. Thus we allow a wider range of potentials but drawthe same conclusion (see the remarks in Section 4) as Weidmann [1971].

THEOREM 2. In (1.1) let b = m, p(x) = 0 ,.) ( )= pkl(z)+ps(x ), where the pkl(x )are differentiable, P 11 P2,> Q and J (P P2)1 2 = c; suppose that either

f a,(p2 1 /p 11 )2"="" or f a2(pi,/p2) L2 = ** and that the functions7= qe/ u)"" and A(z) = n(pu /p 1) satisfy L = A,+A3 where A1-O and ,1',A L [a,00). Suppose that a/ph, and Ph2/Pk1 can be decomposed into long-range, short-range, and oscillating potentials as in Theorem 1. Then (1.1) liesin the limit point case (and thus T is selfadjoint) and the spectrum of T is con-tinuously differentiable in (-Cc, W).

198

We outline the proofs of the theorems in Section 3. More details and furthergeneralizations may be found in Hinton and Shaw [1984a].

We shall require a characterization of the spectrum in terms of m (X). Hin-ton and Shaw [1982 and 1984b] extended the classification theorem of Chau-dhuri and Everitt [1968] to the setting of Hamiltonian systems, which include(1.1) as a special case. We are concerned here mainly with the fact that m(X)continues analytically into each interval of the resolvent set, and that the iso-lated eigenvalues of T are (isolated) poles of m(A); see Hinton and Shaw [1982].

Owing to the fact that m(X) = m (X) [Hinton and Shaw, 1982] for nonreal A,it follows that m(X) continues analytically to a real interval if and only if it has areal limit there as lm(A)-O. In particular, if m(A) is real and meromorphic onan interval, then the spectrum is discrete there, with the eigenvalues coincidingwith poles of mn(k).

To illustrate the connection of m(N) with the theorems stated above, wegive two simple examples. First, consider the constant coefficient equation

g'= X_+c X c117,gOsz<(0, (2.4)

which is a special case of (1.3 and which is equivalent to the scalar equationy "+(A2-c3 )y = 0. Computing and 0 of (2.1) and evaluating the limit (2.2) forLm(X) > 0, we find for this example that m(A) =i[i[(A-c)/ (A+c)]3. This expres-sion is real if and only if -c s X ! c, so the spectrum consists of (-m,-c ]u[c ,mo).This is an instance of Theorem with p2 (z) = c , p1 (z) = -c, p (z) = 0. A niceexample of Theorem 2 is the one in which a(z) = a2 (z) = a(z), p (x) = 0, andp 1(z) = p2 (z) = b(z), for the solutions of

0 ( \ a(x)+b (z)l-Aa(z)-b (z) 0 JJ

are available explicitly and have the form 9 = [) exp(tifo(Aa +b)ds). Thehypotheses of Theorem 2 are satisfied if we take p,, = 0, aa = 1, and assumethat, say, b(z)~z-2, Z-. It is then an easy matter to compute m(A) = i forIm(A) > 0. Thus mn(X) = -i in the lower half plane, and this means that mv(X) isanalytic nowhere on the real line; i.e., the spectrum is continuouslydifferentiable in (-ac,oc) with p(A) = A/ i (see Formula 2.3).

3. O ines of ProofsThe systems of Theorems 1 and 2 can be nut intocommon form by a change

of variable in Theorem 2. Let f (z) and 1tz) be differentiable functions on[a,b), with f (z) monotone increasing and unbounded. Let n2(z) = 1/77 1(z), andintroduce new independent and dependent variables t = f (z) and

9(z) = [10z) z) (t), to = f(a) c t <oo.

Then (1.1) becomes

f t)= 1 (-vli'/n1) (n2/m1)(Xa2+p2) et 31'( -f'- ) (-n1/12)(Xa 1+P1) (-n2'/ 12)'-

where dots represent differentiation with respect to t, and where the expres-sions in the matrix are evaluated at z = f -1(t). In Theorem 2, recall that wehave p i = P i+P a, Pe = P21+P22. We now make the special choices

199

f (z) = f[p1 1(s)p2 i(s)]1/2 ds, r1(z) = [p2 1(z)/p 11(z)]1/4 .

Then (3.1) becomes

r -m' (t( ') 1 +A(a2/P 2 1 )+ (Ps/P21)]2y(t). (3.2)(t) = -1-(ai/pii)(p1 2 /P) -72'/ (2f!')

Note that the diagonal members of (3.2) are negatives of each other because72 = 1/71. Both (3.2) , under the assumptions of Theorem 2, and the system ofTheorem 1 (with z = t) have the general form

l 6(t )a( +rt ,)

2(t)_= -(A+r1(t,) -A(t) p f(t), (3.3)

where a(X) and b (X) are entire functions of X which are real for real A,A(t) = A1(t)+As(t) decomposes into long- and short-range terms as in thehypothesis on p () in Theorem 1, and where r (tA)= rk1(t)+rk2(t)+rk3(t)+AXsk1(t)+sk2(t)+sks(t)] with the rk1 and Ski long-range

terms, r9 and ss short-range terms, and rk2 and sk2 oscillating terms.Under hypotheses that include Theorem'r 1 and 2, we studied the asymp-

totic form of the solutions of (3.3). We state here the main conclusions as theyapply to the present paper, and refer to the earlier work [Hinton and Shaw,1964a] for technical assumptions and proofs. Let us define

po'(t A) = [(a (A)+r21(t)+As2 1 (t))(b (X)-r1 1(t)-Xs 11(t))-A? (t)]1' 2 ,and let U be a compact subset of the complex A-plane such that a (A)b (A) s 0 forAE U. Thus for Theorem 1 we have a (A) = A-AA, b (A) = A-A 1, and U can be anycompact subset of the plane that omits Al and A2. In Theorem 2, a (A) = b (A) = 1from (3.2), so U is any compact subset of the plane. We prove [Hinton and Shaw,1984a] that for each solution 9 of (3.3) there exists a function A(X), analytic onthe interior of U and continuous on U, such that if f Im po = o,

(A)+r21 (t)+As21 (t )+0(1)2(t) = A(A)exp(-ifpods) -pE(tA)-Al(t)+0(1) ] t-. (3.4)

to

for AEU. In the cases considered here we have Im (tA) > 0 if Im(A) > 0 and t issufficiently large. Moreover, pe(t,A)~[aa(A)b (A)]v ; hence the exponential termdominates the asymptotic formula (3.4) for Im(A) > Q.

Let f. and f denote the transformed solutions of (3.3) corresponding tothe solutions i(Z,5) and O(z,A) with initial values (2.1). Let Ae(A) and A,(X) bethe A(X) coefficients appearing in (3.4) for 4, and ,, respectively. Since ourbasic change of variables has first component yi(x) = 71(x)z 1 (t), then (2.2) and(3.4) imply

) (zA) (Z)_(tA) A(A)>.b)= (ziA)m= ~- (z.)1(t ,A) = ~ A,(A)

We now have to consider the behavior of the A(A) coefficient when A andM(t ,A) are real, this case not being Covered by (3.4). We proved [Hinton andShaw, 1984a] that if AE U is real and f Imp, < eo, then

ReAe(A)ImA,(A)-ReA,(A)1mAO(A) = [4a (A)(ac(A)b (A)) 2]- 1. (3.6)

Note that for the values of p determined by Theorems 1 and 2, pA(tA) iseither purely real or purely imaginary for t sufficiently large and for real AE U.

We are now ready to prove Theorem 1. Let AE(XX 2 ), and pick U to be acompact set omitting X1 and A2 but containing A. Then po(t ,A) is purely ima-ginary, and it follows (see Hinton and Shaw [1984]) that the solutions 2 of (3.3)are real. In particular, A(X) and A,(A) are reel for all AE(X1 ,X2 ). Hencem(A) = -A,(A)/A,(A) continues analytically through (XX2) save for zeros ofA,(A) which coincide with its poles. That is, mn(X) is meromorphic in (X1, 2 ), andso the spectrum of T is discrete there. If instead A > X or A < A2 , then po(t ,A) isreal, and we are under case (3.6). Now by (3.6)

Imm (A) = -Im 4 = ,(3.7)A() IA,(A)|

which is also finite and nonzero by the right side of (3.6). Since m(A) has a non-real value on the real line in (-=,X 1 ) and (A2 ,oo), then it cannot be analytic there.Since (3.7) implies lin em(A+ir) = 0, then the intervals (-co,A 1) and (2,0) lie in

the continuous spectrum; see Hinton and Shaw [1982]. Continuousdifferentiability of the spectral function is a consequence of the Titchmarsh-Kodaira formula (2.3). This proves Theorem 1.

The proof of Theorem 2 is identical to the second part of the above proofbecause pe(t ,A) is real for large t and real A.

4. Htemark3 and Extensians

1. The hypotheses regarding fai(42/pi)1/2 or f a a(p i/p2i)1/2 are presentin Theorem 2 to ensure that the limit point case prevails. In the limit circle casethe spectrum is discrete.

2. Turning to two singular endpoint problems, we indicate briefly here howTheorem 1 may be extended to the case in which the endpoint z = a is singular.Analogous remarks apply to Theorem 2. Thus, suppose we have a system

1'(z) =_ 4 z) pz) V(z), a < z < c, (4.1)

which is singular at z = a. If the end z = a is limit point, define the operator Tjust as in (1.2) except that the condition B(V) = 0 is dropped. If z = a is limitcircle, we take T to be a selfadjoint extension of the minimal operator associ-ated with (4.1). Denote by i.(A) the Titchmarsh-Weyl coefficient for (4.1) atz = a; i.e.,

dazA)m,(A) = -lim , (4.2)

where 4 and P are the solutions of (4.1) taking values (2.1) at some basepointac, a < ao < m, instead of at z = a. The limit (4.2) exists and defines an analyticfunction for Im(A) # 0 in the limit point case. If the limit-circle case prevails,the limit in (4.2) must be taken through a subsequence, and so for simplicity wewill consider only the limit point case at z = a. It is beneficial to consider "half-line operators" T. and T.. associated with (4.1), as restricted to the intervals(a,a0 ) and (ao,ao), respectively. Here T., is defined just as T in Theorem 1 exceptthat ao replaces a, and T is defined similarly. Then m(A) describes the spec-trum of T.., and v% (A) describes that of T6, in the sense of the proof of Theorem1.

The whole line operator T has Titchmarsh-Weyl coefficient a 2x2 matrixfunction

M () ()-m (a]-11 (ma (X)+m (X))/ 2 (43M(X) = [.().(T)] (m(X)+m(X))/ 2 ma (X)m() 2] (43)

see Hinton and Shaw [1984a and 1984b] and Coddington and Levinson [1955].The spectral function 'r(X) is a 2x2 Hermitian, nondecreasing (in the positivedefinite sense), right continuous function which bears the same relationship toM(X) as p(X) does to m(X) from (2.3). When M(X) has a nonreal and finite limitat the real axis, the spectrum of T is continuous there (see Hinton and Shaw[1984b]) and therefore continuously differentiable by the matrix analogy of(2.3).

It is not difficult to prove (see Hinton and Shaw [1984a]) that if either ma (X)or m(X) has a nonreal and finite limit on some interval J, then so does M(X),regardless of the behavior of the other m-coefficient. In essence, this isbecause Imm(X) > 0 and Imm(?) < 0 for Im(X) > 0, and if one of these inequali-ties preserves its strictness as Im(X)-*, then M(X) can be shown to have a non-real limit. Therefore if A belongs to the continuous spectrum of either T.. or Ta,then it lies in the continuous spectrum of T. Consequently the whole line opera-tor T arising from Theorem 1 has continuously differentiable spectrum in(-o,X1)u( 2 ,o), quite independently of the behavior at the singular end z = a.To claim discreteness of the spectrum in ( 1,1X2), however, we would needdetailed knowledge of the singular behavior at z = a; see Theorem 3 in Hintonand Shaw [1984a].

References

Chaudhuri, J., and Everitt, W. N. 1988. "On the spectrum of ordinary secondorder differential operators." Proc. Roy. Soc. Edinburgh 68A:95-119.

Coddington, E. A., and Levinson, N. 1955. Theory of Ordinary Differential Equa-tions. McGraw-Hill, New York.

Erdelyi, A. 1963. "Note on a paper of Titchmarsh." Quart. J. Math. Oxford14:147-152.

Hinton, D. B., and Shaw, J. K. 1981. "On Titchmarsh-Weyl m-functions for linearHamiltonian systems." J. Differential Equations 40(3):316-342.

Hinton, D. B., and Shaw, J. K. 1982. "On the spectrum of a singular Hamiltoniansystem." Quaes. Math. 5:29-81.

Hinton, D. B., and Shaw, J. K. 1984a. "Absolutely continuous spectra of Dirac sys-tems with long range, short range and oscillating potentials." Quart. J.Math. Oxford (to appear).

Hinton, D. B., and Shaw, J. K. 1984b. "On the spectrum of a singular Hamiltoniansystem II." Quaes. Math. (submitted).

Kogan, V. I., and Rofe-Beketov, F. S. 1974. "On square-integrable solutions ofsymmetric systems of differential equations of arbitrary order." Proc. Roy.Soc. Edinburgh 74A:5-39.

Levitan, B. M., and Sargsjan, I. S. 1975. Introduction to Spectral Theory: Self-adjoint Ordinary Differential Operators. Translations of Mathematical

Monographs 39, American Mathematical Society, Providence.

Titchmarsh, E. C. 1961. "On the nature of the spectrum in problems of relativis-tic quantum mechanics." Quart. J. Math. Oxford 2(12):227-240.

Titchmarsh, E. C. 1962. "On the nature of the spectrum in problems of relativis-tic quantum mechanics III." Quart. J. Math. Oxford 2(13):255-263.

Weidmann, J. 1971. "Oszillationsmethoden fur systeme gewohnlicher differentialGleichungen." Math. Z. 119:349-373.

WeidmL.nn, J. 1982. "Absolut stetiges Spektrum bei Sturm-Liouville Operatorenund Dirac-systeme." Math. Z. 180:423-427.

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J. Hooker, Southern Illinois UniversityD. Jabon, University of ChicagoM. A. Kon, Boston UniversityA. M. Krall, The Pennsylvania State UniversityMan Kam Kwong, Northern Illinois UniversityH. Langer, Technical University, DresdenM. Lapidus, University of Southern CaliforniaA. B. Mingareili, University of Ottawa, CanadaH. Niessen, Universitaet Essen, West GermanyL. A. Raphael, Howard UniversityB. Schultze, Universitaet Essen, West GermanyJ. K. Shaw, Virginia TechA. Zetti, Northern Illinois University