recursive construction for a class of radial functions. i ...hp0117/publications/jmathphys_43...e...

Let

h

ions.

. Theil-l.eseso

ic

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 43, NUMBER 5 MAY 2002

Downloaded 0

Recursive construction for a class of radial functions.I. Ordinary space

Thomas Guhra)

Matematisk Fysik, LTH, Lunds Universitet, Box 118, 22100 Lund, Sweden,and Max Planck Institut fu¨r Kernphysik, Postfach 103980, 69029 Heidelberg, Germany

Heiner Kohlerb)

Departamento de Fı´sica, Universidad Auto´noma de Madrid, Madrid, Spain,and Max Planck Institut fu¨r Kernphysik, Postfach 103980, 69029 Heidelberg, Germany

~Received 19 October 2001; accepted for publication 4 February 2002!

A class of spherical functions is studied which can be viewed as the matrix gener-alization of Bessel functions. We derive a recursive structure for these functions.We show that they are only special cases of more general radial functions whichalso have a properly generalized, recursive structure. Some explicit resultsare worked out. For the first time, we identify a subclass of such radial functionswhich consist of a finite number of terms only. ©2002 American Institute ofPhysics.@DOI: 10.1063/1.1463709#

I. INTRODUCTION

In 1957, Harish-Chandra1 derived a famous formula for a certain class of group integrals.G be a compact semisimple Lie group and leta andb be elements of its Cartan subalgebraH ,then

EUPG

dm~U !exp~ tr U21aUb!51

uWu (wPW

exp~ tr w~a!b!

P~a!P~w~b!!. ~1.1!

Here, dm(U) is the invariant measure,P(a) is the product of all positive roots ofH , andW is theWeyl reflection group ofG with uWu elementsw.

This result depends crucially on the condition thata andb are in the Cartan subalgebraH . Inother words,U21aUb has to be in the algebra of the group. If one replacesa andb in the integralon the left-hand side with more general matricesx andk which are not inH , formula~1.1! is notvalid anymore. Thespherical functionsintroduced by Gelfand2,3 form an important class of sucintegrals which are, in general, not covered by Harish-Chandra’s result~1.1!. In another work,Harish-Chandra4 studies in great detail the harmonic analysis involving these spherical functIn a more physics oriented contribution, Olshanetsky and Perelomov5 discussed them in theframework of quantum integrable systems.

Here, we wish to address spherical functions of the following kind: we takex and k asdiagonal matrices containing the eigenvalues of a Hamiltonian in a matrix representationHamiltonian is diagonalized by the integration matrixU. In particular, we assume that the Hamtonian U21xU or, equivalently,UkU21 is real-symmetric, Hermitian, or Hermitian self-duaThus, G is the orthogonal, the unitary or the unitary-symplectic group. We will refer to thspherical functions asmatrix Bessel functions. We notice that the unitary case is special: since ithappens that the eigenvaluesx andk do lie in the Cartan subalgebraH , the result~1.1! appliesand coincides with the Itzykson–Zuber formula.6 In the orthogonal and the unitary-symplectcases, however, formula~1.1! is not valid.

a!Electronic mail: [email protected]!Electronic mail: [email protected]

27070022-2488/2002/43(5)/2707/34/$19.00 © 2002 American Institute of Physics

4 Apr 2005 to 147.142.186.54. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp

here,. How-Besselmatrix

for

whichland

Theyuchnnately,f the

iance,-Jackindex

ntainan be

of ther thenda-is an

atrixstrate

ce, weIII, wes.

some

2708 J. Math. Phys., Vol. 43, No. 5, May 2002 T. Guhr and H. Kohler

Downloaded 0

We choose the term matrix Bessel function for the spherical functions to be discussedbecause they can be viewed as a natural extension of the ordinary vector Bessel functionsever, due to the rich features of the spherical functions, other extensions relating to ordinaryfunctions are equally natural. Related functions have been discussed and terms similar toBessel functions have already been used by Hertz,7 by Gross and Kunze,8,9 by Holman,10 and byOkounkov and Olshanski.11 Kontsevich12 introduced the matrix Airy functions.

Duistermaat and Heckman13 developed a stationary phase approach involving localizationa class of spherical functions, see also the treatise by Szabo.14

Remarkably, our matrix Bessel functions are only special cases of more general objectswe call radial functions. Moreover, there is an important connection to the Calogero–Suthermodels which we will discuss separately, see the following.

The matrix Bessel functions are of considerable interest for applications in physics.appear in random matrix theory,15–17which models spectral fluctuations of complex systems, sas quantum chaotic ones. In particular, they are the kernels of Dyson’s Brownian motio18,19

describing crossover transitions between different symmetry or invariance classes. Unfortuonly the case of broken time-reversal invariance can be treated explicitly with the help oItzykson–Zuber formula. In the physically important cases of conserved time-reversal invarthe kernels are not known analytically, as argued previously. Muirhead20 discusses spherical functions in the framework of multivariate statistical theory. In his book, an expansion in terms ofpolynomials for the orthogonal case can be found. Such an expansion for arbitrary Dysonwas recently worked out by Okounkov and Olshanski.11

The goal of the present paper is to explore the structure of the radial functions which cothe matrix Bessel functions as special cases. In particular, we show how explicit results cobtained. The paper is organized as follows. In Sec. II, we briefly review some propertiesvector Bessel functions. In doing so we wish to help the reader in developing an intuition fomatrix Bessel functions which we introduce in Sec. III. In Sec. IV, we state and derive a fumental recursive structure for matrix Bessel functions. We show in Sec. V that this recursioniterative solution of general radial functions which contain group integrals defining the mBessel functions as special case. Sections IV and V are our main results. In Sec. VI we illuhow the recursion can lead to closed and explicit formulas. Because of its special importandiscuss the connection to Calogero–Sutherland models separately in Sec. VII. In Sec. Vsummarize and conclude. Various aspects and calculations are collected in the appendice

II. VECTOR BESSEL FUNCTIONS REVISITED

Before turning to the matrix case, we compile, for the convenience of the reader,well-known results for the vector case.

In a real, d dimensional space withd52,3,4,..., we consider a position vectorrW5(x1 ,...,xd) and a wave vectorkW5(k1 ,...,kd). The plane wave exp(ikW•rW) satisfies the waveequation

D exp~ ikW•rW !52kW2 exp~ ikW•rW !, ~2.1!

where we define the Laplacian as in the physics literature,

D5]2

]rW2 5(i 51

d]2

]xi2 . ~2.2!

The zeroth-order Bessel function in this space is the angular average of the plane wave,

x (d)~kr !5E dV exp~ ikW•rW !, ~2.3!


s

sure.

d

s.

f

-

e ofvially

2709J. Math. Phys., Vol. 43, No. 5, May 2002 Recursive construction. I

Downloaded 0

over the solid angleV, defining the orientation of eitherrW or kW . In our context, it is advantageouto takeV as the solid angle ofkW . Obviously, only the relative angle betweenrW andkW matters andx (d)(kr) can only depend on the product of the lengthsr 5urWu andk5ukW u of the two vectors. Wenormalize the measure dV with the volume 2pd/2/G(d/2) of the unit sphere, i.e., we have

E dV51. ~2.4!

Thus, by construction, we also have

x (d)~0!51. ~2.5!

It is convenient to viewrW as the azimuthal direction of the coordinate system in which we meaV. Thus, in these spherical coordinates, one findskW•rW5kr cosq whereq is the azimuthal angleThe measure dV contains sind22 q and one has

x (d)~kr !5G~d/2!

ApG~~d21!/2!E

0

p

exp~ ikr cosq!sind22 q dq52(d22)/2G~d/2!J(d22)/2~kr !

~kr !(d22)/2 ,

~2.6!

whereJn(z) is the standard Bessel function21 of ordern. The functions~2.6! are often referred toas zonal functions.

There is a remarkable difference for the functionsx (d)(kr) if one compares even and oddimensions. For example, one has ind52 dimensionsx (2)(kr)5J0(kr) and ind53 dimensionsx (3)(kr)5(p/2)1/2J1/2(kr)/(kr)1/25 j 0(kr) with the spherical Bessel functionj 0(z) of zerothorder.21 In d52 dimensions,J0(z) is a complicated infinite series in the argumentz, in d53dimensions, however,j 0(z) is the simple ratioj 0(z)5sinz/z. One easily sees how this generalizeUpon introducingj5cosq as integration variable in Eq.~2.6!, one finds the representation

x (d)~kr !5G~d/2!

ApG~~d21!/2!E

21

11

exp~ ikr j!~12j2!(d23)/2dj. ~2.7!

In dimensionsd>3, this can be cast into the form

x (d)~kr !52G~d/2!

ApG~~d21!/2!(m50

` S ~d23!/2m D ]2m

]~kr !2m

sinkr

kr. ~2.8!

For even d, the exponent (d23)/2 is a fraction21/2,11/2,13/2,..., and thefunction (12j2)(d23)/2 in the integrand in Eq.~2.7! is an infinite power series. This yields, ford54,6,8,..., thecomplicated power series~2.8! involving an infinite number of inverse powers okr. However, if d is odd, the exponent (d23)/2 is an integer 0,1,2,..., and thefunction (12j2)(d23)/2 is a finite polynomial of order (d23)/2 in j2. Thus,x (d)(kr) acquires a comparatively simple structure, because it only contains a finite number of inverse powers ofkr. Formally,this means that for oddd all binomial coefficients form.(d23)/2 are zero.

The differential equation for the functionsx (d)(kr) is easily obtained by averaging Eq.~2.1!over the solid angleV of kW , i.e., by integrating both sides,

DE dV exp~ ikW•rW !52kW2E dV exp~ ikW•rW !. ~2.9!

We notice that the LaplacianD commutes with the integral, because the former is in the spacthe position vector, the latter in the space of the wave vector. Moreover, the integral tricommutes withkW25k2. Hence, one arrives at


i-and

sting

efore

rnativeole ineneral

someC.


Downloaded 0

D rx(d)~kr !52k2x (d)~kr !. ~2.10!

Sincex (d)(kr) depends exclusively on radial variables, we replaced the full LaplacianD with itsradial part

D r51

r d21

]

]rr d21

]

]r5

]2

]r 2 1d21

r

]

]r. ~2.11!

In general, there are two fundamental solutionsx1(d)(kr) andx2

(d)(kr) of the differential equation~2.10! which behave as exp(6ikr)/(kr)(d21)/2 for large argumentskr. Thus, to obtain the fullsolutions, one can make the Hankel ansatz

x6(d)~kr !5

exp~6 ikr !

~kr !(d21)/2 w6(d)~kr !. ~2.12!

Here,w6(d)(kr) is a function with the propertyw6

(d)(kr)→1 for kr→`. The differential equationfollows easily from Eq.~2.11! and is given by

S ]2

]r 2 6 i2k]

]r2

d21

2 S d21

221D 1

r 2Dw6(d)~kr !50. ~2.13!

For d>3, one uses the ansatz as an asymptotic power series

w6(d)~kr !5 (

m50

`am

~6kr !m , ~2.14!

which yields a recursion for the coefficients

am1151

i2~m11! S m~m11!2d21

2 S d21

221D Dam , ~2.15!

with the starting valuea051. A special situation occurs when the integer running indexm reachesthe critical valuemc5(d23)/2. If d is odd, mc is integer and the recursion terminates atm5mc , i.e., one hasam50, m.mc . Thus, the asymptotic series becomes afinite polynomial ininverse powers ofkr. However, ifd is even,mc is half-odd integer and the series cannot termnate, it is alwaysinfinite. This explains the different structure of the Bessel functions in evenodd dimensional spaces from the viewpoint of the differential equation.

In Appendix A we discuss an alternative integral representation which has an intereanalog in the matrix space.

III. MATRIX BESSEL FUNCTIONS

We compile the basics features of the matrix spaces we want to work with in Sec. III A bwe define the matrix Bessel functions as group integrals in Sec. III B.

Two general aspects are shifted into the appendices. First, we present an interesting alteintegral representation in Appendix B. Second, the matrix Bessel functions play a crucial rharmonic analysis or, equivalently, in Fourier–Bessel analysis in matrix spaces. For the gtheory, we refer the reader to Harish-Chandra’s treatise in Ref. 4 and to Helgason’s book.3 How-ever, to achieve our goal of being explicit, we collect, for the convenience of the reader,results for the Fourier–Bessel analysis of invariant functions in matrix spaces in Appendix

A. Basics and notation

We introduceN3N matricesH whose elementsHnm , n,m51,...,N are real, complex, orquaternion variables. In other words, each elementHnm has b real componentsHnm

(a) , a50,...,(b21) with b51,2,4, respectively,


athat

eels

e’s

aveera-

le

e


Downloaded 0

Hnm5 (a50

b21

Hnm(a)t (a). ~3.1!

Here, we use the basist (a), a50,...,(b21). We havet (0)51 for the real case withb51. Forthe complex case withb52, we havet (0)51 andt (1)5 i . Finally, we have

t (0)5F1 0

0 1G , t (1)5F 0 11

21 0 G ,~3.2!

t (2)5F 0 2 i

2 i 0 G , t (3)5F1 i 0

0 2 i Gin the quaternion case forb54 where thet (a), a51,2,3 are the Pauli matrices. We notice ththe totalH is a 2N32N matrix for b54. However, here and in the following, the dimensions twe use always refer to the number of matrix elements such asHnm . These are scalar forb51,2 and quaternion forb54. The labelb is often referred to as Dyson index.

We assume that the matrixH is real symmetric, Hermitian, or Hermitian self-dual in the thrcasesb51,2,4. We always writeH†5H to indicate this symmetry. There areN independent reavariablesHnn5Hnn

(0) , n51,...,N on the diagonal andbN(N21)/2 independent real variableHnm

(a) , a50,...,(b21),1<n,m<N outside the diagonal. We write the volume element ofH inthe form

d@H#5 )n51

N

dHnn(0) )

n,m)a50

b21

dHnm(a). ~3.3!

The matrixH is diagonalized by the matrixU, with columnsUn , n51,...,N . Depending on thevalue ofb, the matrixU is either orthogonal, unitary, or unitary-symplectic. Following Gilmornotation,22 we write UPU(N;b) with U(N;1)5SO(N), U(N;2)5U(N) and U(N;4)5USp(2N). The volume of these groups is given by

vol U~N;b!5 )n51

N2pbn/2

G~bn/2!5

2NpbN(N11)/4

)n51N G~bn/2!

. ~3.4!

We use it to normalize the invariant measure dm(U) of UPU(N;b) to unity,

E dm~U !51. ~3.5!

The N real eigenvaluesxn , n51, . . . ,N of H are ordered in the diagonal matrixx. We havex5diag(x1,...,xN) for b51 andb52. Forb54, the eigenvalues are doubly degenerate and we hx5diag(x1,x1,...,xN ,xN). Physically, this doubling of the eigenvalues is due to Kramer’s degencies. Thus, the diagonalization reads

H5U†xU, with Hnm5Un†xUm . ~3.6!

The diagonalizing matrixU has the propertyU215U†. The volume element in eigenvalue–angcoordinates is given by15,23

d@H#5CN(b)uDN~x!ubd@x#dm~U ! ~3.7!

where d@x# denotes the product of all differentialsdxn . We have introduced the Vandermonddeterminant


nd the

eul to

e. For

The

trix

ngular


Downloaded 0

DN~x!5 )n,m

~xn2xm!. ~3.8!

The normalization constant

CN(b)5

pbN(N21)/4

N!

GN~b/2!

)n51N G~bn/2!

~3.9!

obtains from the constants given in Mehta’s book15 and from Eq.~3.4!.To avoid inconveniences and to ensure a compact notation, we define the trace Tr a

determinant Det with Tr5tr and Det5det for b51,2 and with

Tr K5 12trK, DetK5AdetK ~3.10!

in the caseb54 for a matrixK with quaternion entries. Ifk denotes the diagonal matrix of theigenvalues of a real-symmetric, Hermitian or Hermitean self-dual matrix, it is also usefdefine the associate matrixk. In all three casesb, it is theN3N matrix k5diag(k1,k2,...,kN), i.e.,we havek5k for b51,2 and no degeneracies forb54.

B. Integral definition and differential equation

As in the case of vector Bessel functions, we start in the matrix case with the plane wavtwo matricesH and K with the same symmetriesH†5H and K†5K, we introduce the matrixplane wave as exp(i Tr HK) where the trace is the proper scalar product in the matrix space.matrix plane wave has the property

1

~2p!NpbN(N21)/2E d@H#exp~ iTr HK !5d~K !, ~3.11!

whered(K) is the product of thed distributions of all independent variables. We define the magradient]/]H and the Laplacian operator

D5Tr]2

]H2 5 (n51

N]2

]Hnn(0)2 1

1

2 (n,m

(a50

b21]2

]Hnm(a)2 , ~3.12!

which acts on the matrix plane wave as

Dexp~ i Tr HK !52Tr K2 exp~ iTr HK !. ~3.13!

We notice that, forb54, inconvenient factors of 2 would occur if we used tr instead of Tr.Analogously to vector Bessel functions, we define the matrix Bessel functions as the a

average

FN(b)~x,k!5E dm~U !exp~ i Tr HK !. ~3.14!

The diagonal matrixk contains the eigenvalues ofK, which is diagonalized by a matrixV suchthat K5V†kV. Due to the invariance of the measure dm(U), the matrixV is absorbed and thefunctionsFN

(b)(x,k) depend on the eigenvaluesx andk only,

FN(b)~x,k!5E dm~U !exp~ i Tr U†xUk!. ~3.15!

Thus, in the scalar product TrHK, solely the relative angles betweenH andK matter. The matrixBessel functions are symmetric in the arguments,


f thehe

hen

tantly,in

est

er,


Downloaded 0

FN(b)~x,k!5FN

(b)~k,x! ~3.16!

and normalized to unity,

FN(b)~x,0!51, FN

(b)~0,k!51 ~3.17!

due to Eq.~3.5!. These are spherical functions in the sense of Ref. 2.As in the vector case, the differential equation is obtained by averaging Eq.~3.13! over the

relative angles,

DE dm~V!exp~ i Tr HK !52Tr K2 E dm~V!exp~ i Tr HK !. ~3.18!

Again, the LaplacianD commutes with the integral, because the former is in the space omatrix H, the latter over the diagonalizing matrixV of K. The integral also commutes witTr K25Tr k2. Due to the symmetry betweenH andK, the integral is obviously identical to thdefinition ~3.15! and we find

DxFN(b)~x,k!52Tr k2 FN

(b)~x,k!. ~3.19!

Since the matrix Bessel functionFN(b)(x,k) depends only on the radial variables, i.e., on t

eigenvalues, we replaced the full Laplacian with its radial partDx . Because of the transformatiorule ~3.7!, it reads

Dx5 (n51

N1

uDN~x!ub]

]xnuDN~x!ub

]

]xn

5 (n51

N]2

]xn2 1 (

n,m

b

xn2xmS ]

]xn2

]

]xmD . ~3.20!

We notice that these steps are fully parallel to the corresponding discussion in Sec. II. Impordue to the symmetry~3.16!, the functionsFN

(b)(x,k) must also solve the differential equationthe kn , n51,...,N, which results from Eq.~3.19! by exchangingx andk. Obviously, this a veryrestrictive requirement.

Comparing the radial operator~2.10! in the vector case and the radial operator~3.20!, we seethat it is theb that corresponds to thespatial dimensiond or, more precisely, tod21. The roleplayed by thematrix dimensionN is a different one. To illustrate this, we study the two simplcases. First, we can formally setN51 and find from the definition~3.14! that F1

(b)(x,k)5exp(ix1k1), whereH115x1 and K115k1 . In this case, the matrixU has dropped out trivially.This reflects simply that the scalar productkW•rW is linear in the relative solid angleV between thevectors whereas the scalar product TrHK is quadratic in the relative diagonalizing matrixU. Thecorresponding radial LaplacianDx for N51 is identical to the CartesianD. Therefore, the caseN51 is too trivial to give any further insight. Second, we setN52 and find straightforwardlyfrom the differential equation~3.19!

F2(b)~x,k!5expS i

~x11x2!~k11k2!

2 D x (b11)S ~x12x2!~k12k2!

2 D , ~3.21!

wherex (d) is the vector Bessel function ind dimensions as defined in Eq.~2.3!. This functionsappears in the solution, because the differencesx12x2 and k12k2 directly correspond to thelengthsurWu and ukW u. In higher matrix dimensionsN, this simple correspondence is lost. Howevwe will see in great detail that the features of the functionsFN

(b)(x,k), in particular whether or notexplicit solutions can be constructed, are more strongly influenced byb than byN.


ential

heore

ucing. The

d by

is the

l

r

d–from

por-

itian

uniqueparam-

een to

r


Downloaded 0

Another point in this context deserves to be underlined. In the vector case, the differequation~2.10! and the solution~2.6! were constructed for integer dimensionsd. However, bothequations are also well defined for any real and positived. Similarly, we observe in the matrixcase that the differential equation~3.19! was derived for the casesb51,2,4. However, neitheritself nor its solution~3.21! for N52 are confined to these casesb51,2,4, they are valid for anyreal and positiveb. Thus, the casesb51,2,4 which correspond to a matrix model, i.e., to tdefining integral~3.14! of the matrix Bessel functions, are only special cases of a much mgeneral problem, namely finding the solutions of the differential equation~3.19! for every integerN and forarbitrary real values ofb. We will return to this in Sec. V.

IV. RECURSION FORMULA

The matrix Bessel functions show a recursive structure which we construct by introdradial Gelfand–Tzetlin coordinates. The result is stated in Sec. IV A and derived in Sec. IV Bcorresponding invariant measure is calculated in Sec. IV C.

A. Statement of the result

The matrix Bessel functions, defined in Eq.~3.15!,

FN(b)~x,k!5E dm~U !exp~ i Tr U†xUk! ~4.1!

depend on the radial space of the eigenvaluesx andk. As before, we writex5diag(x1,...,xN) andk5diag(k1,...,kN) for b51,2 and, for b54, we write x5diag(x1,x1,...,xN ,xN) and k5diag(k1,k1,...,kN ,kN). We emphasize that the radial spaces do not lie in the manifolds coverethe groups U(N;b). However, we will show that the group integral~4.1! can be exactly mappedonto a recursive structure which acts exclusively in the radial space. This remarkable featuremain result of this section.

Under rather general circumstances, the matrix Bessel functionsFN(b)(x,k) can be calculated

iteratively by the explicitrecursion formula

FN(b)~x,k!5E dm~x8,x!exp~ i ~Tr x2Tr x8!kN!FN21

(b) ~x8,k!, ~4.2!

whereFN21(b) (x8,k) is the group integral~4.1! over U(N21;b). We have introduced the diagona

matrix k5diag(k1,...,kN21) for b51,2 and k5diag(k1,k1,...,kN21,kN21) for b54 such thatk5diag(k,kN) for b51,2 andk5diag(k,kN ,kN) for b54. Importantly, theN21 integration vari-ablesxn8 , n51,...,N21, ordered in the diagonal matrixx85diag(x18 ,...,xN218 ) for b51,2 andx8

5diag(x18 ,x18 ,...,xN218 ,xN218 ) for b54 are arguments ofFN21(b) (x8,k). Moreover, we notice that thei

further appearance in the exponential is a simple one due to the trace.The coordinatesx8 are constructed in the spirit of, but they are different from, the Gelfan

Tzetlin coordinates of Refs. 24 and 25. To clearly distinguish these two sets of coordinateseach other, we refer to the latter asangularGelfand–Tzetlin coordinates and to the variablesx8 asradial Gelfand–Tzetlin coordinates. The difference is at first sight minor, but of crucial imtance. In the angular case,x is in the Cartan subalgebra belonging to U(N;b). In the radial case,however,x is in theradial space of the eigenvalues of the real-symmetric, Hermitian, or Hermself-dual matrixH, which are the arguments of the functions~4.1!. While the angular Gelfand–Tzetlin coordinates never leave the group space, the radial ones establish an exact andrelation between the group and the radial space. The radial Gelfand–Tzetlin coordinates reetrize the sphere that is described by theNth columnUN of the matrixUPU(N;b). The recursionformula~4.2! can only be constructed in the radial coordinatesx8, but not in the angular ones. Thradial and the angular Gelfand–Tzetlin coordinates are, in general, different. They happcoincide for b52, i.e., for the unitary group U(N). This illustrates, in the framework of ourecursion formula, the special role played by the unitary group.


y no

sel

s the


Downloaded 0

The invariant measure dm(x8,x) is, apart from phase angles, the invariant measure dm(UN)on the sphere in question, expressed in the radial coordinatesx8. It only contains algebraicfunctions and reads explicitly

dm~x8,x!52N21G~Nb/2!

pN(b22)(b24)/6

DN21~x8!

DNb21~x! S 2)

n,m~xn2xm8 ! D (b22)/2

d@x8#. ~4.3!

The normalization constant obtains from results in Gilmore’s book.22 It ensures normalization tounity according to Eq.~3.5!. The domain of integration is compact and given by

xn<xn8<xn11 , n51,...,~N21!, ~4.4!

reflecting a ‘‘betweenness condition’’ for the radial Gelfand–Tzetlin coordinates. This is whabsolute value signs appear in the measure~4.3!.

The general recursion formula~4.2! states an iterative way for constructing the matrix BesfunctionFN

(b)(x,k) for arbitraryN from the matrix Bessel functionF2(b)(x,k) for N52 which can

usually be obtained trivially. We remark that the recursion formula allows one to expresmatrix Bessel functions in the form

FN(b)~x,k!5E )

n51

N21

dm~x(n),x(n21)!

3exp~ i ~Tr x(n21)2Tr x(n)!kN2n11! exp~ ix1(N21)k1!, ~4.5!

where we have introduced the radial Gelfand–Tzetlin coordinatesxm(n) , m51,...,N2n on N21

levelsn51,...,(N21). We definex(0)5x andx(1)5x8.

B. Derivation

We introduce a matrixV5diag(V,V0) with VPU(N21;b) and V0PU(1;b) such thatVPU(N21;b) ^ U(1;b),U(N;b) and multiply the right-hand side of the definition~4.1! with

15E dm~V!5E dm~V0! E dm~V!. ~4.6!

The invariance of the Haar measure dm(U) allows us to replaceU with UV† and to write

FN(b)~x,k!5E dm~V0!E dm~V!E dm~U ! exp~ i Tr U†xUV†kV!. ~4.7!

We collect the firstN21 columnsUn of U in the N3(N21) rectangular matrixB such thatB5@U1 U2¯UN21# andU5@B UN#. We notice that

B†B51N21 ,~4.8!

BB†5 (n51

N21

UnUn†51N2UNUN

† .

As already stated in Sec. III A, the elements of a vector or a matrix are scalar forb51,2 andquaternion forb54. In this sense, we also write 1N as the unit matrix forb54 because itselements aret (0). By defining the (N21)3(N21) square matricesH5B†xB andK5V†kV wemay rewrite the trace in Eq.~4.7! as

Tr U†xUV†kV5Tr HK1HNNkN ~4.9!


m

lysates.se.

uations


Downloaded 0

with HNN5UN† xUN according to Eq.~3.6!. We notice thatV0 has dropped out. Since the first ter

of the right-hand side of Eq.~4.9! depends only on the firstN21 columnsUn collected inB andthe second term depends only onUN , we use the decomposition

dm~U !5dm~B!dm~UN! ~4.10!

of the measure to cast Eq.~4.7! into the form

FN(b)~x,k!5E dm~UN!exp~ iH NNkN! E dm~V! E dm~B!exp~ i Tr HK !, ~4.11!

where we have already done the trivial integration overV0 .The difficulty to overcome lies in the decomposition~4.10!. While dm(UN) is simply the

invariant measure on the sphere described byUN , the measure dm(B) is rather complicated.Pictorially speaking, the degrees of freedom in dm(B) have always to know that they are localorthogonal toUN . Thus, dm(B) depends onUN . Luckily, there is one distinct set of coordinatethat is perfectly suited to this situation. It is the system of the radial Gelfand–Tzetlin coordinWe construct it by transferring the methods of Ref. 26 for the angular case to the radial ca

TheN3N matrix (1N2UNUN† ) is a projector onto the (N21)3(N21) space obtained from

the originalN3N space by slicing off the vectorUN . We project the radial coordinatesx onto thisspace and study its spectrum. The defining equation reads

~1N2UNUN† ! x ~1N2UNUN

† ! En85xn8 En8 , n51,...,N21. ~4.12!

Equation~4.12! determines theN21 radial Gelfand–Tzetlin coordinatesxn8 and the correspondingvectorsEn8 as eigenvalues and eigenvectors of the matrix (1N2UNUN

† ) x (1N2UNUN† ) which has

the rankN21. Since we have by constructionUN† En850, we may as well write

~1N2UNUN† ! x En85xn8 En8 , n51,...,N21. ~4.13!

The eigenvaluesxn8 ,n51,...,N are obtained from the characteristic equation

05Det~~1N2UNUN† !x2xn8!

5Det~x2xn8!det~1N2~x2xn8!21UNUN† x!

5Det~x2xn8! S 12UN† x

x2xn8UND

52xn8 Det~x2xn8! TrUN† 1N

x2xn8UN . ~4.14!

Together with the normalization TrUN† UN51, this yields theN equations

15Tr UN† UN5 (

n51

N

(a50

b21

UnN(a)2,

~4.15!

05Tr UN† 1N

x2xn8UN5 (

m51

N

(a50

b21 UmN(a)2

xm2xn8, n51,...,N21.

In these formulas, the trace Tr is only needed in the symplectic case. We notice that the eqfor the variablesx8 depend on the variablesx as parameters. We emphasize once more thatx inthese equations is in the radial space and, in general, not in the Cartan subalgebra of U(N;b).


ates

q.

.

ve


Downloaded 0

At this point, it is not clear yet why the introduction of the radial Gelfand–Tzetlin coordinis at all helpful. The great advantage will reveal itself when we express the matrixH and thematrix elementHNN in the trace~4.11! in these coordinates. To this end, we first multiply E~4.12! from the right withEn8

† and sum overn,

~1N2UNUN† ! x ~1N2UNUN

† !5 (n51

N21

xn8 En8En8† , ~4.16!

where we used the completeness relation

(n51

N21

En8En8† 1 UNUN

† 51N . ~4.17!

Taking the trace of the spectral expansion~4.16! we find immediately

Tr x2Tr x85Tr UN† xUN5HNN . ~4.18!

This is a remarkably simple result. An analogous expression exists for theNN matrix element ofthe unitary group in the theory of angular Gelfand–Tzetlin coordinates for the unitary group24,25

Here we have shown that Eq.~4.18! is a general feature in every radial space.We now turn to the (N21)3(N21) matrix H. Its N21 eigenvaluesyn , n51,...,N21 are

determined by the characteristic equation

05Det~H2yn!5Det~B†xB2yn!

521

ynDet~BB†x2yn!

521

ynDet~~1N2UNUN

† !x2yn!, ~4.19!

where we used Eq.~4.8! and reexpressed an (N21)3(N21) determinant as anN3N determi-nant. The comparison of Eq.~4.19! with Eq. ~4.14! shows that, most advantageously, we hayn[xn8 ,n51,...,N21. Thus we may write

H5B†xB5U†x8U ~4.20!

by introducing the (N21)3(N21) squarematrix U which diagonalizesH. Obviously,U mustbe a complicated function of theN3(N21) rectangularmatrix B, i.e., of the columnsUn ,n51,...,N21. However, all we need to know is thatU must be in the group U(N21;b) because,by construction,H has the symmetryH†5H.

Collecting everything, we cast Eq.~4.11! into the form

FN(b)~x,k!5E dm~x8,x!exp~ i ~Tr x2Trx8!kN!

3E dm~V!E dm~B!exp~ i Tr U†x8UV†kV!. ~4.21!

We may now use the invariance of the Haar measure dm(V) to absorbU such that


ng

ation-

sults

e

in


Downloaded 0

FN(b)~x,k!5E dm~x8,x!exp~ i ~Tr x2Tr x8!kN!

3E dm~V!exp~ i Tr x8V†kV!E dm~B!. ~4.22!

Thus, the integration overB is trivial and yields unity due to our normalization. The remainiintegration overV gives precisely the matrix Bessel functionFN21

(b) (x8,k). This completes thederivation of the recursion formula in Sec. IV A. The reader experienced with group integrhas realized that the introduction of the matrixV5diag(V,V0) was not strictly necessary. Alternatively, one could have shown that the measure dm(B) can be identified with dm(U) and have donethe corresponding integral. However, we believe that the introduction ofV makes this part of thederivation more transparent.

C. Invariant measure

The invariant measure dm(UN) has to be expressed in terms of the radial coordinatesx8. Tothis end, we first have to solve Eq.~4.15! for the moduli squared of the vectorUN as a function ofthe new coordinatesx8. Since Eq.~4.15! for the total moduli square for allb coincides with theequation for theangularGelfand–Tzetlin coordinates of the unitary group, we can use the reas derived in Refs. 24 and 25. We have in the three cases

uUnNu25 (a50

b21

~UnN(a)!25

)m51N21~xn2xm8 !

)mÞn~xn2xm!. ~4.23!

The betweenness condition~4.4! follows from the positive definiteness of this expression. Wparametrize the remaining degrees of freedom ofUnN in the casesb52,4. We setUnN

(0)

5cosgn(1) andUnN

(1)5singn(1) in the caseb52 and

UnN5F coscn exp~ ign(1)! sincn exp~ ign

(2)!

2sincn exp~2 ign(2)! coscn exp~2 ign

(1)!G ~4.24!

for b54 in the basis~3.2!. The invariant length element reads

Tr dUN† dUN5 (

n51

N

(a50

b21

~dUnN(a)!2

5 (n51

N S 1

4uUnNu2 ~duUnNu2!2

1(i 51

b/2

uUnNu2~dgn( i )!21db4uUnNu2~d coscn!2D . ~4.25!

To express the differential duUnNu2 in terms of the dxn8 , we again take advantage of the resultsRefs. 24 and 25,

(n51

N1

4uUnNu2 ~duUnNu2!25 (n51

N21)m51

N21~xm8 2xn8!

4)m51N ~xm2xn8!

~dxn8!2. ~4.26!

From these equations, we can read off the metricg in the basis of the coordinatesxn8 ,gn( i ) andcn .

Conveniently, it is diagonal. The determinant ofg is given by


s

ationneralf then. In

eseneral

as

o realoups.

ns

-


Downloaded 0

detg5DN21

2 ~x8!

DN2b22~x! )n,m

~xn2xm8 !(b22), ~4.27!

which yields the invariant measure dm(UN) in terms of thexn8 and of the additional coordinategn

( j ) andcn . These angles can be integrated out trivially. This yields Eq.~4.3!.

V. RADIAL FUNCTIONS FOR ARBITRARY b

Remarkably, the recursion introduced in Sec. IV, is the iterative solution of the radial equfor arbitrary values ofb. Thus, the matrix Bessel functions are special cases of more gefunctions which we want to refer to as radial functions. We give the precise formulation oproblem in Sec. V A and show in Sec. V B that the recursion is the general iterative solutioSec. V C, we discuss a Hankel ansatz for the radial functions.

A. Definition by the differential equation

In Sec. III B, we defined the matrix Bessel function through the group integral~3.14! or,equivalently, the group integral~3.15!. This definition confines the dimensionb to the valuesb51,2,4, corresponding to the groups U(N;b). However, discussing the simplest caseN52, wealready saw in Sec. III B thatF2

(b) is well defined forarbitrary values ofb. This was a simpleconsequence of the explicit form~3.21! which expressesF2

(b) in terms of the Bessel functionx (b11). The latter is known to be well defined for arbitraryb. Hence, we conclude that the casb51,2,4 which relate to matrices and groups are embedded into a space of far more gfunctions.

It seems natural that this phenomenon also extends toN.2. The problem has to be posedfollows: We seek the solutionsFN

(b) of the differential equation

DxFN(b)~x,k!52 (

n51

N

kn2 FN

(b)~x,k!, ~5.1!

where the operator is given by

Dx5 (n51

N]2

]xn2 1 (

n,m

b

xn2xmS ]

]xn2

]

]xmD . ~5.2!

Here,b is arbitrary. For technical reasons, however, we restrict ourselves for the time being tand positive values ofb. We make no reference whatsoever to matrices, eigenvalues, and grTo emphasize this, we viewx andk as sets ofN variablesxn , n51,...,N andkn , n51,...,N forevery positiveb. We do not use traces.

We require that the solutions are symmetric in the argument

FN(b)~x,k!5FN

(b)~k,x! ~5.3!

and normalized

FN(b)~0,k!51 or FN

(b)~x,0!51 ~5.4!

at the originx50 and k50. Due to the symmetry, one of the two normalization conditiosuffices.

In the sequel, we want to refer to the functionsFN(b)(x,k) for arbitraryb as radial functions

while we reserve the termmatrix Bessel functionsto the casesb51,2,4 where the direct connection to matrices and Lie groups exists.


s

l

v-

it

ve

ries


Downloaded 0

B. Recursive solution

We claim that the solutions are, for arbitraryb, given as an iteration inN by the recursionformula

FN(b)~x,k!5E dm~x8,x!expS i S (

n51

N

x2 (n51

N21

x8D kND FN21(b) ~x8,k!, ~5.5!

whereFN21(b) (x8,k) is the solution of the differential equation~5.2! for N21. Here,k denotes the

set of variableskn , n51,...,(N21) and x8 the set of integration variablesxn8 , n51,...,(N21). The integration measure

dm~x8,x!5GN(b) DN21~x8!

DNb21~x! S 2)

n,m~xn2xm8 ! D (b22)/2

d@x8# ~5.6!

is the continuation of Eq.~4.3! to arbitrary positiveb. The normalization constant

GN(b)52N21

G~Nb/2!

GN~b/2!~5.7!

is also the continuation of the constant in Eq.~4.3!. We calculate it in Appendix F. As in the caseb51,2,4, the inequalities

xn<xn8<xn11 , n51,...,~N21! ~5.8!

define the domain of integration.We stress that we derived the recursion formula~5.5! in Sec. IV for the casesb51,2,4. To

prove that it is the iterative solution for arbitrary positiveb, we show that it solves the differentiaequation~5.1!. The keystone for the proof is the identity

DxFN(b)~x,k!52kN

2 FN(b)~x,k!1E dm~x8,x!expS i S (

n51

N

xn2 (n51

N21

xn8D kNDDx8FN21(b) ~x8,k!,

~5.9!

which is derived in Appendix D. Equation~5.9! establishes a not immediately obvious, but neertheless natural connection between, on the one hand, the action of the LaplacianDx in the Nvariablesxn on the radial function inN dimensions, i.e., on the recursion integral~5.5!, and, on theother hand, the recursion integral over the LaplacianDx8 in the N21 variablesxn8 acting on theradial function inN21 dimensions. There is a compensation term which is just2kN

2 FN(b)(x,k).

Thus, we can prove the eigenvalue equation~5.1! by induction: assuming that it is correct forN21, identity ~5.9! implies Eq.~5.1! for N. The induction starts withN52 where the eigenvalueequation~5.1! is clearly valid for arbitraryb as shown in Sec. III B by deriving the explicsolution ~3.21!.

The symmetry relation~5.3! is nontrivial. In the matrix casesb51,2,4, it is obvious from theintegral definitions~3.14! and~3.15!. For arbitraryb, we cannot use this argument, we only hathe recursion~5.5!. In Appendix E, we prove the symmetry relation~5.3! by an explicit change ofvariables.

The normalizationFN(b)(x,0)51 in Eq. ~5.4! follows directly from the normalization of the

measure~5.6!. The symmetry relation~5.3! then also yieldsFN(b)(0,k)51.

Regarding the domain ofb, a comment is in order. We have seen in Sec. III B that forN52 the matrix Bessel function is well defined for arbitrary complexb. This should also be true foour recursion formula~5.5!. However, forb<0 nonintegrable singularities arise at the boundarin the integral in Eq.~5.5!. At the same time the normalization constant becomes zero forb50,


la fores to

riableshials

dition,ble to. Anerted

t partly,

neral,n to

ionsver

s

,

cesrries


Downloaded 0

22,24,..., compensating the singularities of the integral. This makes the recursion formub<0 not ill-defined but it gets more difficult to treat. Therefore, we have restricted ourselvpositive values ofb.

In the work of Okounkov and Olshanski11 an expansion of the radial functions for arbitrarybin Jack polynomials is derived. The series run over sets of partitions$l%. These authors alsoderive a recursion formula for the Jack polynomials depending on one set of continuous vax, say, and belonging to such partitions$l%. It is related to, but different from ours, whicinvolves two sets of continuous varablesx andk. The crucial difference rests in the exponentfunction which is present in our formula~5.5!, but not in the formula of Ref. 11. Importantly, it ithis exponential term which makes sure that the symmetry condition~5.3! is fulfilled on all levelsof the recursion. Since the Jack polynomials themselves do not obey such a symmetry conthere is no exponential term in the recursion formula of Ref. 11. However, it must be possiderive the recursion formula for the radial functions from the one for the Jack polynomialsinteresting, although probably not very elegant approach would be the following: If one insthe recursion formula for the Jack polynomials into the expansion11 of the radial functions in termsof these Jack polynomials, one ought to see that the series over the partitions can, at leasbe resummedto yield the exponential function present in the recursion formula~5.5!. This isremarkable and could be very helpful for the application of Jack polynomials, because, in geresummations over partitions are known to be difficult and involved. For the connectioCalogero–Sutherland models, we refer the reader to Sec. VII.

C. Hankel ansatz

In the spirit of Eq.~2.12! for the vector case, we make a Hankel ansatz for our radial functfor arbitrary positiveb. We also do this in view of the applications in Sec. VI. Since the sum othe kn

2 on the right-hand side of the eigenvalue equation~5.1! is invariant under all permutationof thekn or, equivalently, their indicesn, we can label a set of solutionsFN,v

(b) (x,k) by an elementv of the permutation groupSN of N objects. For these solutions, we make the ansatz

FN,v(b) ~x,k!5

exp~ i (n51N xnkv(n)!

uDN~x!DN~k!ub/2 WN,v(b) ~x,k!, ~5.10!

wherev(k) is the diagonal matrix constructed fromk by permuting thekn , or the indicesn. Thefull solution FN

(b)(x,k), satisfying the constraints~5.3! and ~5.4!, is then, apart from possiblenormalization constants, given as the linear combination

FN(b)~x,k!5

1

N! (vPSN

~21!p(v)FN,v(b) ~x,k! ~5.11!

of the functions~5.10!. Here,p(v) is the parity of the permutation.We find for the functionWN,v

(b) (x,k) the differential equation

Lx,v(k) WN,v(b) ~x,k!50, ~5.12!

where the operator is given by

Lx,v(k)5 (n51

N]2

]xn2 1 i2(

n51

N

kv(n)

]

]xn2bS b

221D (

n,m

1

~xn2xm!2 . ~5.13!

This differential equation generalizes Eq.~2.13! to the matrix case forb51,2,4 and, furthermorethe latter to general radial functions for arbitraryb. Due to the symmetry~5.3!, the differentialequation~5.12! must also hold ifx andv(k) are interchanged.

The functionsWN,v(b) (x,k) are translation invariant, i.e., they depend only on the differen

(xn2xm). We show this in Appendix G. Again, because of the symmetry, this argument ca


.

slation

s

an

atless

lasis

p the

ula

ial


Downloaded 0

over to k, and WN,v(b) (x,k) depends only on the differences (kn2km) as well. Moreover, the

symmetry implies that it depends only on the products (kv(n)2kv(m))(xn2xm).It is the last term of the operatorLx,v(k) that makes the differential equation~5.12! so difficult.

In the sequel, we are only interested in solutionsWN,v(b) (x,k) which are translation invariant

Moreover, in the caseb52, when the last term of the operatorLx,v(k) vanishes, oscillatorysolutions can be possible. We exclude them because, first, they are, in general, not traninvariant and, second, they can be absorbed in the oscillatory part of the ansatz~5.10!. Thus, wesimply haveWN,v

(2) (x,k)51. This is the Itzykson–Zuber case6 corresponding to unitary matriceUPU(N). For arbitraryb, it is obvious from the differential operator thatWN,v

(b) (x,k)→1 if uxn

2xmu→` for all pairsn,m. Once more, this must also be true ifukn2kmu→`. Thus, we expectthat WN,v

(b) (x,k) is some kind of asymptotic series, generalizing Eq.~2.14! in the vector case.Hence, we rederive a known result by concluding that the leading contribution in

asymptotic expansion of the functions~5.10! is given by

FN,v(b) ~x,k!;

exp~ i (n51N xnkv(n)!

uDN~x!DN~k!ub/2 . ~5.14!

According to Eq.~5.11!, this means that

FN(b)~x,k!;

det@exp~ ixnkm!#n,m51, . . . ,N

uDN~x!DN~k!ub/2 ~5.15!

is the asymptotic behavior of the radial functionsFN(b)(x,k) if the differencesuxn2xmu and

ukn2kmu are large for all pairsn,m.Collecting all pieces of information, we make the ansatz

WN,v(b) ~x,k!5(

$m%

am12m13¯m(N21)N

)n,m~~kv(n)2kv(m)!~xn2xm!!mnm~5.16!

with coefficientsam12m13¯m(N21)Nthat depend onN(N21)/2 integer indicesmnm , as many as

there are differences. The summation is over the set of these indices. The presence of thekn makesit very difficult to solve Eq.~5.12! with the ansatz~5.16!. In the vector case, one easily sees ththe differential equation~2.13! in r can be transformed into an equation in the dimensionvariableskr such thatk does not appear anymore. This leads to the simple recursion~2.15! for thecoefficients. Here, in the matrix case, thekn cannot easily be absorbed and the recursion formufor the coefficients will depend on thekn in a nontrivial way. However, in some simple cases, itpossible to solve them. These difficulties were an important motivation for us to develomethods which we introduced in Sec. IV.

VI. APPLICATIONS

Can we obtain an explicit formula for the radial functions by using the recursion form~5.5!?—At least in some cases, this ought to be possible. Here, we present some results.

For the sake of completeness, we comment once more on the special caseb52, i.e., theunitary case. Obviously, the measure~5.6! simplifies enormously. This is so because the radGelfand–Tzetlin coordinates coincide with the angular ones. Thus, the caseb52 is identical tothe rederivation of the Itzykson–Zuber integral by Shatashvili.25

We now consider the orthogonal caseb51. The recursion formula reads

FN(1)~x,k!5GN

(1)E DN21~x8!

A2)n,m~xn2xm8 !expS i S (

n51

N

xn2 (n51

N21

x8D kND FN21(1) ~x8,k!d@x8#.

~6.1!


bvious

rent

inwe

alntialials.alues

given

s

an


Downloaded 0

The square roots appearing in the measure make a further evaluation very difficult. As is ofrom the trivial caseN52, given in Eq.~3.21!, the functionFN

(1)(x,k) will be an infinite series forall values ofN. However, we expect that, due to the different construction, this series is diffefrom the expansion inzonal functionswhich was obtained by Muirhead.20

Obviously, there is a pattern emerging. The integration measure~5.6! is purelyrational for alleven and positive values ofb. This is reminiscent of the situation for vector Bessel functionsodd dimensionsd, which consist of afinite number of terms, as discussed in Sec. II. Hence,conjecture that the radial functionsFN

(b)(x,k) can also be written as afinite sum, exclusivelycontaining exponential and rational functions.

For all other values ofb, the measure~5.6! is algebraic, but not rational, and the radifunctionsFN

(b)(x,k) must beinfinite series. Nevertheless, these infinite series contain exponeand rational functions. Thus, they are different from expansions in terms of zonal polynom

To furnish our conjecture about the form of the radial functions for even and positive vof b with an illustrative example, we turn to the unitary-symplectic caseb54. The results givenhere in the sequel were first derived by the present authors in Ref. 27. Later, they were alsoby Brezin and Hikami in Ref. 28.

To simplify the notation we avoid the imaginary unit by writing

FN(4)~2 ix,k!5E

UPUSp(2N)exp~Tr u21xuk!dm~U !, ~6.2!

wherex5diag(x1,x1, . . . ,xN ,xN) andk5diag(k1,k1, . . . ,kN ,kN) are diagonal matrices with Kramerdegeneracies. The starting point of the recursion is the smallest nontrivial caseN52, i.e., thegroup USp~4!. We obtain after an elementary calculation

F2(4)~2 ix,k!5G2

(4) (vPS2

S 1

D22~x!D2

2~v~k!!2

2

D23~x!D2

3~v~k!! Dexp~Tr xv~k!!. ~6.3!

The sum runs over the elements of the permutation groupSN for N52. Inserting Eq.~6.3! into therecursion formula, we find for USp~6!, the next step in the recursion,

F3(4)~2 ix,k!5G3

(4)G2(4) (

vPS2

Ex1

x2dx18E

x2

x3dx28

) i 513 ) j 51

2 ~xi2xj8!

D33~x!D2

2~v~ k!!exp~~Tr x2Tr x8!k31Tr xv~ k!!

3S 1

D2~x8!2

2

D22~x8!D2~v~ k!!

D . ~6.4!

Although the integrand is finite everywhere, in particular atx185x285x2 , the denominatorsD2(x8)andD2

2(x8) raise a technical difficulty. The key to remove them is to use the identity

2

D22~x8!

52S ]

]x182

]

]x28D 1

D2~x8!~6.5!

and to observe that the product) i 513 ) j 51

2 (xi2xj8) annihilates all boundary terms. Hence, we cintegrate by parts and arrive at

F3(4)~2 ix,k!5G3

(4)G2(4) (

vPS2

1

D33~x!D2

3~v~ k!!E

x1

x2dx18E

x2

x3dx28(

i 51

3

)j 51j Þ i

3

~xj2x18!~xj2x28!

3exp~~Tr x2Tr x8!k31Tr xv~ k!!, ~6.6!


canq.

r

tz as

dive the

y

n

es of

efIn theand also


Downloaded 0

where no denominator is left. Due to the permutation symmetry of the original integral, werestrict ourselves to the unity elemente of the permutation group in the further evaluation of E~6.6!. Thus we need only to consider the limitsxi8→xi , i 51,2 while integrating by parts. Aftecollecting orders ink we find

F3,e(4)~2 ix,k!5G3

(4)G2(4) 1

D33~x!D3

3~k!S 2D3~x!D3~k!12(

i , j

3 S D3~x!D3~k!

~xi2xj !~ki2kj !D

24(i , j

3

~xi2xj !~ki2kj !112D exp~Tr xk!. ~6.7!

By introducing the composite variables

zv( i j )5~xi2xj !~kv( i )2kv( j )!, i , j 51,...,3, vPS3 , ~6.8!

we can expressF3(4)(2 ix,k) in a compact form as

F3(4)~2 ix,k!5G3

(4)G2(4) (

vPS3

1

D33~x!D3

3~v~k!!S 41)

i , j

3

~22zv( i j )!D exp~Tr xv~k!!. ~6.9!

So far, we have not been able to extend this procedure to all values ofN.However, we succeeded in calculatingF4

(4)(2 ix,k), i.e., the case of the group USp~8!, by anhybrid method which combines information obtained from the recursion with a Hankel ansadescribed previously. We extend the right hand side of Eq.~6.9! for N53 to N54 and use thisexpression as an ansatz for the functionWN,v

(4) (x,k). As it turns out, a correction term is needeand, furthermore a correction to the correction. Fortunately, there is a structure to this. We gdetails in Appendix H. We emphasize that the knowledge ofF3

(4)(2 ix,k) is essential for thishybrid procedure, in particular the fact thatF3

(4)(2 ix,k) contains only linear terms in evercomposite variablezv( i j ) . Up to a normalization,F4

(4)(2 ix,k) is given by

F4(4)~2 ix,k!5 (

vPS4

1

D43~x!D4

3~v~k!! S )i , j

~22zv( i j )!122 (l ,m,n

)i , jÞ lmÞ lnÞmn

~22zv( i j )!

123(l ,mk,n

)i , j ,Þ lk,Þ ln

Þmk,Þmn,Þkn

~22zv( i j )!D exp~Tr xv~k!!. ~6.10!

Comparing this result with Eq.~6.9! we notice that, once more, the composite variableszv( i j ) enteronly linearly in the polynomial part ofF4

(4)(2 ix,k). Similarly, the spherical Bessel functioj 1(z), which is the counterpart ofFN

(4)(x,k) in the vector case given in Eqs.~2.6! and~3.21!, hasa polynomial part linear inz. We expect that such analogies are also present for higher valub and the dimensiond.

Formula ~6.10! indicates a general structure forFN(4)(x,k). The leading term is always th

generating function of the elementary symmetric functions inz. To this term combinations oother symmetric functions are added, where certain combinations of indices are cut out.supersymmetric case, we could apply the present method in even more complicated casesfind explicit results.40


uallyappli-

ndator byughe

rminedr

es to

’s

non-

b-

pace of

erator

ra–s to a

d sym-ndyel

bles,

nc-


Downloaded 0

VII. CONNECTION TO CALOGERO–SUTHERLAND MODELS

The radial functions are related to, but different from, the eigenfunctions which are usemployed in models of the Calogero–Sutherland type. Since this issue is so important forcations and so often raised in discussions, we briefly collect the main points.

The radial Laplace operator defined in Eq.~5.2! is closely related to the Calogero–SutherlaHamiltonians. In general one can always cast a Fokker–Planck operator in a Hamilton operadjunction29 with the square root of the stationary probability distribution defined throDxPeq(x)50. ChoosingPeq(x)5uDN(x)u2b/2, the operator~5.2! can be associated with thHamiltonian

HD52 (n51

N]2

]xn2 1

b~b22!

2 (n,m

N1

~xn2xm!2 . ~7.1!

It describes a scattering system with a continuous spectrum, the large time behavior is deteby the states near the ground state. Apart from a sign, this operatorHD coincides with the operatoLx,0 in Eq. ~5.13! for k50. We also notice that the interaction vanishes forb52.

To have a well-defined thermodynamic limit one often confines the motion of the particla circle. This yields the Calogero–Sutherland Hamiltonian

HCS52 (n51

N]2

]xn2 1

b~b22!

2 (n,m

N~p/N!2

sin2~p~xn2xm!/N!, ~7.2!

which can also be derived directly from Dyson’s circular ensembles.15 Another way of confiningthe particles is by a harmonic potential. This leads to the Calogero Hamiltonian30

HC52 (n51

N]2

]xn2 1

b~b22!

2 (n,m

N1

~xm2xm!2 11

16 (n51

N

xn2. ~7.3!

In the thermodynamic limit the particle densityR1(x) of the ground state is described by Wignersemicircle law. The mean particle level spacingD51/R1(0) scales asD}1/AN. Therefore in thethermodynamic limit the harmonic confining term in Eq.~7.3! vanishes on the scale of the mealevel spacing. On thisunfolded scalethe correlation functions become independent of the cfinement mechanism. The three HamiltoniansHC , HD , and HCS are known to be integrablesystems for arbitraryb.31 However, the three valuesb51,2,4 are distinguished, since they estalish a connection to the random matrix ensembles. Indeed, for these values ofb they belong to amuch wider class of integrable systems, which can be constructed by means of the root sa simple Lie algebra or—still more generally—of a Kac–Moody algebra.5 This class comprisesHamiltonians which can be derived by an adjunction procedure from a Laplace–Beltrami opof a group acting in a symmetric space. This space has positive curvature forHCS and zerocurvature forHD ,HC . In Refs. 32 and 33 it was pointed out that the Dorokhov–Mello–PereyKumar equation for scattering matrices with broken time reversal symmetry correspondLaplace–Beltrami operator in a symmetric space of negative curvature.

EigenfunctionscN,E(b) (x) of the HamiltoniansHC ,HD ,HCS with eigenenergyE for arbitraryb

are known. Essentially, these solutions are products of the ground state wave function anmetric polynomials in the coordinatesx of the N particles. In case of the Calogero–SutherlaHamiltonianHCS, these polynomials are the Jack polynomials.34–36 In this approach, the energeigenvaluesE are labeled by a partition of lengthN. The crucial difference to the matrix Bessfunctions is that the Jack polynomials are symmetric polynomials in one set of variablesx onlywhereas the matrix Bessel functionsFN

(b)(x,k) are symmetric in two sets of variablesx and k.Importantly, they are, in addition, symmetric under interchange of the two sets of variaFN

(b)(x,k)5FN(b)(k,x). This is reflected in the fact that the operatorLx,v(k) emerging in the

Hankel ansatz depends onk while HD does not. Due to their symmetry, the matrix Bessel fu


taine

thegies

lly be

rbi-

em ass in thele and,ential

ted tonsfor-zetlinfully

at firstrative

e hasrsion

con-nal

oach byhradial

onsver,

apping

e builtm-


Downloaded 0

tionsFN(b)(x,k) are, at least forHD , the more natural eigenfunctions. This is so, because, to ob

orthogonality conditions, one has to sum thecN,E(b) (x) over an infinite number of partitions. On th

other hand, orthogonality relations are an inherent feature of theFN(b)(x,k) due to their meaning

in the Fourier–Bessel analysis, as discussed in Appendix C.In other words, the functionscN,E

(b) (x) can be viewed as a basis in an expansion ofFN

(b)(x,k). We can consider the variablesk as a set of real numbers corresponding to the enerE labeling the eigenstates ofHD . The matrix Bessel functions~3.15! are solutions of the Schro¨-dinger equation with HamiltonianHD for the coupling parametersb51,2,4. The recursion for-mula~5.5! represents an analytic continuation of these integral solutions to arbitrary positiveb. Allthese functionsFN

(b)(x,k) have for arbitraryb additional features: the symmetry inx andk, whichhas no analog in the functionscN,E

(b) (x). The merit of our recursion formula lies in the fact that,apriori , no infinite resummation is required to obtain functions of the typeFN

(b)(x,k). NeverthelessForrester35 and Forrester and Nagao37 showed that such resummed expressions can successfuused in certain cases. They treated the case of Poissonian initial conditions37 for the Calogero–Sutherland HamiltonianHCS and derived exact expressions for the correlation functions for atrary b for one or two particles. This is also related to the works of Muirhead20 and Pandey.38

VIII. SUMMARY AND CONCLUSION

We presented a recursive construction for certain spherical functions. We referred to thmatrix Bessel functions because, first, they are a natural extension of vector Bessel functionsense that the integration over a group corresponds to the integration over a solid angsecond, they satisfy a partial differential equation generalizing the Bessel ordinary differequation. For matrices, the indexb labeling the groups appears analogous to the dimensiond inthe case of vectors. The introduction of radial Gelfand–Tzetlin coordinates, which are relabut different from the ordinary angular ones, was crucial for the recursion. The Cayley tramation ought to provide a connection between the angular and the radial Gelfand–Tcoordinates.39 As evident from its construction, the recursion maps an integral over a grouponto an iteration which exclusively takes place in the radial space.

Remarkably, the recursion turned out to be far more general than was to be expected,sight, from the proof which involved Lie groups. We showed that our recursion is also the itesolution of the corresponding partial differential equation for arbitrary values ofb. We introducedthe term radial functions for this generalization of matrix Bessel functions. We expect that onto employ the theory of quantum groups to give a group theoretical derivation of the recuformula for arbitrary values ofb.

Using the recursion formula, we discussed the structure of radial Bessel functions. Wejectured that, for evenb, they can be written as finite sums involving only exponential and ratiofunctions. We illustrated that by working out, forb54, the cases ofN53 and N54 distincteigenvalues. Further evaluation of explicit formulas for arbitraryN and, maybe, for all evenbdoes not seem impossible. Work is in progress. The extension of the stationary phase apprDuistermaat and Heckman13 to higher orders could, for evenb, be an alternative to derive sucexplicit results, because the expansion terminates. In this context, we mention that thefunctions for higher values ofb are, to some extent, but not fully, the higher order radial functifor lower values ofb. This also generalizes the situation for ordinary Bessel functions. Howethere are many more higher order radial functions; they are not at all exhausted by this mbetween values ofb.

In the present contribution, we only focused on ordinary spaces, i.e., spaces which arupon commuting numbers. In a second study40 we also address superspaces which involve comuting and anticommuting variables.


nd R.rtantsenbergstitute

e

e

in of

btain

er

y.it in


Downloaded 0

ACKNOWLEDGMENTS

The authors thank Y. Fjodorov, F. Leyvraz, G. Olshanski, W. Schempp, T. Seligman, aWeissauer for fruitful discussions at various stages of this work and for pointing out imporeferences. Financial support from the Deutsche Forschungsgemeinschaft, to T.G. as a Heifellow and H.K. as a doctoral stipend is acknowledged. H.K. also thanks the Max–Planck-Infor financial support.

APPENDIX A: ALTERNATIVE INTEGRAL REPRESENTATION FOR VECTOR BESSELFUNCTIONS

The Bessel functionsx (d)(kr) are defined as an integral over angles in Eq.~2.1!, but they canalso be written as integrals over the entire or half real axis.21 To make possible an instructivcomparison with the matrix case, we quote and rederive the representation

x (d)~kr !5G~d/2!

i2p S 2

ikr D(d22)/2E

2`

1`

expS 2 ikr

2 S t11

t D D dt

td/2 . ~A1!

The singularities have to be treated properly.The position vector in thed dimensional space isrW5reW r , where the vectoreW r parametrizes

the unit sphere. To integrate over its orientation, i.e., over the solid angleV, one can reexpress thmeasure as

dV5G~d/2!

pd/2 d~eW r221!dder . ~A2!

Here, the vectoreW r is reinterpreted: its components live on the entire real axis and the domaintegration is the fulld dimensional space with the Cartesian measure dder . The d distributionconfines the vector to the unit sphere. Writing this distribution as a Fourier transform, we ofrom Eq. ~2.1!

x (d)~kr !5G~d/2!

pd/2

1

2p E2`

1`

dtE dder exp~ i t ~eW r221!!exp~ ikW•reW r !

5G~d/2!

2p E2`

1` exp~2 i t !

~ i t !d/2 expS 2 i~kr !2

4t Ddt, ~A3!

where the integral overeW r gave a Gaussian ind dimensions. The contour for the integration ovt has to be chosen appropriately. Upon a trivial change of variables, this result yields Eq.~A1!.

APPENDIX B: ALTERNATIVE INTEGRAL REPRESENTATION FOR MATRIX BESSELFUNCTIONS

The matrix Bessel functionsFN(b)(x,k) for b51,2,4 can be be written in an alternative wa

Although we can hardly believe that this representation is completely new, we could not findthe literature. Similar to Eq.~A1! in the vector case, we can write

FN(b)~x,k!5AN

(b)E d@T#exp~ i Tr T! Det2b/2~x^ k2T^ 1N!, ~B1!

where 1N is theN3N unit matrix. The normalization constant is given by

AN(b)5

i bN2/2pbN(N21)/4

bN1bN(N21)/2 )n51

N

G~bn/2!. ~B2!


for

sel

real

f

a

sr

r

is

ns


Downloaded 0

The matrixT in Eq. ~B1! is real symmetric, Hermitian, or Hermitian self-dual, respectively,b51,2,4. The measure d@T# is Cartesian and given by Eq.~3.3!. All independent variables inTare integrated over the entire real axis. To ensure convergence, the diagonal elements ofT have tobe given a proper imaginary increment. We notice thatx andT areN3N matrices forb51,2 and2N32N for b54 with doubly degenerated eigenvalues. The matrixk is, in all three casesb, justthe N3N matrix k5diag(k1,k2,...,kN), as defined following Eq.~3.10!.

The integral representation~B1! leads to an interesting integral equation for the matrix Besfunctions,

FN(b)~x,k!5BN

(b) Det12b/2x E d@ t#uDN~ t !ubFN

(b)~x,t !

)n,m~km2tn!b/2 , ~B3!

where the normalization constant reads

BN(b)5

i bN2/2GN~b/2!

~2p!NN!. ~B4!

The tn in Eq. ~B3! have a proper imaginary increment and their domain of integration is theaxis. Due to the symmetry relation~3.16!, the variablesx andk can be interchanged in Eqs.~B1!and ~B3!.

It is not difficult to see from the integral equation~B3! that the product in the denominator oits right-hand side can, in the caseb52, be written as

BN(2)

)n,m~km2tn!5

det@d~xn2tm!#n,m51,...,N

uDN~k!DN~ t !u1/2 . ~B5!

For bÞ2, the term Det12b/2x contributes. Nevertheless, the product still shares features withddistribution.

To derive this alternative integral representation, we proceed analogously to Eq.~A2! byrewriting the invariant measure ofUPU(N;b) usingd distributions. The invariance simply meanthat all columnsUn , n51,...,N are orthonormal, TrUn

†Um5dnm . The trace Tr is only needed fob54, because the entries ofU are quaternions in this case. Thus, we may write

dm~U !5MN(b) d@U# )

n51

N

d~Tr Un†Un21! )

n,md~Tr Un

†Um!, ~B6!

where d@U# is the Cartesian measure of all entries ofU and the integration is for all variables ovethe entire real axis. The constantMN

(b) will be determined later. Ullah41,42used such forms for themeasure to work out certain probability density functions. The bilinear forms in thed distributionshaveb components fornÞm,

Un†Um5 (

a50

b21

@Un†Um# (a)t (a). ~B7!

We notice that@Un†Un# (a)50 for a.0 in the casen5m, because the length of every vector

real. Thus, because of Eq.~B7!, thed distributions in the measure~B6! have to be products ofddistributions for every nonzero component@Un

†Um# (a). We now introduce Fourier representatio

d~@Un†Um# (a)!5

1

p E2`

1`

dTnm(a) exp~2 i2@Un

†Um# (a)Tnm(a)!,

~B8!


n the

g

y Fyo-

e

f

alize


Downloaded 0

d~@Un†Un# (0)21!5

1

2p E2`

1`

dTnn(0) exp~2 i ~@Un

†Un# (0)21!Tnm(0)!

for nÞm andn5m, respectively. The Fourier variables form the elements

Tnm5 (a50

b21

Tnm(a) t (a) ~B9!

of a matrixT which is real-symmetric, Hermitian or Hermitian self-dual according tob51,2,4.We notice that the diagonal elementsTnn5Tnn

(0) are always real,

d~Tr Un†Um!5

1

pb E dbTnm exp~2 i Tr Un†~Tnm^ 1N!Um2 i Tr Um

† ~Tnm* ^ 1N!Un!,

~B10!

d~Tr Un†Un21!5

1

2p E2`

1`

dTnn exp~ i Tr Tnn2 i Tr Un†~Tnn^ 1N!Un!

for nÞm andn5m, as previously. Just as for the trace Tr, the direct product is only needed icaseb54.

We order the columnsUn , n51,...,N of the matrixU in a vectorUW 5(U1 ,U2 ,...,UN)T withN2 elements. Forb51,2, the elements are scalars; forb54, they are quaternions. Collectineverything, we can rewrite the measure~B6! in the form

dm~U !5MN

(b)d@U#

~2p!NpbN(N21)/2E d@T#exp~ i Tr T2 i Tr UW †~T^ 1N!UW !. ~B11!

For the unitary case, a related Fourier integral form for integration measures was used bdorov and Khoruzhenko43 in the context of quantum chaotic scattering. To use the measure~B11!in the integral~3.15! for the matrix Bessel functionsFN

(b)(x,k), we also take advantage of threlation

Tr U21xUk5Tr UW †~x^ k!UW , ~B12!

which allows us to write

FN(b)~x,k!5

MN(b)

~2p!NpbN(N21)/2E d@U#E d@T#exp~ i Tr T!exp~ i Tr UW †~x^ k2T^ 1N!UW !

5MN

(b)i bN2pbN/2

~2p!N E d@T#exp~ i Tr T!Det2b/2~x^ k2T^ 1N!. ~B13!

Thus, the integration overU could be done as a Gaussian one and gave the result~B1!. Obviously,the Gaussian integrals overUW only converge, if the diagonal elements ofT have a proper imagi-nary increment.

Formula ~B13! yields immediately the integral equation~B3!. Upon making the change ovariables

T5x1/2T8x1/2,

implying d@T#5Det11b(N21)/2x d@T8#, ~B14!

we bring x into the exponential function and remove it from the determinant. We diagonT85V821t8V8 and find


pecial

.ure

d

byma-

tion,


Downloaded 0

Det2b/2~x^ k2T^ 1N!5Det2bN/2x )n,m

~km2tn8!2b/2. ~B15!

The integral overV8 is then just the integral definition~3.15! of the matrix Bessel functionFN

(b)(x,t8) and we arrive at Eq.~B3!.The normalization constants remain to be derived. Conveniently, they nicely relate to a s

form of Selberg’s integral which is given in Eq.~17.5.2! of Mehta’s book,15

JN5E d@ t#uDN~ t !u2g )n51

N

~a11 i t n!2b1~a22 i t n!2b2

5~2p!N

~a11a2!(b11b2)N2gN(N21)2N

3 )n50

N21G~11~n11!g!G~b11b22~N1n21!g21!

G~11g!G~b12ng!G~b22ng!. ~B16!

We now putx50 or k50 and haveFN(b)(0,k)51 or FN

(b)(x,0)51 on the left-hand side of Eq~B13!. We diagonalizeT5V21tV and use the invariance of the integral. Employing the meas~3.7! and the constantCN

(b) given in Eq.~3.9!, we find the condition

15MN

(b)CN(b)pbN/2

~2p!N E d@ t#uDN~ t !ub )n51

Nexp~ i t n!

~ i t n!bN/2 . ~B17!

We map this onto Selberg’s integral~B16! by settingg5b/2, b15bN/2, anda25b2 , by using

lima2→`

a2a2

~a22 i t n!a25exp~ i t n! ~B18!

and by consideringa2Na2JN in the limits a1→0 and a2→`. With the help of some standar

asymptotic formulas for theG function, we obtainMN(b) and, eventually, the constantsAN

(b) andBN

(b) in Eqs.~B2! and ~B4!.

APPENDIX C: FOURIER–BESSEL ANALYSIS

The Fourier–Bessel analysis involving matrix Bessel functions was discussedHarish-Chandra4 in a general and formal way. To show the connection to our results, we sumrize here some essential features of the Fourier–Bessel analysis on an explicit level.

We write the Fourier transform of a functionf (H) as

F~K !5DN(b)E d@H#exp~ i Tr HK ! f ~H !, ~C1!

where the matricesH andK have the same symmetries. If we choose a symmetric normaliza

DN(b)5

1

~2p!N/2pbN(N21)/4, ~C2!

we can write the inverse transform as

f ~H !5DN(b)E d@K#exp~2 iTrKH !F~K !. ~C3!


e

side of

-of


Downloaded 0

We notice that, according to Eq.~3.11!, the Fourier transform of the constantDN(b) is the d

distributiond(K) and vice versa.If f is an invariant function such thatf (H)5 f (x), its Fourier transform turns out to b

invariant as well,F(K)5F(k). Introducing eigenvalue-angle coordinates, we easily find

F~k!5DN(b)CN

(b)E d@x#uDN~x!ubFN(b)~x,k! f ~x! ~C4!

for the Fourier transform and

f ~x!5DN(b)CN

(b)E d@k# uDN~k!ubFN(b)* ~k,x!F~k! ~C5!

for its inverse. We now insert the transform~C4! into the inverse~C5! and conclude that

~DN(b)CN

(b)!2E d@k#uDN~k!ub FN(b)~x,k! FN

(b)* ~k,y!5det@d~xn2ym!#n,m51,...,N

uDN~x!DN~y!ub/2 . ~C6!

This is the analog of Hankel’s expansion of thed distribution. From Eq.~C6!, the formula

E dm~U !d~U†xU2y!51

CN(b)

det@d~xn2ym!#n,m51,...,N

uDN~x!DN~y!ub/2 ~C7!

obtains. To see this, we introduce a matrixG having the same symmetries asH and write

d~H2G!5~DN(b)!2E d@K#exp~2 i Tr K~H2G!!. ~C8!

Averaging over the diagonalizing matrixU of H yields

E dm~U !d~H2G!5~DN(b)!2E d@K# FN

(b)* ~x,k!exp~ i Tr KG!, ~C9!

by using the invariance of the measure. We now introduce eigenvalue-angle coordinates forK anddo the integral overV, the diagonalizing matrix ofK,

E dm~U !d~H2G!5~DN(b)!2CN

(b)E d@k#uDN~k!ub FN(b)* ~x,k!FN

(b)~k,y!, ~C10!

where we have, once more, employed the invariance of the measure. Since the right-handthis equation does only depend on the eigenvaluesy of G, we may replaceG on the left-hand sidewith y. Together with Eq.~C6!, this gives formula~C7!.

For the convolution in matrix space of two functionsf 1(H) and f 2(H), we straightforwardlyfind the generalization of the standard convolution theorem,

f ~H !5E d@G# f 1~G! f 2~H2G!5E d@K#exp~2 i Tr HK !F1~K !F2~K !, ~C11!

whereG has the same symmetries asH. The functionsF1(K) andF2(K) are the Fourier transforms of f 1(H) and f 2(H), respectively. If the functions are invariant, the second equation~C11! acquires the form

f ~x!5CN(b)E d@k#uDN~k!ub FN

(b)* ~x,k!F1~k!F2~k!. ~C12!


he

culate


Downloaded 0

On the other hand, we find from the first equation of~C11!

f ~x!5CN(b)E d@y#uDN~y!ub f 1~y! f 2~x,y!, ~C13!

where y are the eigenvalues ofG. This formula is a convolution in the curved space of teigenvalues. The second function is given by

f 2~x,y!5E dm~U ! f 2~x2U†yU!. ~C14!

We insert the Fourier integral forF1(k) according to Eq.~C4! into Eq. ~C12!, compare with Eq.~C13! and obtain the Fourier decomposition

f 2~x,y!5DN(b)CN

(b)E d@k#uDN~k!ubFN(b)* ~x,k!F2~k!FN

(b)~k,y!. ~C15!

Formulas~C6! and ~C7! can be viewed as special cases of these results.

APPENDIX D: ACTION OF THE LAPLACIAN ON THE RADIAL FUNCTIONS FORARBITRARY b

We make the notation more compact by defining

m~x8,x!d@x8#5dm~x8,x!expS i S (n51

N

xn2 (n51

N21

xn8D kND , ~D1!

where the measure is given in Eq.~5.6!. To prove the identity~5.9!, we write the integral usingQfunctions. The left hand side of Eq.~5.9! reads

DxE m~x8,x!FN21(b) ~x8,k!)

i . jQ~xi2xj8!)

j > lQ~xj82xl !d@x8#, ~D2!

where now the integration domain is the real axis for all variables. Thus, we can directly calthe action of the operatorDx onto the integral. We find


i . jQ~xi2xj8!)

j > lQ~xj82xl !d@x8#

5E FN21(b) ~x8,k!)

i . jQ~xi2xj8!)

j > lQ~xj82xl !

3S Dx8(2)

1b (nÞm

1

~xn82xm8 !2 2kN2 D m~x8,x!d@x8#

1E FN21(b) ~x8,k!m~x8,x!Dx)

i . jQ~xi2xj8!)

j > lQ~xj82x!d@x8#

12E F~x8,k! (n51

N]

]xnm~x8,x!

]

]xn)i . j

Q~xi2xj8!)j > l

Q~xj82xl !d@x8#, ~D3!

where we define the operator

Dx8(2)

5 (n51

N]2

]xn22 (

n,m

b

xn2xmS ]

]xn2

]

]xmD . ~D4!


t

f


Downloaded 0

By a series of integrations by parts, the operatorDx8(2) acting onm(x8,x) is transformed toDx8

acting only onF(x8,k). At taking the derivative of theQ functions, we notice that only adjacenlevels contribute, because otherwise terms likeQ(xi2xj ) with i , j arise which annihilate theintegral due to the chosen ordering. Therefore, we can write

]

]xn)i . j

Q~xi2xj8!)j > l

Q~xj82xl !5) ~QÞnn8,QÞ(n21)8n!~d~xn2xn8!Q~xn218 2xn!

2d~xn218 2xn!Q~xn2xn8!!, ~D5!

where )(QÞnn8 ,QÞ(n21)8n) is short-hand for the product on the left-hand side of Eq.~D5!without the two factorsQ(xn218 2xn)Q(xn2xn8). Importantly, this product is symmetric inxn218andxn8 . The second derivatives yield

]

]xn)i . j

Q~xi2xj8!)j > l

Q~xj82xl !5) ~QÞnn8 ,QÞ(n21)8n!~d8~xn2xn8!Q~xn218 2xn!

1d8~xn218 2xn!Q~xn2xn8!1d~xn218 2xn!d~xn2xn8!!.

~D6!

The last term vanishes upon integration, since it is symmetric inxn218 andxn8 , whereas the rest othe integrand is antisymmetric due to the Vandermonde determinantDN21(x8) in the measure~5.6!. Differentiation with respect toxn8 gives

]

]xn8)i . j

Q~xi2xj8!)j > l

Q~xj82xl !5) ~QÞn8(n11) ,QÞnn8!~d~xn82xn11!Q~xn2xn8!

2d~xn2xn8!Q~xn82xn11!!. ~D7!

Integration by parts of the first term of the right hand side of Eq.~D3! yields


i . jQ~xi2xj8!)

j > lQ~xj82xl !d@x8#

5E m~x8,x!Dx8FN21(b) ~x8,k!d@x8#2kN

2 E m~x8,x!FN21(b) ~x8,k!d@x8#

12E FN21(b) ~x8,k! (

n51

N21 S) ~QÞn8(n11) ,QÞnn8!

3~d~xn2xn8!Q~xn82xn11!1d~xn82xn11!Q~xn2xn8!!

3S ]

]xn1

]

]xn81

1

2 (mÞn

b

xn2xm12

1

2 (mÞn

b

xn82xm8D D m~x8,x!d@x8#. ~D8!

Inserting in Eq.~D8! the functionm(x8,x) as given in Eq.~D1! and~5.6! we find after a straight-forward calculation


e

r


Downloaded 0

DxE m~x8,x!FN21(b) ~x8,k!d@x8#

5E m~x8,x!Dx8FN21(b) ~x8,k!d@x8#2kN

2 E m~x8,x!FN21(b) ~x8,k!d@x8#

12E FN21(b) ~x8,k! (

n51

N21 S) ~QÞn8(n11) ,QÞnn8!~g~xn8 ;yn ;x,x8!2g~xn8 ;xn ;x,x8!!

3~d~xn2xn8!Q~xn82xn11!1d~xn82xn11!Q~xn2xn8!! D m~x8,x!d@x8#, ~D9!

with

g~xn ;xn8 ;x,x8!5~b/221!S (m51

N211

xn2xm82 (

mÞn

1

xn2xmD ,

~D10!

g~xn8 ;xn ;x,x8!5~b/221!S (mÞn

1

xn82xm82 (

m51

N1

xn82xmD .

We now can perform the integration of thed distributions in Eq.~D9!. We notice that the differ-ence (g(xn8 ;xn ;x,x8)2g(xn8 ;xn ;x,x8)) vanishes linearly, wheneverxn8 approaches one of thboundaries of its integration domain. Thus the second integral in Eq.~D9! yields zero as long asthe measure diverges less than (xn2xn8)

21 when xn8 approachesxn . This is always the case fob.0. Collecting everything, we arrive at the identity~5.9!.

APPENDIX E: SYMMETRY OF THE RADIAL FUNCTIONS FOR ARBITRARY b

Applying the recursion formula~5.5! to all N21 levels, we can extend Eq.~4.5! to arbitraryb and write

FN(b)~x,k!5E )

n51

N21

dm~x(n),x(n21)!expS i S (m51

N2n11

xm(n21)2 (

m51

N2n

xm(n)D kN2n11D exp~ ix1

(N21)k1!,

~E1!

wherex(0)5x. Analogously, we also find

FN(b)~k,x!5E )

n51

N21

dm~k(n),k(n21)!expS i S (m51

N2n11

km(n21)2 (

m51

N2n

km(n)D xN2n11D exp~ ik1

(N21)x1!

~E2!

with k(0)5k for the solution of the differential equation which results from Eq.~5.1! by inter-changingx andk. We have to show that these two radial functions~E1! and~E2! are identical. Tothis end, we change in Eq.~E1! on thenth level the variablesxm

(n) ,m51,...,(N2n) to km(n) ,m

51,...,(N2n) by setting

) l 51N2n21~xm

(n21)2xl(n)!

) lÞm~xm(n21)2xl

(n21)!5r m

(n)5) l 51

N2n21~km(n21)2kl

(n)!

) lÞm~km(n21)2kl

(n21)!~E3!

for n51,...,(N21). These are, on thenth level, N2n11 equations for making a change ofN2n variables. However, one has

(m51

N2n11

r m(n)51 ~E4!


ich

esWe

piece

p

of Eq.

tential

proof


Downloaded 0

on all levels which eliminates one of theN2n11 equations.Of course, the substitution~E3! is motivated by the radial Gelfand–Tzetlin coordinates wh

we introduced to construct the recursion formula forb51,2,4. In this case, ther m(n) are the moduli

squared of a column of a matrixUPU(N2n;b). Here, we do not use this connection to matricand groups. We simply view Eq.~E3! as a standard change of variables in an integral.underline that Eq.~E3! does not involveb at all. Equation~E4! is just the normalization of acolumn ofU for b51,2,4. Since it is independent ofb, it also holds for arbitraryb. One can alsoverify Eq. ~E4! by a direct calculation.

The original domains of integration arexm(n21)<xm

(n)<xm11(n21) . In these boundaries, ther m

(n) arepositive definite. Hence, to satisfy this when changing the variables, we must havekm

(n21)<km(n)

<km11(n21) for the new domains of integration.To work out the measure in the new variableskm

(n) , we interpret Eq.~E3! as a change to theintegration variablesr m

(n) , too. This yields immediately

DN2n~x(n)!

DN2n11~x(n21)!d@x(n)#5dm~r (n)!5

DN2n~k(n)!

DN2n11~k(n21)!d@k(n)#. ~E5!

The first equality sign goes back to the radial Gelfand–Tzetlin coordinates. We may use thisof information, because it is independent ofb. The second equality sign is simply due to Eq.~E3!.Using this result, we find for the full andb dependent measure

dm~x(n),x(n21)!5GN2n11(b) S ) l ,m~xm

(n21)2xl(n)!

DN2n112 ~x(n21)!

D (b22)/2 DN2n~x(n)!

DN2n11~x(n21)!d@x(n)#

5GN2n11(b) S )

m51

N2n11

r m(n)D (b22)/2

DN2n~x(n)!

DN2n11~x(n21)!d@x(n)#

5GN2n11(b) S P l ,m~km

(n21)2kl(n)!

DN2n112 ~k(n21)!

D (b22)/2

3DN2n~k(n)!

DN2n11~k(n21)!d@k(n)#5dm~k(n),k(n21)!. ~E6!

For b51,2,4, this result is a direct consequence of the invariance of the group measure dm(U).Here, we have derived it for arbitraryb. This, in turn, implies that the invariance of the groumeasure dm(U) is embedded into and reflects much more general features.

We now collect all these intermediate results and plug them into Eq.~E1!. Apart from theexpressions in the exponential functions, we have full agreement with the right-hand side~E2!. Hence, it remains to be shown that the change of variables~E3! leads to the identity

(n51

N21 S (m51

N2n11

xm(n21)2 (

m51

N2n

xm(n)D kN2n111x1

(N21)k1

5 (n51

N21 S (m51

N2n11

km(n21)2 (

m51

N2n

km(n)D xN2n111k1

(N21)x1 . ~E7!

Since the symmetry relation~5.3! holds forb51,2,4, we know that Eq.~E7! must be true in thesecases. However, as Eq.~E7! does not involveb at all, it must also be valid for arbitraryb.Inserting this into the right-hand side of Eq.~E1!, we recover Eq.~E2!, as desired. We notice thathis line of arguing cannot be spoiled by any other contribution to the argument of the exponfunctions, because all other terms in the integrand are purely algebraic. This completes theof the symmetry relation~5.3! for arbitraryb.


plextes

a-

have


Downloaded 0

APPENDIX F: CALCULATION OF THE NORMALIZATION CONSTANT GN„b…

In Appendix E, we introduced the coordinatesr n85r n(1) ,n51,...,N on the first level of the

recursion. They are the moduli squared of the coordinates on the unit sphere in the comNdimensional space. Thus, it is natural to use the following type of hyper spherical coordina

Ar n85cosqn )n51

n21

sinqn, n51,...,~N21!,

~F1!

Ar N8 5sinqN21 )n51

N22

sinqn ,

where the positive semidefiniteness of ther n8 restricts the domain of integration to 0<qn

,p/2, n51,...,(N21). Thus, we integrate over a (2N)th segment of the unit sphere. The mesure

dm~r 8!5 )n51

N21

sin2(N2n)21qn cosqn dqn ~F2!

is, apart from the phase angles, the measure on the unit sphere. Collecting everything, we

15E dm~x8,x!5GN(b) E S )

n51

N

Ar n8D b22

dm~r 8!

5GN(b) )

n51

N21 E0

p/2

sin(N2n)b21qn cosb21qn dqn

5GN(b) )

n51

N21G~~N2n!b/2!G~b/2!

2G~~N2n11!b/2!5GN

(b) GN~b/2!

2N21G~Nb/2!, ~F3!

where the integral overqn is just Euler’s integral of the first kind.

APPENDIX G: TRANSLATION INVARIANCE OF WN,v„b…

„x ,k …

We shift everyxn in the the recursion formula~5.5! for arbitraryb by a constantx and obtain

FN(b)~x1 x,k!5E dm~x8,x1 x! expS i S (

n51

N

x1Nx2 (n51

N21

x8D kNDFN21(b) ~x8,k! ~G1!

with xn1 x<xn8<xn111 x as the domains of integration. The change of variablesxn8→xn81 xremovesx from the measure given in Eq.~5.6! and the domains of integration, we find

FN(b)~x1 x,k!5exp~ i xkN! E dm~x8,x! expS i S (

n51

N

x2 (n51

N21

x8D kNDFN21(b) ~x81 x,k!. ~G2!

We want to employ an induction. We assume that the radial functions for arbitraryb have theproperty

FN(b)~x1 x,k!5expS i x (

n51

N

knDFN(b)~x,k!. ~G3!


s

ation

y

wers

o


Downloaded 0

If this is correct forN21, formula~G2! implies that it is also true forN. The induction starts withN52 where the correctness of Eq.~G3! is immediately obvious from the explicit solution~3.21!for arbitraryb. Thus, Eq.~G3! is valid for all N.

Since thekn are arbitrary and since the sum over allkn is invariant under the permutationv(k), the property~G3! must also be true for every functionFN,v

(b) (x,k) with vPSN . We comparethis with

FN,v(b) ~x1 x,k!5expS i x (

n51

N

knD exp~ i (n51N xnkv(n)!

uDN~x!DN~k!ub/2 WN,v(b) ~x1 x,k!, ~G4!

which results from the Hankel ansatz~5.10!. Hence, we conclude that we necessarily have

WN,v(b) ~x1 x,k!5WN,v

(b) ~x,k!. ~G5!

This is the translation invariance.

APPENDIX H: CALCULATION OF F4„4…

„x ,k …

We perform the calculation forF4(4)(2 ix,k) to avoid inconvenient factors ofi . The operator

Lx,v(k) defined in Eq.~5.13! splits into two parts. The first part

Dx,v(k)5 (n51

N]2

]xn2 24 (

n,m

1

~xn2xm!2 ~H1!

does not change the order ink, while the second one,

Lx,v(k)52(n51

N

kv(n)

]

]xn, ~H2!

raises the order ink by one. Since we can restrict ourselves to one element of the permutgroup, we discuss only the identity permutation in the sequel. The symmetry ofx andk togetherwith the result forF3

(4)(x,k) suggests that one try an expansion in the composite variablezi j asdefined in Eq.~6.8!. To this end we define the elementary symmetric functions

en~z!5 (i 1 j 1, i 2 j 2,¯, i n j n

)l 51

n

zi l j l. ~H3!

Here, we assume the following ordering of the composite index$ i l j l%, i l, j l . We say$ i l j l%,$ i mj m% if i l, i m or i l5 i m and j l, j m . All indices run to N. The highest order elementarsymmetric function is of orderN(N21)/2 and is given byDN(x)DN(k). The asymptotic formula~5.14! yields the leading term for large arguments. It is the starting point for a recursion in poof z21,

WN(4)~z!5 (

n50

N(N21)/2

pn~z21!, ~H4!

wherepn(z) is a symmetric function of ordern in xi andki . We investigate the action of the twoperators defined in Eqs.~H1! and ~H2! and find

Lx,ken~z21!522 (n,m

N1

~xn2xm!2 en21~zÞnm21 !, ~H5!


mixed

es oftruction

be


Downloaded 0

Dx,ken~z21!524 (n,m

N1

~xn2xm!2 en~zÞnm21 !22 (

n,mkÞnkÞm

N1

~xn2xm!2 znk21zmk

21en22~zÞnmÞnkÞmk

21 !. ~H6!

The functionen(zÞnm) is the elementary symmetric functionen(z) with all terms containingznm

omitted. Forn50,1,2 we simply havepn(z21)5(22)nen(z21). For n>3 the last term in Eq.~H5! causes corrections to the elementary symmetric functions. This arises due to thederivatives which have to be taken into account in the action ofDx,k onto en(z21) for n>3.Because of this term the Hankel ansatz becomes increasingly cumbersome as higher valuNare considered. More and more correction terms have to be constructed. So far, the conswas only possible forN54. To construct the correction terms explicitly for the caseN54, wedefine a new set of symmetric functions as follows:

f n~z21!5 (k, l ,m

N

zkl21zkm

21zlm21en23~zÞkl

ÞkmÞ lm

21 !. ~H7!

Again we have to investigate the action ofLx,k and Dx,k on f n(z21). We find

Dx,kf 3~z21!524 (n,m

N1

~xn2xm!2 f 3~zÞnm21 ! ~H8!

and

Lx,kf 3~z21!522 (n,mkÞnkÞm

N1

~xn2xm!2 znk21zmk

21, ~H9!

thus f 3(z21) is the desired correction term. We have

p3~z21!5223~e3~z21!1 12 f 3~z21!!. ~H10!

Fortunately, due to Eq.~H8! in the next step the correction term itself does not have tocorrected and we find

p4~z21!524~e4~z21!1 12 f 4~z21!!. ~H11!

Up to now these results are valid for arbitraryN. The action ofDx,k onto the symmetric functionf 4(z21) is not as simple as Eq.~H8!. After a series of manipulations we arrive at

Dx,kf 4~z21!524(n,m

N1

~xn2xm!2 f 4~zÞnm21 ! ~H12!

22 (n,mkÞnkÞm

N1

~xn2xm!2 znk21zmk

21f 2~zÞnmÞnkÞmk

21 !. ~H13!

The contribution~H6! has to be added to this expression stemming from the action ofDx,k ontoe4(z21). On the other hand we calculate


way oflve anre.

g that,


Downloaded 0

Lx,kf 5~z21!522(n,m

N1

~xn2xm!2 f 4~zÞnm21 ! ~H14!

22 (n,mkÞnkÞm

N1

~xn2xm!2 znk21zmk

21e2~zÞnmÞnkÞmk

21 !. ~H15!

Thus, we have to find yet another correction term to compensate the second term in Eq.~H13!. Wedefine

f 58~z21!5 (i 1, i 2, i 3, i 4

)r , j

zi r i j

21(r , j

zi r i j5 (

j ,kl ,m

zjl21zjm

21zkl21zkm

21zlm21 ~H16!

and see thatLx,kf 58(z21) yields exactly the desired second term of Eq.~H13!. Pushing forward

this procedure becomes more complicated step by step. There seems to be no obviousconstructing the additional terms. Apparently for higher orders the correction terms also invoincreasing amount of indices. Nevertheless forN54 we are already at the end of the proceduThen the general expression

p5~z21!5225~e5~z21!1 12 f 5~z21!1 1

4 f 58~z21!! ~H17!

reduces to

p5~z21!5272e5~z21!. ~H18!

The last step can readily be done, since the action ofDx,k onto e5(z21) is already known by Eq.~H6!. Thus we arrive at

p6~z21!5288e6~z21!. ~H19!

Importantly, we have

Dx,ke6~z21!5Dx,k

1

D4~x!D4~k!50. ~H20!

That means, the sequence finishes after the sixth step. Collecting everything and observinfor N54, f 5(z)52e5(z) and f 6(z)54e6(z), we obtain

W4(4)~x,k!5 (

n51

6

~22!nen~z21!1 (n53

6

~22!n21f n~z21!28e5~z21!196e6~z21!. ~H21!

This can be rewritten in a more compact way as

W4(4)~x,k!5

1

D4~x!D4~k!

3S )i , j

~22zi j !122 (l ,m,n

)i , jÞ lmÞ lnÞmn

~22zi j !123(l ,mk,n

)i , j Þ lk Þ ln

Þmk Þmn Þkn

~22zi j !D ,

~H22!

which yields Eq.~6.10!.


,


Downloaded 0

1 Harish-Chandra, Am. J. Math.79, 87 ~1957!.2I. M. Gelfand, Dokl. Akad. Nauk SSSR70, 5 ~1950!.3S. Helgason,Groups and Geometric Analysis~Academic, San Diego, 1984!.4 Harish-Chandra, Am. J. Math.80, 241 ~1958!.5M. A. Olshanetsky and A. M. Perelomov, Phys. Rep.94, 313 ~1983!.6C. Itzykson and J. B. Zuber, J. Math. Phys.21, 411 ~1980!.7C. S. Hertz, Ann. Math.61, 474 ~1955!.8K. I. Gross and R. A. Kunze, J. Funct. Anal.22, 73 ~1976!.9K. I. Gross and R. A. Kunze, J. Funct. Anal.25, 1 ~1977!.

10W. J. Holman, J. Math. Phys.21, 1977~1980!.11A. Okounkov and G. Olshanski, Math. Res. Lett.4, 69 ~1997!.12M. Kontsevich, Commun. Math. Phys.147, 1 ~1992!.13J. J. Duistermaat and G. J. Heckman, Invent. Math.69, 259 ~1982!.14R. J. Szabo,Equivariant Cohomology and Localization of Path Integrals, Lecture Notes in Physics, New Series M

monographs; 63~Springer, Heidelberg, 2000!.15M. L. Mehta,Random Matrices, 2nd ed.~Academic, San Diego, 1990!.16F. Haake,Quantum Signatures of Chaos, 2nd ed.~Springer, Berlin, 2001!.17T. Guhr, A. Muller-Groeling, and H. A. Weidenmu¨ller, Phys. Rep.299, 189 ~1998!.18F. J. Dyson, J. Math. Phys.1, 140 ~1962!.19F. J. Dyson, J. Math. Phys.1, 1191~1962!.20R. J. Muirhead,Aspects of Multivariate Statistical Theory~Wiley, New York, 1982!.21M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions, 9th ed.~Dover, New York, 1970!.22R. Gilmore,Lie Groups, Lie Algebras and Some of their Applications~Wiley, New York, 1974!.23L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains~American Physical

Society, Providence, 1963!.24I. M. Gelfand and M. L. Tzetlin, Dokl. Akad. Nauk SSSR71, 825 ~1950!.25S. L. Shatashvili, Commun. Math. Phys.154, 421 ~1993!.26T. Guhr, Commun. Math. Phys.176, 555 ~1996!.27T. Guhr and H. Kohler, math-ph/0011007.28E. Brezin and S. Hikami, math-ph/0103012; Commun. Math. Phys.~in press!.29H. Risken,The Fokker–Planck Equation~Springer, Berlin, 1989!.30F. Calogero, J. Math. Phys.10, 2191~1969!.31B. M. Sutherland, Phys. Rev. A5, 1372~1972!.32C. W. Beenakker and B. Rejaei, Phys. Rev. B49, 7499~1994!.33M. Caselle, Phys. Rev. Lett.74, 2776~1995!.34R. P. Stanley, Adv. Math.77, 76 ~1989!.35P. J. Forrester, Mod. Phys. Lett. B9, 359 ~1995!.36Z. N. C. Ha, Nucl. Phys. B435, 604 ~1995!.37P. J. Forrester and T. Nagao, Nucl. Phys. B532, 733 ~1998!.38A. Pandey, Chaos, Solitons Fractals5, 1275~1995!.39G. Olshanski~private communication!.40T. Guhr and H. Kohler, J. Math. Phys.43, 2741~2002!, following paper; math-ph/0012047.41N. Ullah, J. Math. Phys.4, 1279~1963!.42N. Ullah, Matrix Ensembles in the Many-Nucleon Problem~Clarendon, Oxford, 1987!.43Y. V. Fyodorov and B. A. Khoruzhenko, Phys. Rev. Lett.83, 65 ~1999!.


recursive construction for a class of radial functions. i ...hp0117/publications/jmathphys_43...e...

Documents