f. nier- bose einstein condensates in the lowest landau level: hamiltonian dynamics

8/3/2019 F. Nier- Bose Einstein condensates in the Lowest Landau Level: Hamiltonian dynamics

1/30

Bose Einstein condensates in the Lowest Landau

Level: Hamiltonian dynamics.

F. NierIRMAR, UMR-CNRS 6625,

Campus de BeaulieuUniversite Rennes 1

35042 Rennes Cedex,France.

March 15, 2006

Abstract

In a previous article with A. Aftalion and X. Blanc, it was shown that thehypercontractivity property of the dilation semigroup in spaces of entire functionswas a key ingredient in the study of the Lowest Landau Level model for fast rotatingBose Einstein condensates. That former work was concerned with the stationnaryconstrained variational problem. This article is about the nonlinear Hamiltoniandynamics and the spectral stability of the constrained minima with motivationsarising from the description of Tkatchenko modes of Bose-Einstein condensates.Again the hypercontractivity property provides a very strong control of the nonlinearterm in the dynamical analysis.

1 Introduction

The Lowest Landau Level energy functional of rapidly rotating Bose-Einstein condensatesin a harmonic trap can be written as

Gh(f) =

C

|z|2f(z)e |z|22h 2 + Na2h2

f(z) e |z|22h 4 L(dz) . (1.1)The number N of atoms in the condensate and the scattering length a are fixed and

h = 1 2h, where h is the ratio of two rotational velocities, is a small parameter.Here and in the sequel L(dz) denotes the Lebesgue measure on C R2. In this scaling,the set of admissible f in the Lowest Landau Level approximation, is the semiclassical

1


2/30

Fock-Bargmann space

Fh =

f L2(C, e

|z|2h L(dz)), s.t f entire

(1.2)

with f2Fh =C

|f(z)|2e|z|2h L(dz) . (1.3)

The equilibrium states which are experimentally observed are within this modelling thesolution of the constrained minimization problem

inf

Gh(f), f Fh, fFh = 1

. (1.4)

We follow the presentation of [ABN1, ABN2] and additional information about the physicsof the problem can be found in [ARVK, Aft, ABD, Ho, WBP].This article focuses on the nonlinear Hamiltonian dynamics

itf = 2fGh(f) (1.5)

and on the spectral analysis of the linearized Hamiltonian around an equilibrium config-

uration. This last problem is motivated by the experimental and numerical study of thevibration modes of condensates, also known as Tkachenko modes, which can be found inthe physics literature [BWCB, SCEMC, Son].

More precisely, after specifying the functional framework and the results of [ABN2]about the minimization problem, the following topics will be studied:

1. Owing the hypercontractivity property already used in [ABN2], the Hamiltonianflow associated with the nonlinear equation (1.5) will be globally defined on Fh.

2. The spectral stability,1 of constrained minimizers will be proved after a naturalmodification of the energy functional. Since a minimum of the functional Gh on

the sphere, fFh = 1, is not always a local minimum of Gh and due to somedegeneracy related with the rotational invariance, the spectral stability is betterstudied by considering a modified energy

Gh0,1(f) = Gh(f) + 0(f

2Fh) + 1(f| zhzfFh)

in which the two additional terms are functions of quantities left invariant by theHamiltonian dynamics. This change of energy functional, amounts to a simpleexplicit time-dependent gauge transformation f0,1(z, t) = e

i1tf(ei2tz, t) wherethe real numbers 1, 2 are determined by the choice of 0, 1, and the initialdata f(z, t = 0) Fh. A good choice of the function 0, 1 allows to prove thatthe spectrum of the modified linearized Hamiltonian is purely imaginary. From

the physical point of view, the introduction of an artificially modified dynamics isrelated with the idea that one is interested in the linearized dynamics up to uniformsolid rotations of the whole condensate. A good intuitive picture is provided by thevibration modes of a swinging bell.

1i.e. the spectrum of the linearized Hamiltonian is purely imaginary

2


3/30

3. The relevance of numerical approximations is considered: the numerical simula-tions in [ABD] consist in minimizing the energy Gh on a space of polynomials withbounded degree, instead of the space Fh. It was checked in [ABN2] that when thedegree of the polynomials is taken sufficiently large, the solution to the finite dimen-sional problem provides a suitable approximation of the minimizer in Fh. Here it isproved that a similar result holds for the linearized Hamiltonian in a norm resolvent

sense. We end with some comments about the numerical stability of the spectralelements in such an approximation, by pointing out that the linearized Hamiltonianis a non anti-adjoint operator with a purely imaginary spectrum.

The appendix gathers standard tools about the classical Hamiltonian dynamics ininfinite dimension and some results are adapted to our specific framework.

2 Preliminaries

In this section, we set up the functional framework. The basic tools introduced in [ABN2]

and the initial result on the minimization problem (1.4) are reviewed. Then, the Kaehlerstructure of Fh, which is the combination of its (real) euclidean structure and its (real)symplectic structure and which is the natural framework for the study of the Hessianand the linearized Hamiltonian flow, is explicitely written. Finally, the Hessian of thefunctional Gh is computed and an estimate is given for HessGh(f) when f is a solutionto (1.4).

2.1 The minimization problem

Here is a very short review of [ABN2]. The Bargmann transform (see for example [Bar,Fol, Mar]) is used with the following normalization

[Bh](z) =1

(h)3/4ez2

2h

R

e(2zy)22h (y) dy ,

with z = xi2

C and S(R). It defines a unitary operator Bh : L2(R, dy) Fh and

the orthogonal projection h = BhBh : L

2(C, e|z|2h L(dz)) Fh is given by

[hf](z) = [BhBhf](z) =

1

h

C

ezzh e

|z|2h f(z) L(dz) .

The harmonic oscillator quantum Hamiltonian (or number operator in the Fock represen-

tation) given by:

Nh =1

2(h22y + y

2 h)

D(Nh) =

u L2(R, dy), yy u L2(R, dy), + 2

,

3


4/30

is transformed into the generator of dilations:

Nh = BhNhBh = z(hz) .

An element f = Bh of Fh considered as an element of L2(C, e |z|2

h L(dz)), satisfies

hzf = hz(hf) = h(zf) .

For the spectral resolution of these two operators, the basis of normalized Hermite func-tions (cnHn(y))nN is tranformed via Bh into the the basis of monomials (cnzn)nN.Then the spaces

Fsh =

f entire, s.t.

C

z2s|f(z)|2e|z|2h L(dz) <

, (2.1)

where z =

1 + |z|2 can identified constructed via the spectral resolution of Nh. Theunion zR Fsh is nothing but Bh [S

(R)] and Fsh is compactly embedded in Fh as soon as

s > 0.Another property which will be used also in this article, is the next consequence of thehypercontractivity property of the semigroup (etN1)t>0 (see for example [Car, Gro, Nel]):

Lemma 2.1. [ABN2]The quantityC

f1(z)f2(z)f3(z)f4(z)e2|z|2

h L(dz)

defines a continuous (2, 2)-linear functional 2 on Fh with norm smaller than1

2h. Hence

for any , {0, 1, 2}, the f

f derivative of the functional

f C

|f(z)|4 e2|z|2

h L(dz)

defines a continuous (2 , 2 )-linear mapping from Fh into Fh

Fh with norm4

2h(2)!(2)! .

The above Lemma and the compactness of the imbedding F1h Fh lead naturally tothe next result (see the proof in [ABN2])

Theorem 2.2. [ABN2]For any fixed h > 0, the minimization problem (1.4) admits asolution in F1

h

. Any minimizer is a solution to the Euler-Lagrange equation

h

|z|2 + N a2he

|z|2h |f|2

f

= 0 (2.2)

2A (2, 2)-linear functional is an R-quadrilinear functional which is C-linear with respect to the twofirst arguments and C-linear with respect to the two last arguments.

4


5/30

where R is the Lagrange multiplier satisfying the uniform estimates

2h3

2N a

< ehLLL 2e

hLLL 2

2h3

2bNa

+ oNa(h

0) , (2.3)

withehLLL = min G

h (f) , f Fh, f

Fh

= 1 .The Euler-Lagrange equation (2.2) can also be written aszhzf + N a

2hh

e

|z|2h |f|2

hf ( h)f = 0 , (2.4)

or zhzf +N a2h

2f(hz)[f

2(21.)] ( h)f = 0 , (2.5)

the operatorf(hz) being defined as the limit limKK

k=0 ak(hz)k iff(z) =

k=0 akz

k.

2.2 The Kaehler structure

The Hessian of the energy functional and the linearized Hamiltonian vector field, areobjects associated with the real euclidean structure or with the real symplectic structureof Fh. Some of their properties are more obvious in this presentation. We specify herethe Kaehler structure associated with the complex Hilbert space Fh.The space Fh can be viewed as a real Hilbert space with the scalar product:

Re f1 | f2Fh

and as a symplectic space with the form

(f1, f2) = Im f1 | f2Fh .

The Euclidean structure on Fh is more convenient when one studies the second variation,while the symplectic structure will be used in Section 3 for the Hamiltonian flow. Thesestructures are completely clarified once the complex conjugation is defined on Fh. Thesimplest way of writing can be done in the orthonormal basis Bh[H

hn ] = cn,hz

n withcn,h =

1(h)1/2hn/2

n!

. Let

fk =nN

fk,ncn,hzn =nN

(fRk,n + ifIk,n)cn,hz

n, k {1, 2}

we get

f1 | f2Fh =nN f1,nf2,nf1 | f2Fh,R =

nN

fR1,nfR2,n + f

In,1f

In,2 =

fR1 , fI1

fR2fI2

,

and (f1, f2) =nN

fI1,nfR2,n f

R1,nf

I2,n =

fR1 , fI1

0 Id Id 0

fR2fI2

.

5


6/30

For the real scalar product Ref1 | f2, we will use the notation fT1 f2 which refers to thematrix representation.The natural definition off (which has to stay in a space of holomorphic functions) followsfrom the writing f(z) =

nN(f

Rn + if

In)cn,hz

n = fR(z) + ifI(z):

f(z) = f(z) .

This definition allows to check the relationships (with f, fi Fh and v(z, z) non necessarilyholomorphic)

f1 | f2Fh =C

e|z|2h f1(z)f2(z) L(dz) , (2.6)

h[v(z, z)] = h

v(z, z)

, h[f(z)] = hf , (2.7)

Gh(f) =

C

e

|z|2h |z|2 f(z)f(z) +

N a2h2

e2|z|2h f

2(z)f2(z)

L(dz) . (2.8)

2.3 Hessian

Of course there are several ways to study the second variation of a function Gh(f), bycomputing the (f, f) coordinates or with the coordinates (fR, fI) introduced before. Thesecond choice put the stress on the real euclidean structure of Fh in which the notion ofHessian makes sense.

Proposition 2.3. Let f = fR + ifI Fh, the Hessian of Gh is a bounded perturbationof 2Nh with form domain F1h = Q(Nh). It defines a closed operator with domain F

2h and,

after writing =

RI

, is equal to

HessGh(f) = A 0

0 A

+ B 0

0 B

+ 0 C

C 0

(2.9)

where A, B and C are the real operators

A = 2(Nh + h) + 4N a2hh

e

|z|2h |f(z)|2

h (2.10)

B = N a2hh

e

|z|2h

f(z)2 + f(z)2

(z)

(2.11)

C = N a2

hhe |z|2h i f2(z) f2(z)(z) . (2.12)In the complexified Hilbert space Fh Fh, it defines a real self-adjoint operator whichhas a compact resolvent (and therefore a discrete spectrum going to infinity) and whosespectrum is bounded from below by C0h1 f

2.

6


7/30

Remark 2.4. a) An operator A is said to be real when = implies A = A. Thisdefinition is used in Fh with the previous definition of the conjugation f f and in thecomplexified space Fh Fh, where the conjugation is defined componentwise.b) Beside the conservation of reality, the explicit expression will also be useful in the finaldiscussion of Subsection 4.3.

Proof: The last expression ofGh

(fR + ifI) suggests to start with the second variationwith respect to f and f:

G2 = (R, I)Hess Gh(f)

RI

= 2ffG

h(f).. + ffGh(f).. + ffG

h(f)..

= 2 | (Nh + h)Fh + 4N a2h

C

e2|z|2h (z) |f|2 (z) L(dz)

+N a2h

C

e2|z|2h f

2(z)2(z) L(dz) + N a2h

C

e2|z|2h 2(z)f2(z) L(dz),

hence

G2 =

R |

2(Nh + h) + 4N ahh

e

|z|2h |f|2

h

R

Fh

+

I |

2(Nh + h) + 4N ahh

e

|z|2h |f|2

h

I

Fh

+N a2h

C

e2|z|2h f

2(z)

R(z)2 I(z)

2 + 2iR(z)I(z)

L(dz)

+N a2h Ce

2|z|2h f2(z)

R(z)

2 I(z)2 2iR(z)I(z)

L(dz).

We thus get:

G2 = R | ARFh + I | AIFh

+N a2h

C

e2|z|2h

f(z)2 + f(z)2

2R(z) L(dz)

N a2h

C

e2|z|2h

f(z)2 + f(z)2

2I(z) L(dz)

+N a2h

C

e2|z|2h i

f(z)2 f(z)2

I(z)R(z) L(dz)

+N a2h C

e 2|z|2

h i f(z)2 f(z)2R(z)I(z) L(dz) .The functions R and I are real elements of Fh and the relations

R,I(z) = R,I(z)

7


8/30

lead to

(R, I)HessGh(f)

RI

= R | ARFh + I | AIFh

+ R | BRFh I | BIFh+ R | CIFh + I | CRFh .

The expression of HessGh is deduced from this once A, B and C are real operators. Thisis a straightforward consequence of (2.7). Lemma 2.1 implies that Hess Gh(f) 2Nh is abounded self-adjoint operator with a norm controlled by C0h

1 f2Fh .

Proposition 2.5. Assume that f F1h is a solution to the minimization problem (1.4)and of the Euler-Lagrange equation (2.2) with Lagrange multiplier > 0. Then we have

PfHessGh(f)Pf 2Pf,

where Pf denotes the orthogonal projector on f for the (real) euclidean structure on

Fh.

Proof: The result is standard for finite dimensional problems. We write a proof inorder to check that Lemma 2.1 provides the suitable norm estimates. Since the derivativesof Gh with order larger than 2 are bounded according to Lemma 2.1, we obtain in thereal representation of elements of Fh

Gh(f + t) = Gh(f) + Gh(f).t +1

2tTHessGh(f)t + O(t3Fh)

for all t F1h. If f solves (2.2) and f + tFh = f = 1, one obtains, while noting thatthe real gradient is equal to 2f,

Gh(f + t) = Gh(f) + 12

tT HessGh(f) 2 Id t + O(t3Fh) .By Proposition 2.3 the operator P(Hess f 2)P is a bounded from below self-adjointoperator with a compact resolvent. If the proposition is not true, it admits a negativeeigenvalue 0 < 0 with a normalized eigenvector . This eigenvector solves the equation

Hess Gh(f) = (2 0) + 1f, 1 R .

It implies F2h f while the Euler-Lagrange (2.2) equation also gives f F2h. We

take

t = t() = f + 1

(f + ) f = + ((1 + 2

)1

1)f + (1 + 2)1 1 and we obtain

Gh(f + t()) = Gh(f) 0

22 + O(3)

with f + t() = 1, which is impossible.

8


9/30

3 The Hamiltonian flow

The Hamilton equations associated with the energy Gh(f) in the Bargmann space Fhsimply reads

itf = 2fGh(f) = 2(Nh + h)f + 2N a

2hh(e

|z|2h |f|2)f . (3.1)

We recall that the symplectic structure is given by

(f1, f2) = Im f1 | f2Fh

according to Subsection 2.2.We first refer to the Appendix for some notations and results about the nonlinear Hamil-tonian flow on Fh. Then, we introduce a modified energy

Gh0,1(f) = Gh(f) + 0(f

2Fh) + 1(f | NhfFh) .

which allows to handle easily the restriction to the sphere fFh = 1 and the degeneracyof the minimization problem due to the rotational symmetry. Finally, we tackle thelinearization problem: Owing to Proposition 2.5 and results of Appendix A, we are ableto select properly the modifier 0 and 1 so that the linearized Hamiltonian has a discretepurely imaginary spectrum (in)nZ with n = n and limn |n| = + .

3.1 The Cauchy problem

We check here that the Hamiltonian flow associated with (3.1) is well defined and admitstwo preserved quantities. We consider the Cauchy problem

itf = 2(Nh + h)f + 2N a2hh(e |z|2

h

|f|

2

)ff(t = 0) = f0 . (3.2)

Proposition 3.1. For any f0 F2h, the Cauchy problem (3.2) admits a unique solutionin C0(R; F2h) C

1(R; Fh).The flow(t)defined by(t)f0 = f(t) admits a unique continuous extension to Fh, so that(t)f0 C0(R; Fh) for all f0 Fh .This flow admits three preserved quantities. By setting f(t) = (t)f0, they are given by:

f0 Fh, t R, f(t)2Fh = f0

2Fh ,

f0 F2h, t R, G

h(f(t)) = Gh(f0) ,

f0 F2h, t R, f(t) | Nhf(t)Fh = f0 | Nhf0Fh .

Proof: We use the results of Proposition A.8 in Appendix A. We simply specifyhere the corresponding notations. The Kaehler space H is the underlying real vector

space of Fh. An element f = fR + ifI of Fh is identified with

fRfI

according to

9


10/30

Subsection 2.2. The real scalar product and the symplectic form on H are the ones definedin Subsection 2.2, while the involution J is the multiplication by i in Fh corresponding

to the matrix

0 IdId 0

. The complexified space HC is simply Fh Fh where a new

multiplication by i is authorized componentwise. The operator A is 2(Nh+h) in Fh, which

corresponds 2Nh + 2h 00 2Nh + 2h in the real space H Fh or its complexificationHC = Fh Fh . It is real, A (D(A) H) H, it has a compact resolvent and it commuteswith J. The function h(f) is the nonlinear part of Gh(f):

h(f) =N a2h

2

C

|f(z)|4 e2|z|2h L(dz)

=N a2h

2

C

(fR(z) ifI(z))2 (fR(z) + ifI(z))e

2|z|2h L(dz)

=N a2h

2

C

fR(z) i fI(z)

2

(fR(z) + ifI(z))e 2|z|2

h L(dz) .

Its expression in terms of fRfI and the continuity property of Lemma 2.1 show that

it is real analytic in H. The last expression, due to fR = fR and fI = fI when f Fh,allows its extension as a real valued, real analytic function, with h(eJf) = h(f), tothe complexified space HC = Fh Fh as it is required in Hypothesis A.2. The relationh(eiNhf) = h(f) for R and f Fh, which is the invariance of the above functionalwith respect to rotations, provides the gauge invariance required in Proposition A.8.

Remark 3.2. Although we are working with an infinite dimensional system, the con-servation of f(t)Fh and f(t) | Nhf(t)Fh can be viewed as a consequence of Noethers

Theorem (see [AbMa][Arn]). These quantities are associated with the invariance with re-spect to the multiplication by a phase factor f eif and with respect to the rotationsf eiNhf, R, of the energy functional Gh .

3.2 A modified Hamiltonian

Modifying the energy functional with the help of preserved quantities in order to studythe stability of equilibrium in Hamiltonian systems is a standard process. This can beviewed for our specific Hamiltonian dynamics as a variation of the Casimir functionalmethod. We refer for example to [HMRW] where many applications are discussed.Here a modified Hamiltonian is introduced for two reasons:

1) We are interested in the stability of a constrained minimum.

2) The minimization problem is degenerate due to the rotational invariance.

We verify at the end of this paragraph that this modification of the energy functionalmakes sense and allows to catch relevant information on the dynamics.

10


11/30

First of all, we note that a solution f to the Euler-Lagrange equation (2.2) withLagrange multiplier , is nothing but a critical point of the functional

Gh0() = Gh() + 0(

2Fh) ,

where 0 is C2 function on R+ such that 0(1) = . Due to the conservation off(t)Fhby the Hamiltonian flow (t) associated with Gh, the new dynamics is well understood interms of classical solutions: for f0 F2h , the Cauchy problem

itg = 2gGh0

(g)g(t = 0) = f0 .

(3.3)

has a unique classical solution g C1(R; Fh) C0(R; F2h). It is equal to

g(t) = e2it0(f02Fh)(t)f0 .

Hence a solution f to the Euler-Lagrange equation (2.2) with Lagrange multiplier ,is transformed by taking 0(1) = into an (unconstrained) equilibrium for the newHamiltonian dynamics. Moreover the new dynamics for all other initial data is obtainedafter applying an elementary change of phase.

The second modification will help to get rid of the degeneracy problem due to therotational invariance.

Definition 3.3. Let f F2h be a solution to the Euler-Lagrange equation (2.2) withLagrange multiplier R and fFh = 1. The functionals 0(

2Fh) and

1( | NhFh) are said to be adapted to (f, ) if

0 and 1 are C2 real-valued functions on [0, +).

0(1) = (fFh = 1) and 1(f | NhfFh) = 0 .

In such a case, the adapted energy is defined by

Gh0,1() = Gh() + 0(

2Fh) + 1( | NhFh) .

Remark 3.4. The conditionf F2h is not a restriction because an element f Fh whichsolves the Euler-Lagrange equation (2.2) necessarily belongs to F2h .

The same argument as above leads to:

Proposition 3.5. Letf be a solution to (2.2) with Lagrange multiplier . Let0(2Fh)

and 1( | NhFh) be adapted to the pair (f, ) according to Definition 3.3. Then forany f0 F2h, the function

(z, t) = e2it0(f02Fh)[(t)f0](e2iht1(f0 |Nhf0Fh )z)

is the unique classical solution ( C0(R; F2h) C1(R; Fh)) of the Cauchy problem

it = 2Gh0,1

()(t = 0) = f0 .

(3.4)

11


12/30

Proof: It is sufficient to write

Gh0,1

() = Gh() + 0(

2Fh) +

1( | NhFh)zhz .

If is a classical solution to (3.4) then

e2i

Rs0

0((s)2Fh) ds(e2ih

Rs01( |NhFh) dsz)

is a classical solution to (3.2). Proposition 3.1 yields the result.

The modification of the energy functional does not change the dynamics of the solu-tion f to the Euler-Lagrange equation (2.2). For other initial data, it is changed by amultiplication by a phase factor and by the action of a uniform solid rotation. These ad-ditional time dependence are slow when the initial data f0 is close to the critical point f.In the spectral stability analysis which follows, this means that we consider the stabilityup to the multiplication by a phase factor (which does not change the modulus of the

wave function |u|2 = |f(z)|2 e|z|2h ) and up to a uniform (and slow) solid rotation.

3.3 Linearized Hamiltonian

Here we consider a solution f to the minimization problem (1.4). Although it is a de-generate constrained minimization problem in infinite dimension, the introduction of amodified energy allows to recover the expected properties of the linearized Hamiltonian.

Theorem 3.6. Assume that f F1h is a solution to the minimization problem (1.4) withLagrange multiplier . Assume moreover that the functional Gh0,1 is adapted to the pair

(f, ) according to Definition 3.3 and set 0 = 0(f

2Fh) and 1 =

1(f| NhfFh).

Then f is a spectrally stable equilibrium for the energy Gh0,1 provided that

0 11

2h 2 , 1 > 0 .

The spectrum (JHessGh0,1(f)) is made of a discrete set of eigenvalues (in)nZ withfinite multiplicity such thatn R, n = n and limn |n| = + .

According to [HMRW], the spectral stability simply says that the spectrum of thelinearized Hamiltonian

0 Id Id 0

HessGh0,1(f) = JHessG

h0,1

(f)

is purely imaginary. According to the notation of Appendix A, J denotes the matrix 0 IdId 0

in the real symplectic space H Fh. We recall that the spectrum of a

linear Hamiltonian has two symmetries with respect to R and iR. Note however that inthe spectrally stable case, a pure imaginary spectrum does not mean that the Hamiltonianis anti-adjoint (see Appendix A).

12


13/30

Lemma 3.7. With the same assumptions as in Theorem 3.6, the Hessian HessGh0,1(f)equals

HessGh0,1(f) = HessGh(f) 2 Id+40

|fRfR| |fRfI||fIfR| |fIfI|

+ 41 |NhfRNhfR| |NhfRNhfI||NhfINhfR| |NhfINhfI| with 0 =

0(f

2Fh) and 1 =

1(f| NhfFh) Except for the lower bound which now

depends on 0 and 1 it shares the properties of Hess Gh stated in Proposition 2.3.

Proof: A direct calculation leads to

THess 0

f2Fh

= 20(f2Fh)

2Fh +

0

f2Fh

4 (Ref| Fh)2

THess 1 (f| NhfFh) = 21(f| NhfFh) |NhFh

+1 (f | NhfFh) 4 (ReNhf| Fh)2

We conclude with 0(f2Fh) = ,

1(f| NhfFh) = 0 and with the identification

between = R + iI with

RI

.

Proof of Theorem 3.6: We refer again to the general framework reviewed in Ap-pendix A, namely Proposition A.5 with A = 2Nh and B = HessG

h0,1

2Nh. Theabove expression for HessGh0,1 combined with Proposition 2.3 and the fact that f F2h = D(Nh), implies that B is a bounded real operator. Proposition A.5 states that thespectral stability can be deduced from

F1h, THessGh0,1(f) 0 .

Any element of F1h can be written + f with f (i.e. Re f, Fh = 0) and R.

We get

( + f)THessGh0,1(f)( + f) =

T(HessGh(f) 2) + 2T(HessGh(f) 2)f + 2fT(HessGh(f) 2)f

+402 f4Fh+41 (Re | NhfFh)

2+81f | NhfFh Re | NhfFh+412f| Nhf

2Fh .

The first term T(Hess Gh(f) 2) is non negative according to Proposition 2.5. Since is real and f the scalar products Tf vanish. Proposition 2.3 (it is shorter to

reproduce the calculation ofG2 in its proof) leads to

T(Hess Gh(f) 2)f = 2Re | (Nh + h)fFh+

(4 + 2)NahRe | h

e

|z|2h |f|2 f

Fh 2Re | fFh

13


14/30

Then the Euler-Lagrange equation (2.2) implies for f

2T(Hess Gh(f) 2)f = 8Re | (Nh + h )fFh = 8Re | NhfFh

and2fT(HessGh(f) 2)f = 82( h) 82f| NhfFh .

Adding all the terms leads to

( + f)THessGh0,1(f)( + f)

8Re | NhfFh + 82( h) 82f| NhfFh + 40

2

+41(Re | NhfFh)2 + 81f| NhfFh Re | NhfFh + 41

2f | Nhf2Fh

42(2 2h + 0)

+412

r2 + 2

f| NhfFh

1

1

r + f | NhfFh

f| NhfFh

2

1

42(2 2h + 0) + 41

2 r + f | NhfFh 1

12

1

21

by setting r =Re |NhfFh

and for 1 = 0 . Finally the last right-hand side is non negative

for 1 > 0 and 0 11 2h + 2 > 0.

4 Approximation by a finite dimensional problem

The approximation of the optimization problem (1.4) by finite dimensional ones, that is Fhis replaced by a set of polynomials with bounded degree, was studied in [ABN2]. Here we

complete this information by showing that such a convergence result can be extended tothe linearized Hamiltonian in the norm resolvent sense. We end this section by recallingthat a more quantitative estimate of the convergence of spectral elements, in such adiscretization process, is a real issue because the linearized Hamiltonian JHessGh(f)is not anti-adjoint.

4.1 Preliminaries

For K N, CK[z] denotes the set of polynomials with degree smaller than or equal toK. Since (cn,hz

n)nN, cnh = 1(h)1/2hn/2n! , is an orthonormal spectral basis for Nh withNhz

n = hnzn, the orthogonal projection h,K onto CK[z] coincides with the orthogonal

spectral projection:h,K = 1[0,hK](Nh) .

We shall use the notation h,K for the imbedding from CK[z] into Fh:

h,K h,K = IdCK [z]

h,K h,K = h,K . (4.1)

14


15/30

We introduce the reduced minimum of the finite dimensional optimization problem:

ehLLL,K = minPCK[z],PFh=1

Gh(P) . (4.2)

Theorem 4.1. 1) The minima ehLLL and ehLLL,K satisfy

K N (h1C2(h), +), 0 < ehLLL,K ehLLL C2(h)

2 + C2(h)3

(1 C2(h)(hK)1)4 (hK)1

where C2(h) =8h3

2bNah + oNa(h

1/2) does not depend on K.If f solves the minimization problem (1.4) then the sequence (fK)KN defined by fK =h,Kf

1 h,Kf, which satisfies fK CK[z] is a minimizing sequence for (1.4).2) If for anyK N, PK CK[z] denotes any solution to (4.2) then the sequence (PK)KNis a minimizing sequence for (1.4). Its accumulation points for the Fh topology aresolutions of (1.4). Moreover if a subsequence (PKn)nN converges to f in Fh then theconvergence also holds in F2h according to:

limn

f PK

Fh

+ Nh

(f PKn

)Fh

= 0 .

4.2 Convergence of the linearized Hamiltonian

We now consider the question of the convergence of the linearized Hamiltonians associatedwith the functional Gh0,1. We forget the term related to the gradient of the functionalssince in the end it will be applied with critical points. The linearized Hamiltonian at apoint Fh is defined as

H() := JHessGh0,1()

withJ

= 0 IdId 0 = nN 0 11 0 in Fh Fh = nN(Czn Czn) . We keepthe notation h,K for the diagonal operator

h,K :=

h,K 0

0 h,K

in Fh Fh .

Due to the commutation h,KJ = Jh,K the restricted linearized Hamiltonian at a pointof Gh0,1

CK [z]

equals

HK(K) = h,K

JHessGh0,1(K)

h,K = Jh,K

Hess Gh0,1(K)

h,K .

For any holomorphic function in an open subset C and any compact regular contour

which does not meet the spectrum (H()) the holomorphic functional calculusprovides the operators

(H()) =1

2i

(z)(z H())1 dz (4.3)

with a corresponding definition for (HK(K)) .

15


16/30

Theorem 4.2. With the notations of Theorem 4.1, let (PKn)nN denote a convergingsubsequence of solutions of (4.2) with K = Kn, and letf denote the limitf = limn PKnwhich is a solution to (1.4). Then for all z C \ (H(f)), the convergence

limn

h,Kn(z IdCKn [z]CKn [z] HKn(PKn))1h,Kn = (z H(f))

1

holds in the norm topology. Hence for any pair(, ) (see (4.3)) such that(H(f)) = the convergence

limn

h,Kn(HKn(PKn))h,Kn = (H(f))

holds in the norm topology.

We start with a lemma derived after the introduction of a Grushin problem (see [SjZw]and references therein), a more flexible variation of the Feshbach method (see [DeJa] andreferences therein).

Lemma 4.3. In a complex Hilbert space (HC, . , .), let A be a self-adjoint operator withdomainD(A) and with a compact resolvent. Let(n)n

N be a sequence of complex numbers

such that limn n = 1 . Let(Bn)nN be a sequence of bounded operators with limit Bin the norm topology as n : limn Bn B = 0 . Letn be the spectral projection1[Tn,Tn](A) with the assumption limn Tn = +. The imbedding Rann H isdenoted by n according to (4.1). Then the limit

limn

n(z IdRann n(nA + Bn)n)1n = (z (A + B))1

holds in the norm topology for all z C \ (A + B).

Proof: 1) We first consider the case n = 1 for all n N . We set

= sup {Bn , n N} { B} .

For z C, we set Anz = z (A + Bn) : D(A) H. After the decomposition H =Rann Ran(1 n) and D(A) = Ran n (D(A) Ran(1 n), it is written:

Anz =

Annnz A

nnnz

Annnz Annnz

=

Annnz B

nnn

Bnnn Annnz

with Xnn = nX

n, X

nn = nX(1 n), Xnn = (1 n)Xn and Xnn = (1

n)X(1 n) . Accordingly we use the notation A,z for z (A + B) with for n Nfixed the corresponding restrictions Ann

,z, A

nn

,z = B

nn, Ann

,z = B

nn and Ann

,z. We follow

[SjZw] for the introduction of the Grushin problem and we set:

Anz =

Anz R

n

Rn+ 0

=

Annnz Bnnn IdRannBnnn Annnz 0IdRann 0 0

: D(A) Rann H Rann

16


17/30

with

Rn =

IdRann0

and Rn+ = (IdRann, 0) .

The Schur complement formula says that if

A1nz = En En+

En

En

+ the operator Anz is invertible if and only if E

n+ is invertible with

A1nz = En En+(E

n+)

1En, (En+)

1 = RnA1nz R

n+ . (4.4)

We now compute A1nz . First note that for any z C there exists n(z) N such that Annnz

is invertible for n n(z) . The second resolvent formula gives:

(Annnz )1

= (z (1 n)(A + Bn)(1 n))1 =

1 + (z Ann)1Bnnn1

(z Ann)1

where the self-adjoint operator Ann = (1 n)A(1 n) has a spectrum included inR \ [Tn, Tn] . This also implies for n n(z):1 + (z Ann)1Bnnn 1 2 and (Annnz )1 2||z| Tn| n 0 .In the former calculation Bn can be replaced by B so that(Annnz )1 2||z| Tn| ,

(Annz)1 2||z| Tn| (4.5)hold for n n(z) .The inverse A1nz is computed by Gauss elimination:

A1nz = 0 0 10 (Annnz )1 (Annnz )1 Bnnn

1 Bnnn (Annnz )

1 Annnz + Bnnn (A

nnnz )

1Bnnn

The Schur complement formula (4.4) yields:

A1nz = (1 n) (Annnz )

1(1 n)+

1

(Annnz )1

Bnnn

Annnz B

nnn (A

nnnz )

1Bnnn

1 1, Bnnn (A

nnnz )

1

(4.6)

when

E

n

+ = Ann

nz Bnn

n (Ann

nz )1

B

nn

n

is invertible.When z C\(A+B), A,z is invertible and the Schur complement formula (4.4) appliedwith Bn replaced by B implies that the operatorEn+ = Annz Bnn (Annz)1 Bnn

17


18/30

is invertible with En+1 = RnA1zRn+uniformly bounded for n n(z) . The second inequality of (4.5) implies

Annz En+ 22

||z| Tn|, for n n(z)

and a uniform bound for (Annz)1 with respect to n n(z). Owing to the convergences

Annnz Annz B Bn

n 0

andEn+ Annnz Bnnn Annn,z1 Bnnn 22||z| Tn| n 0 ,

the same is true for (Annnz )1 and

En+ and we get

(En+)

1 (Annnz )1

Cz

Bnnn

Annn,z

1

Bnnn

2Cz2

||z| Tn|n

0 . (4.7)

We infer from (4.5), (4.6) and (4.7) the estimateA1nz n(Annnz )1n Cz||z| Tn| n 0 .The second resolvent formula also givesA1nz A1z = (z A Bn)1 (z A B)1 n 0 ,which yields the result for n = 1 .2) For the general case, it is enough to write

(z n(nA + Bn n)1 = 1

n( z

n n(A + 1

nBn)

n)1 = 1

n(z n(A +Bn)n)1

with Bn = 1n Bn + (1 1n ) Id and to apply the result of Part 1).Proof of Theorem 4.2: we recall that

Gh0,1(f) = Gh(f) + 0

f2Fh

+ 1(f| NhfFh)

and its Hessian at a point f F+2h equals according to Lemma 3.7 (without assuming1(f | NhfFh) = 0):

HessGh

0,1(f) = HessGh

(f) 20(f2

Fh) Id 21(f | NhfFh)Nh

+ 40(f2Fh)


+ 41(f| NhfFh)

|NhfRNhfR| |NhfRNhfI||NhfINhfR| |NhfINhfI|

.

18


19/30

When f solves (1.4) and for well chosen 0 and 1, it writes:

HessGh0,1(f) = HessGh(f) 2 Id+40


+41

|NhfRNhfR| |NhfRNhfI||NhfINhfR| |NhfINhfI| = 2 Nh 0

0 Nh + B ,

where B L(Fh Fh) . Meanwhile we obtain for PKn (with limn PKn = f in F2h):Hess Gh0,1(PKn) = Hess G

h(PKn) 20(PKn

2Fh) Id 2

1(PKn | NhPKnFh)Nh

+40(PKn2Fh)

|PKn,RPKn,R| |PKn,RPKn,I||PKn,IPKn,R| |PKn,IPKn,I|

+41(PKn | NhPKnFh)

|NhPKn,RNhPKn,R| |NhPKn,RNhPKn,I||NhPKn,INhPKn,R| |NhPKn,INhPKn,I|

= 2(1 1(PKn | NhPKnFh)) Nh 00 Nh + Bn

withlimn

1(PKn | NhPKnFh) = 0, limn

Bn B = 0 .By recalling

H(f) = JHess Gh0,1(f) and H(PKn) = JHessGh0,1

(PKn)

and by applying Lemma 4.3 with

A = 2iJ Nh 00 Nh = 0 2iNh2iNh 0 ,n = (1

1(PKn | NhPKnFh))

B = iJB , Bn = iJBnand n =

h,Kn 0

0 h,Kn

= h,Kn, Tn = 2hKn ,

one gets

limn

h,Kn(z IdCKn [z]CKn [z] h,KnH(PKn)h,Kn)

1h,Kn = (z H(f))1

for all z C \ (H(f)) . We conclude withHKn(PKn) = h,KnH(PKn)

h,Kn .

Finally the convergence of the spectral elements h,Kn(HKn(PKn))h,Kn comes from thefact that all the convergence estimates are locally uniform in z for h > 0 fixed.

19


20/30

4.3 Remarks about the stability of spectral quantities

Contrary to Theorem 4.1, the results of Theorem 4.2 about the stability of spectral quan-tities does not provide any quantitative estimate. For a fixed not so small value ofh > 0and when only a fixed finite number of spectral elements are computed, such a quanti-tative estimate is not crucial. It becomes definitely an issue when h > 0 gets small or

if one is interested in a large number of spectral elements. For example, the behaviourof the sequence (in)nZ of eigenvalues of the linearized Hamiltonian JHessGh0,1(f)stated in Theorem 3.6, limn |n| = +, can be stated more accurately since the prob-lem amounts to looking at a bounded perturbation of the harmonic oscillator quantumHamiltonian. Nevertheless this behaviour seems difficult to recover in numerical simula-tions3

A very likely explanation is that the linearized Hamiltonian JHessGh0,1(f) isnot anti-adjoint. Whatever the choices of the functions 0 and 1 are, the HessianHessGh0,1(f) is a finite rank perturbation of Hess G

h(f) according to Lemma 3.7. But

the commutator

J, HessGh(f)

can be computed from (2.9) and equals

2C 2B2B 2C

.The operators B and C, defined in (2.11)(2.12), are non vanishing Hilbert-Schmidt op-erators but with infinite rank. Hence the linear Hamiltonian JHessGh0,1(f) is notanti-adjoint in spite of a purely imaginary spectrum.The stability of the spectrum of non self-adjoint (or non normal) operators with respectto perturbations enters in the theory of pseudospectral estimates and it is known thatthere can be a big gap between the knowledge of the spectrum and a good control of resol-vent estimates with dramatic consequences in numerical computations. Such an analysisfor pseudo-differential operators has had a great development in the recent years and we

refer the reader to [Tre][Dav1][Dav2][DSZ][Hag1][Hag2][HerNi][HelNi][Pra][Zwo]. In orderto perform such an analysis of the linearized Hamiltonian JHess Gh(f), a better infor-mation on the minimizer f than the one provided in [AfBl][ABN1][ABN2] is necessary.

A Specific infinite dimensional Hamiltonian systems

Our aim is not here to give a complete account on infinite dimensional Hamiltoniansystems. We refer the reader for example to [ChMa][Kuk][BHK] for a more general pre-sentation or different points of view. We simply briefly point out the properties which arerelevant to our problem.

We consider a separable Kaehler space (H, (. | .), ): (H, (. | .)) is a real Hilbert spacewhile is a symplectic form compatible with (. | .). We recall that the last conditionmeans that there exists a continuous R-linear (skew-adjoint) operator on H such that

3according to discussions with A. Aftalion and X. Blanc.

20


21/30

J2 = 1 and(X, Y) = (JX| Y) = (X| JY) .

Before going further, it is useful to introduce the complexified Hilbert space HC with thescalar product

(f1 + if2 | g1 + ig2)C = (f1 | g1) + (f2 | g2) + i (f1 | g2) i (g1 | f2) .

In this framework the operator J becomes a skew-adjoint bounded involution

J = J, J2 = Id

which differs from i Id .

Remark A.1. It is important to note here that the complexified Hilbert space HC hasnothing to do with the natural complex structure associated with J. In fact, the complexi-

fied space has no relationship with the symplectic structure on H. It is introduced only inorder to provide the framework for spectral theory. More precisely, consider the example

where H = R2n = Rnx Rn is endowed with

the scalar product: (X| X) =

x

.

x

= xx +

and the symplectic form: (X, X) = x x .

Let J =

0 11 0

. After the identification between X R2n and z = x + i Cn, the

real scalar product happens to be the real part of the complex scalar product (X| X) =Re z.z and the symplectic form the opposite of the imaginary part (X, X) = Im z.z ,while the operator J is translated into the multiplication by i . Instead, the complexified

space HC equals C2n and allows the action of J =

0 11 0

and the componentwise

multiplication by the complex scalar i .In the case of our analysis, the Kaehler space is the complex Hilbert space Fh. We have,

as a set, H = Fh, after the identification between f = fR + ifI Fh and

fRfI

H

while HC equals Fh Fh .As in the study of second variations, some properties and symmetries of a linearizedHamiltonian are more obvious when working with the real structure (the complexificationbeing added only in order to apply spectral theory) .

The energy functional is given by

H(f) =1

2(f | Af)

C+

1

2(f | Bf)

C+ h(f), f D(A) HC

where the operators A, B and the nonlinear function h satisfy the next assumptions:

21


22/30

Hypothesis A.2. The operator (A, D(A)) is a non negative self-adjoint opera-tor on HC, with a compact resolvent, which commutes with J and which is real:A (D(A) H) H.

The operator B is a bounded real (BH H) self-adjoint operator on HC (nonnecessarily commuting with J).

The function h : HC R is real analytic and satisfies the gauge invarianceh(eJf) = h(f) for all R and all f HC .

The Hamilton equation can be written astf = JH(f) = JAf JBf Jh(f)f(t = 0) = f0 H (or HC) .

(A.1)

where denotes the gradient with respect to the scalar product (. | .) in the real case andthe gradient with respect to the real scalar product Re (. | .)

Cin the complex case. As

usual an equilibrium is a critical point of H.

Proposition A.3. Assume Hypothesis A.2. Then the initial value problem (A.1) admitsa unique mild global solution for any f0 HC. Moreover the flow defined by f(t) = (t)f0

for f0 HC and t R satisfies

f0 H, t R, (t)f0 H

and f0 HC, t R, (t)f0HC = e[B,J]t/2 f0HC .

Proof: 1) The linear case with B = 0: Since the operator A has a compactresolvent, commutes with the involution J and is real, it admits an orthonormal basis ofreal eigenvectors {n H, Jn H, n N} with

An = nn and AJn = nJn

so that n R and limn |n| = +.The operator iJA is self-adjoint with domain D(A) and writes in HC = nN(Cn

CJn) as the block diagonal operator

iJA = nN

0 in

in 0

.

Hence the equation

itf = iJAfis solved by the unitary strongly continuous group

eit(iJA)

tR = (e

tJA )tR, whichadmits the explicit block diagonal expression

etJA = nN

cos(tn) sin(tn)

sin(tn) cos(tn)

.

22


23/30

Hence this linear evolution preserves the HC-norm, the domain D(A) and reality.2) Local existence for the nonlinear case: The Duhamel formula

f(t) = etJAf0 t0

e(ts)JAJ(Bf(s) + h(f(s))) ds .

and the analyticity assumption on h allow to use the standard fixed point argumentin C0([0, Tf0] ; HC) . The fixed point provides the real analyticity of f(t) with respectto f0 HC . Finally, the uniqueness in C

0 ([0, Tf0]; HC) and the fact that the integralequation can be solved in C0([0, Tf0]; H) ensures that f(t) H for all t [0, Tf0] as soonas f0 H .3) Approximation with a bounded generator: In order to establish the preservedquantities, we approximate the linear operator by bounded ones. Let A = 1[0,](A)A.By the spectral theorem, we get

(etJA etJA)f2

(eit(iJA) eit(iJA))f2

=

1 eit

2

df()

where df is the spectral measure of the given element f Fh with respect to the self-ajoint operator iJA . Hence by dominated convergence we get:

t R, s-lim

etJA = etJA .

We write the difference between the two integral equations:

f(t) = etJAf0

t0

e(ts)JAJ(Bf(s) + h(f(s))) ds

and f(t) = etJAf0

t

0

e(ts)JAJ(Bf(s) + h(f(s))) ds

as

f(t) f(t) =

etJA etJA

f0

t0

e(ts)JA e(ts)JA

J(Bf(s) +h(f(s))) ds

t0

e(ts)JAJB(f(s) f(s)) ds t0

e(ts)JAJ(h(f(s)) h(f(s))) ds .

For a fixed f0 and a fixed t [0, Tf0], the analyticity assumption on h and the fact thatetJA and etJA are unitary operators, lead to

f

f

2

f0() +

Cf0 0,t f(s) f(s)2 ds

with lim f0() = 0 . By the Gronwall Lemma, we obtain for any f0 HC theexistence of Tf0 such that

t [0, Tf0], lim

f(t) f(t) = 0 .

23


24/30

4) Upper bound for the norm and global existence: According to the first step,we can reduce the analysis to the case where A is a bounded operator. Then the local intime mild solution is a classical solution for any f0 HC. We compute

t f2 = (tf | f)C + (f| tf)C = 2Re (JAf + JBf + Jh(f) | f)C

= 2Re (f| Jh(f))C

(JBf | f)C + (f| JBf)C

2Re (f| Jh(f))C + [J, B] f2

Here we differentiate the gauge invariance ofh, h(eJf) = f,:

0 =d

dh(eJf)

=0

= Re (h(f) | Jf)C

.

We have proved t f2 [B, J] f2 when f solves (A.1) with A L(HC). The

inequality f(t) e[B,J]t/Ef0 can be extended to the case of unbounded A accordingto step 3).Finally this norm control provides the existence of a global in time solution, Tf0 = +,

for any f0 HC .We now consider the conservation of energy under the additional assumption that the

flow (t) preserves the domain D(A). This will be checked in the proof of Proposition A.8below. We refer to [ChMa] for a more general statement.

Proposition A.4. Under Hypothesis A.2 and if the flow (t) preserves the domainD(A)in the sense that the solution f to (A.1) belongs to C0 (R; D(A)) when f0 D(A), thenthe equality

H(f(t)) = H((t)f0) = H(f0)

holds for any t R and any f0 D(A) .

Proof: If f C0 (R; D(A)), then the mild solution to (A.1) is a strong solution,f C1(R; HC). Since the gradient of H equals Af + Bf + h(f) we write:

tH(f) = Re (Af + Bf + h(f) | tf)C= Re ((A + B)f | J(A + B)f)

CRe ((A + B)f| Jh(f))

CRe (h(f) | J(A + B)f)

C

Re (h(f) | Jh(f))C

= 0 .

We now give some applications in specific cases arising in our analysis.

Proposition A.5. Under Hypothesis A.2 withh = 0, then the Hamiltonian vectorJ(A+B) defines a linear closed unbounded operator on HC with domainD(J(A+B)) = D(A).It has a compact resolvent and its spectrum has symmetries with respect to the two axesR and iR.Moreover if the energy H(f) = 12(f| (A + B)f) is non negative for all f D(A) H then(J(A + B)) iR .

24


25/30

Remark A.6. Note that although (J(A + B)) iR, the operator J(A + B) is not anti-adjoint (even in finite dimension), except when J and A + B commute, [J, B] = 0. The

finite dimensional version of the final result is a very specific case of the classification ofquadratic Hamiltonian functions which is reviewed in [Arn]-Appendix 6.Especially when applying this Proposition, a good identification of the Kaehler structureon H and the two different complex structures on HC is useful.

Proof: Since the operator is a bounded perturbation of JA, the first statementsare standard (see [ReSi]). Concerning the symmetries of the spectrum, the followingequivalences hold:

( (J(A + B))) ((J(A + B) ) invertible)

J((A + B)J )J1 invertible

((A + B)J ) = (J(A + B) ) invertible

(J(A + B))

and provide the symmetry with respect to iR

. For the second symmetry, we introduce theconjugate u of any vector u HC as the symmetric vector with respect the real subspaceH . Since J, A and B are real operators, we obtain 4 for any u H, u = 0,

(J(A + B)u = u)

J(A + B)u = u

.

Since J(A + B) has a compact resolvent, its spectrum is thus symmetric with respectto R .Finally, assume that the energy H(f) = 1

2(f| (A + B)f) is non negative for all f

D(A) H. Since (A + B, D(A)) is self-adjoint on HC, this means that A + B is a nonnegative operator and we get

(H(u) = 0, u HC) (u Ker(A + B)) .

Let (J(A + B)) be a non zero eigenvalue with eigenvector u0 = 0. Since u0 D(A + B) = D(A), u(t) = etJ(A+B)u0 C0(R; D(A)) and according to Proposition A.4,the energy is conserved:

e2Ret (u0 | (A + B)u0)C =

etu0 | (A + B)etu0C

= (u(t) | (A + B)u(t))C

= (u0 | (A + B)u0)C .

Since = 0, we get u0 Ker(A + B), (u0 | (A + B)u0)C = 0 and Re = 0 .

Proposition A.7. Under Hypothesis A.2, a sufficient condition for an equilibrium to bespectrally stable, is that it is a local minimum.

4Especially for this argument, it is preferable to forget the complex structure on H identifying J withthe multiplication by i .

25


26/30

Proof: According to [HMRW], an equilibrium is spectrally stable when the spectrumof the linearized Hamiltonian is included in iR . At a critical point f ofH, the linearizedHamiltonian equals

JA JB JHess h(f) = J(A + B + Hess h(f)) .

Iff

is a minimum forH

thenA

+B

+ Hess h

(f

) is non negative owing to the analyticityproperty ofh:

H(f + ) =1

2( | (A + B + Hess h(f))) + O(3) .

By replacing B by B + Hess h(f) in Proposition A.5, we get (J(A + B + Hess h(f))) iR .

We end this appendix with a specific nonlinear Hamiltonian for which Noethers the-orem ([Arn][AbMa]) can be stated in a very explicit form.

Proposition A.8. Assume Hypothesis A.2 with B = 0 and with the additional gauge

invariance forh: R, f H, h(eJAf) = h(f) .

Then for any f0 D(A) H, the solution f to (A.1) satisfies

f C1 (R; H) C0(R; D(A))

t R, H(f(t)) = H(f0) , (f(t) | Af(t))C = (f0 | Af0)Ch(f(t)) = h(f0) and f(t) = f0.

Proof: The new gauge invariance implies that for any f, H and R:

h(f) + (

h(f) | ) + O(

2

) =h

(f + ) =h

(e

JA

(f + ))= h(eJAf) +h(eJAf) | eJA+ O(2)

= h(f) +

eJAh(eJAf) |

+ O(2) .

Hence we getf H, R , h(eJAf) = eJAh(f) .

Thus the solution to (A.1) with f0 H, f(t) = (t)f0, also satisfies

R, t R, eJAf(t) = (t)(eJAf0) .

The regularity of the flow with respect to initial data allows to say that for any t R,

eJAf(t) is differentiable with respect to when f0 D(A). This yields

(f0 D(A))

f(.) = (.)f0 C0(R; D(A))

.

Proposition A.4 givest R, H(f(t)) = H(f0)

26


27/30

when f0 D(A) . It is enough to check that the quantity (f(t) | Af(t)) does not vary. Bydifferentiating the new gauge invariance with respect to we get now for any g D(A)H:

0 =d

dh(eJAg)

=0

= (h(g) | JAg) .

For f(t) = (t)f0 with f0 D(A) H we compute:

t (f(t) | Af(t)) = 2Re (tf | Af(t))

= 2Re (JAf(t) | Af(t)) + 2Re (Jh(f(t)) | Af(t))

= 0 + 2 (h(f(t)) | JAf(t)) = 0 .

Finally, the fact that f(t) = f0 is a direct consequence of the equality h(eJf) = h(f).

Remark A.9. For the specific Hamiltonian considered in Proposition A.8, one can thinkabout several criteria for the formal stability and the nonlinear stability. This would

require additional discussions and again specific assumptions according to [HMRW]. Wedo not consider such criteria in our analysis.

Acknowledgments: This work started while the author had a sabbatical semester atCNRS and after the organization of the workshop Nonlinear spectral problems and meanfields models supported by the French ACI Systemes hors-equilibre classiques et quan-tiques. He was also supported by the ESF network SPECT for a short visit in the ErwinSchrodinger Institute and he thanks J. Yngvason for his hospitality. Finally, the authorwarmly thanks A. Aftalion and X. Blanc for the numerous discussions which stimulatedhis interest in the models for Bose-Einstein condensates.

References

[ARVK] Abo-Shaeer JR, Raman C,Vogels JM, Ketterle W, Observation of Vortex Latticesin Bose-Einstein Condensates (2001) Science 292, 476-479.

[AbMa] R. Abraham, J.E. Marsden Foundations of Mechanics. Addison-Wesley Publish-ing Company (1985).

[Aft] A. Aftalion, Vortices in Bose Einstein condensates. Progress in Nonlinear Differen-tial Equations and Their Applications, Vol. 67, Birkhauser, 2006.

[AfBl] A. Aftalion, X. Blanc Vortex lattices in rotating Bose Einstein condensates.Preprint.

[ABD] A. Aftalion, X. Blanc and J. Dalibard, Vortex patterns in fast rotating Bose Ein-stein condensates, Phys. Rev. A 71, 023611 (2005).

27


28/30

[ABN1] A. Aftalion, X. Blanc and F.Nier, Vortex distribution in the lowest Landau Level,Phys. Rev. A (2006) to appear.

[ABN2] A. Aftalion, X. Blanc and F. Nier A Lowest Landau Level functional andBargmann spaces in Bose Einstein Condensates. preprint (2006).

[Arn] V. Arnold Mathematical Methods of Classical Mechanics. Graduate Texts in Math-ematics, Springer (1989).

[Bar] V. Bargmann, On a Hilbert space of analytic functions and an associated integraltransform. Comm. Pure Appl. Math. 14 1961 187214.

[BHK] G. Benettin, J. Henrard and S. Kuksin Hamiltonian dynamic theory and applica-tions. Lect. Notes in Mathematics 1861, CIME Foundation Springer-Verlag (2005).

[BWCB] L. O. Baksmaty, S. J. Woo, S. Choi, and N. P. Bigelow, Tkachenko Waves inRapidly Rotating Bose-Einstein Condensates, Phys. Rev. Lett. 92, 160405 (2004).

[Car] E. Carlen Some Integral Identities and Inequalities for Entire Functions and TheirApplication to the Coherent State Transform. J. Funct. Analysis 97 (1991).

[ChMa] P.R. Chernoff and J.E. Marsden Properties of infinite dimensional Hamiltoniansystems. Lect. Notes in Mathematics 425, Springer-Verlag (1974).

[Dav1] E.B. Davies. Pseudospectra, the harmonic oscillator and complex resonances. R.Soc. London Ser. A Math. Phys. Eng. Sci. 455, p. 585-599 (1999).

[Dav2] E.B. Davies. Non-self-adjoint differential operators. Bull. London Math. Soc.34(5), p. 513-532 (2002).

[DeJa] J. Derezinski and V. Jaksic Spectral theory of Pauli-Fierz operators. J. Funct.Anal. 180(2) (2001), pp 243327.

[DSZ] N. Dencker, J. Sjostrand, and M. Zworski. Pseudospectra of semi-classical(pseudo)differential operators. Comm. Pure Appl. Math. 57 (3), p. 384-415 (2004).

[Fol] G.B Folland Harmonic Analysis in Phase Space. Princeton University Press, 1989.

[GiJa] S.M.Girvin, T.Jach, Formalism for the quantum Hall effect: Hilbert space of ana-lytic functions, Phys. Rev. B, 29 (1984) 5617-5625.

[Gro] L. Gross Hypercontractivity on complex manifolds. Acta Math. 182, (1999).

[Hag1] M. Hager Instabilite spectrale semiclassique pour des operateurs non auto-adjoints I : un modele. Preprint http://hal.ccsd.cnrs.fr

[Hag2] M. Hager Instabilite spectrale semiclassique pour des operateurs non auto-adjoints II. Preprint http://hal.ccsd.cnrs.fr

28


29/30

[HelNi] B. Helffer and F. Nier Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians. Lect. Notes in Math. Vol. 1862, Springer(2005).

[HerNi] F. Herau and F. Nier. Isotropic hypoellipticity and trend to the equilibriumfor the Fokker-Planck equation with high degree potential. Archive for Rational

Mechanics and Analysis 171 (2), p. 151-218 (2004).[Ho] T. L. Ho, Bose-Einstein Condensates with Large Number of Vortices, Phys. Rev.

Lett. 87, 060403 (2001).

[HMRW] D. Holm, J.E. Marsden, T. Ratiu and A. Weinstein Nonlinear stability of fluidand plasma equilibria. Physics Reports 123(1-2) (1996), pp 1116.

[Kuk] S. Kuksin Hamiltonian perturbations of infinite dimensional linear systems withimaginary spectrum. Funct. Anal. Appl. 21(3) (1987) pp 192205.

[Mar] A. Martinez An Introduction to Semiclassical Analysis and Microlocal Analysis.

Universitext, Springer-Verlag, (2002).

[Nel] E. Nelson A quartic interaction in two dimensions in Mathematical Theory of Ele-mentary Particles 6973, MIT Press (1966), 6973.

[Pra] K. Pravda-Starov Pseudospectrum for a class of semiclassical operators. preprint.

[ReSi] M. Reed and B. Simon. Method of Modern Mathematical Physics. Academic press,(1975).

[SCEMC] V. Schweikhard, I. Coddington, P. Engels, V. P. Mogendorff, and E. A. Cornell,Rapidly Rotating Bose-Einstein Condensates in and near the Lowest Landau Level,

Phys. Rev. Lett. 92, 040404 (2004).

[SHMacD] J. Sinova, C. B. Hanna, and A. H. MacDonald, Quantum Melting and Absenceof Bose-Einstein Condensation in Two-Dimensional Vortex Matter, Phys. Rev. Lett.89, 030403 (2002).

[SjZw] J. Sjostrand and M. Zworski Elementary linear algebra for advanced spectralproblems. eprint arXiv:math/0312166 (2003).

[Son] E.B. Sonin, Ground state and Tkachenko modes of a rapidly rotating Bose Einsteincondensate in the lowest Landau level state, Phys. Rev. A 72, 021606(R) (2005).

[Tre] L.N. Trefethen. Pseudospectra of linear operators. Siam Review 39 (3), p. 383-406(1997).

[WBP] G. Watanabe, G. Baym and C. J. Pethick, Landau levels and the Thomas-Fermistructure of rapidly rotating Bose-Einstein condensates, Phys. Rev. Lett. 93, 190401(2004).

29


30/30

[Zwo] M. Zworski. Numerical linear algebra and solvability of partial differential equa-tions. Comm. Math. Phys. 229 (2), p. 293-307 (2002).

f. nier- bose einstein condensates in the lowest landau level: hamiltonian dynamics

Documents