Strong nonresonance of Schrödinger operators and an averaging theorem

ELSEVIER

Physica D 86 (1995) 349-362

Strong nonresonance of Schr6dinger operators and an averaging theorem T. Kappeler 1, S. Kuksin Ohio State University, Columbus, OH 43210, USA and Institute for Information Transmission Problems, Moscow, Russia

Received 31 August 1994; revised 23 February 1995; accepted 1 March 1995 Communicated by H. Flaschka

Abstract

We prove that linear Schr6dinger operators -A + q on a torus or on a bounded smooth domain in R a considered with Dirichlet boundary conditions, have a strongly nonresonant spectrum for any potential q of generic type (generic in the sense of Kolmogorov measure). As a consequence, a Krylov-Bogolubov averaging theorem holds for nonlinear perturbations of the corresponding Schr6dinger evolution equations.

I. Introduction

Consider the time dependent Schr6dinger equation for a complex valued function u = u(t,x)

(x~O;t~) iOtu = - Au + q(x) u

(1.1)

and the following nonlinear perturbation:

iO,u = - A u + q(x) u +

g(lul% u ,

(1.2)

where q(x) is a bounded, real valued potential, e is a perturbation parameter and g:N---~ R is a smooth real-valued function. Concerning the space domain 12 we consider two cases: e i t h e r / 2 is a d-torus qrd, or O is a bounded domain in N d with smooth boundary. In the later case we impose Dirichlet boundary conditions on the function u(t, x),

u(t,x)=O

(x~O0).

(1.3)

(Condition (1.3) is empty when g2 = -ga as 0$2 = ~.) Solutions u(t, x) of (1.1), (1.3) are supposed to be in C(R; H ) , u(t, . ) ~ H, where H = (H, (., . ) ) C Lz($2) is a Hilbert space of functions u: O--* C, which will be specified later. Denote by (h/ -= hi(q))i_,~ the eigenvalues corresponding to (1.1), (1.3) and 1

Partially supported by NSF.

0167-2789/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved S S D I 0167-2789(95)00115-8

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choose an L2-orthogonal system of eigenfunctions ~oj(x)=-~oj(x, q) corresponding to the eigenvalues Aj. For any N E N, a solution us(t , x) = u(t, x) is said to be an N-frequency solution of (l.1), (1.3) if it is of the form N

us(t, x) = ~ ~j e%' q~j(x; q) j=l

with arbitrary nonzero numbers fij E C. In analogy to the finite dimensional situation (cf. [2]) we define the following averaged N-frequency solutions:

Definition 1. A 1-parameter family of functions UN;E(t , X) in C(~; H ) of the form N

Un,~(t, x) = ~ t~j e i~j;~` ~j(x; q)

(1.4)

j=l

depending continuously on the perturbation parameter e, is said to be an averaged trajectory of the nonlinear problem (1.2), (1.3) if there exist a > 0 independent of e, a positive number T~ = T,(a; Un,,) and a solution u,(t, x) of (1.2), (1.3) in C([0, T~]; H ) such that (AV,)

sup O~t<-Te

Ilus;At, ") - u~(t, .)]],

= O(e")

and T~ = O ( l o g ] e - ' l / e ) .

Definition 2. A Hilbert space H = (H, (., .)) of functions u: J2---~ C, H C L2(j2), is said to be admissible for (1.2), (1.3) if the following properties hold: ( H I ) the linear problem (1.1), (1.3) is well posed in H (i.e., for initial data in H there exists a unique solution of (1.1), (1.3) in C(ff~; H)); (H2) for u in H, g(lu(.)] 2) u is in H and the map H--*H, u(. ) ,-, g(lu(. )l 2) u is C =. Moreover this map is real analytic in Re u(.), Im u(.) in the case where g is real analytic. (H3) The eigenfunctions q~j(-, q) are elements in H and form a basis of H. In many situations H can be chosen as follows: in case g2 = ql-d, H is the Sobolev space HS(-~ d) with

s > d/2. In case J2 is a bounded domain in •d with 1--- 0 (1-
TN;~:={Uj~lZJ~°J(',q):z,~C;lzJ]=~=

(1--
Denote by u the natural Haar measure on TN,~. In a neighbourhood of TN;~ in F n the arguments Oi(z 1. . . . , z n ) = Arg(zj)C ff~/2~r~_ are well defined. So Ojo~-n are well defined smooth functions in a neighbourhood U of TN;~ in H. Due to (H3), H is dense in Lz(J2)= L2(j2; C). Therefore the embedding H ~L2(J2) induces a dual embedding L2(J2)~H* and the real scalar product (., .) in L2(j2),

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( (u, v) = Re ~ u(x) 6(x) dx, g2

induces a pairing H x H*--~ ~. Given a real-valued smooth functional f" G---~ ~, defined in some open set G C H , its LZ-gradient 7 f = ~f/~a: O---~H* is given by (v, Vf(u)) = d,f(v)

(u ~ t~, v E H).

Definition 3. Given two smooth, real valued functionals f~, f2 defined on an open set G C H, introduce for u in (? the following Poisson brackets:

If,, Ll(u) = (i VL (u), VL(u)). Finally, with g~(y) = f~' g(y') dy' where g is smooth and real valued, define for u in H

h(u) :-- ½ f

g,(lu(x)l~)dx

(1.5)

12

and note that 7h(u)=g(]u(')l z) u(.). In [6], one of us proved the following result:

Theorem 1. Let H be any admissible Hilbert space and assume that g is real analytic. Given N ~ N, denote by A/;~ the real numbers defined by Aj; = A i ( q ) + e

f [h,0joTr~v]dx ,

(1.6)

TNU"

where h is given as in (1.5). Then ux:,(t, x), as defined in (1.4), is an averaged trajectory of (1.2), (1.3) so that (AV,) holds for any 0 < a < 1, if the following two nonresonance conditions are satisfied for the vector au(q)= (A~(q),..., aN(q)): there exist # > 0 and L > 0 such that (NRI)

PN [AN(q)'ot ] >-- lot}

(NR2)

IaN(q)'~ --'Xjl-

(v,~ ~ zN\{0}) (1 + P[a[) L (Va E Z N ; W _ N +

1).

Remark. The assumption that g is real analytic was imposed in [6] to simplify the proof. The result remains true for sufficiently smooth nonlinearities. It is shown in [6] that sequences of real numbers (rb)j> ~ of generic type with asymptotic estimates of the form of the Weyl asymptotics rIj = C] 2/d + o ( j 2/a)

(1.7)

satisfy the nonresonance conditions (NR1), (NR2).More precisely given a sequence (r/~°))j_>N+~, satisfying (1.7), the set of all vectors (r/j)~j< u E A N SO that (NR1) or (NR2) with L = 3d are violated for all p > 0 has zero Lebesgue measure. In this paper we prove that given any N - - 1 and L - > 3 d the spectrum (Aj(q))/> I of - A + q satisfies (NR1), (NR2) for potentials q(x) of generic type. More precisely, introduce for any r > 0 and X > 1

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K(r) :=

q(x) = ~ 0j~0j(x) : qi E C;

toil ~

rJ -x

vj

1 ,

j=l

where q~j(x) -= ~(x; q - 0). Denote by /z, the Kolmogorov measure on K(r),

jx j=l

with 3~j denoting the Lebesgue measure on the interval [-rj -x, rj-X].

Theorem 2. Let X > 1, r > 0 , N -> 1 and L = 3d. Then for /z,-almost every potential q in K(r) the spectrum (Aj(q))m 1 satisfies the nonresonance conditions (NR1), (NR2). Theorem 2 is proved in Section 3 and in Section 2 some auxiliary results are presented.

Remark. We point out that Theorem 1 says nothing about the behavior of the considered solutions for t > 1/e b with b > 1. If d = 1 and O is an interval, KAM-theory for (1.2), (1.3) asserts the existence of many solutions which are quasiperiodic in time t. For q in K(r) denote by Orb,(q) the union of all orbits in H of t-quasiperiodic solutions. Then for any fixed p E H, the distance distn( p, Orb,(q)) converges to zero in/zr-measure, i.e. for any 6 > 0 lim tZr{q E K(r): distil( p, Orbs(q)) > ~} = 0 e----*0

(cf Theorem 4.1 in [7]). This result can be viewed as an averaging theorem for large time. However, the result requires that the spatial dimension be one and little is known for d---2.

2. Auxiliary results Let g2 be a bounded, smooth domain in ~d or /2 = T d. Consider the Laplacian - A on O with Dirichlet boundary conditions and denote by (h~°))i_>1 the Dirichlet eigenvalues of - A written in increasing order and with multiplicities. According to Weyl's law, the h~°) satisfy the following asymptotics: (2.1)

A~ °) = C d j 2/a + o ( j 2 / a ) ,

where C a is the volume of 12. Fix an L2-orthogonal basis of eigenfunctions (~Oj)j> 1 corresponding to the eigenvalues (~(0),~ t,,j ~j_>~ and introduce for s E ~ the complex Hilbert spaces H;(g2) :

q = ~ 0j~j: ~' j=l

2<

-

(2.2)

j

Notice that, due to (2.1), q E H~(g2) implies that AS/2q E LZ(g2). Moreover HSo(T d) = H'(T d) and, for s > d/2, H'o(g2 ) ~ Co(O ) where C0(g2) denotes the Banach space of continuous real valued functions on g2 which vanish on the boundary 012. Fix s > d/2 and, for convenience, introduce F := H'o(O ). For any subset B C F and any subset J C 77, introduce zrj : F---* F and Bj C F as

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e~

(2.3) (2.4)

Bj := {-a'j(q) : q ~ B } . In particular, Fj = 1rj(F). Introduce for arbitrary, but fixed X > ½ + s/d and any r > 0 K(r):=

q = ~'~ 0jq~j: 10jl <

,

(2.5)

]=1

endowed with the Tychonoff topology. Then K(r)~ F = H ~ ( O ) compactly.

Definition. The cr-algebra "~r of cylindrical sets of K(r) is the it-algebra generated by cylinders of the form Bj + K(r)N~j where B is any open set in K(r) and J is any finite subset of N. Denote by ~r the Kolmogorov measure defined on -~r. It is the following infinite product of probability measures: "r=

]=1

'

(2.6)

where ,/j denotes the Lebesgue measure on the interval [-r] -~, r]-X]. More generally, for any subset J C N, denote by IXr.J the Kolmogorov measure on cylindrical subsets of K(r)j:

Note that ~r = ~r.J ® ~.~\J"

Lemma 2.1. Let U be an open connected set in Ho(I2 ). Let f " U---> • be a function with the following properties: (i) f is real analytic; (ii) f does not vanish identically on U. Then the closed set f - l ( 0 ) M K(r) is of measure zero, i.e.

txr(f - '(0) f-) K(r)) = 0.

(2.8)

Proof. We only have to consider the case where U f3 K(r) ~s 9. As f does not vanish identically, there exists po E U with f(po)#O. As U is open, 7r[1,Mip0 is in U for M sufficiently large and limm-.~ f(Trll,glPo) = f(Po) # 0. Thus there exists M o -> 1 so that ~rtl,MolPo~ U and f(~rtl,M0lPo ) # 0. As f is real analytic this implies that {q E Uj: f ( q ) # 0 } is open and dense in U~ where J = [1, M] with M >-M o. As K(r)fq U ~1~ we conclude that K(r)71 Uj is a nonempty open set of Uj. Therefore {qEK(r) M Uj: f ( q ) ~ 0 } is an open and dense subset of K(r) M Uj. Choose q l E K ( r ) n U j with 2e := If(q01 > 0. As K(r) is a compact subset of H~(/2), f is uniformly continuous on K(r). Thus for e > 0, given as above, there exists J -- [1, M] _~ [1, M0) with q~ + K(r)~\j C_U so that for any q', q" K(r) f3 U with 7rjq' = 7bq" one has If(q') - f ( q " ) [ <- e . Therefore, for any given p E K(r)N\j, If(ql +P)] >- e and so

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q~ @ {q ~ K(r)j: q +p ~ U, f(q +p) ~ 0 ) . Using that f is real analytic we conclude that for any p E K(r)~j the set {q E K(r)j : q + p @ U, f(q + p) # 0 } is open and dense in { q E K(r)j " q + p E U }. As a consequence, for any p E K(r)~\j

tZrj((q E K(r)j: q + p E U; f(q + p ) = 0}) = 0. Using Fubini's theorem, (2.9) leads to (2.8).

(2.9) []

Let q be in F--- HD(O), and denote by (A/= A j ( q ) ) ~ the Dirichlet eigenvalues of - A + q, written in increasing order and with multiplicities, A0(q) < A,(q) -< Az(q) - < ' . . . Introduce for J C N Ej := (q ~ F" Aj(q) is a simple eigenvalue for any j E J } . For convenience, we write E N = EI~,N 1 and observe that

e N = {q @ F: Al(q) < A2(q) < - " < AN(q) < AN+~(q) --<'" "}. Let V be a subset of F. Then V is called a real analytic variety of finite definition, or, for short, real analytic variety (cf. [4] or [8]), if there exists a finite or infinite sequence {V~ } of subsets of V such that

V= U vk

(2.10)

k~l

is a disjoint, locally finite union, and for q in V~ there exists an open neighbourhood U of q in F and a finite number of real analytic functions fl, • • •, f, defined on U such that an element p in U satisfies

pEVkifffj(p)=O

(2.11)

for 1 - < j - < n .

For a real analytic variety V the codimension of V is defined as follows: codim V "= sup{in ~ codim Vk" V= U Vk satisfies (2.10) and (2.11)}, k_>l

where codim Vk := inf{codimqVk : q ~ Vk} and codimq Vk :=

inf U open nbh

ofqinF

[sup{ sup

p~UnVk

dim(dpf~ . . . . . dpf.)}],

where the supremum has to be taken over all finite sequences of real analytic functions ft . . . . .

~: g---, R. Lemma 2.2. Let J C_[~ be finite. Then (1) b-NEj is a real analytic variety of codimension at least 2. (2) Ej is open, dense and connected. (3) /xr((FkEj) D K(r)) = 0 for any r > 0.

fn, with

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Proof. For (1) and (2) cf. [4]. Statement (3) follows from Lemma 2.1 applied to a stratification of the real analytic variety b ~ s.

[]

Now introduce the map

AN: EN---~ N, q~Au(q)=(A,(q),...,AN(q) ) .

(2.12)

Note that A N is a real analytic map. The following result is certainly well known (cf. e.g. [3] or [5]). For the convenience of the reader we include its proof.

Lemma 2.3. Let N E t~. There exists a potential q in C~(I2) so that the first N Dirichlet eigenvalues of - A + q on ~ are simple (therefore q is in EN) a.nd so that dqAN is onto. Proof. Choose N + 1 open discs, / 2 , , . . . , ON+l, of the form Oj = xj + 12j(O, rj) C_a , where g2j(0, rj) is a disc in 0Ofqg2j=0

~d of radius rj and center 0, in such a way that

(1-
g~kfqg2j=0

(j~k).

(2.13)

Denote the Dirichlet eigenvalues of the Laplacian - A on g2j (1 <---j--
<...

and for each j choose an LZ-orthogonal system of eigenfunctions ~bj;,, ~bj;2,..., corresponding to the eigenvalues ,,j;,x(°),-j;2x(°),. . . . Choose the radii r, > ' ' ' > r N > r N + ~ in such a way that ~(0)_1;1< " " " < A(°~ N + I ; , < A (°~ 1,2" Denote by O' the complement

and define a sequence of potentials q~ with support in ~ ' , q~ = n 1n, , where le, denotes the characteristic function of the set 0 ' . Denote the Dirichlet eigenvalues of - A + q~ on ~ by ()tn; k

:=

)tk(qn))k>l

and let (~0n;k)k_>1 be an L2-orthogonal system of eigenfunctions corresponding to the eigenvalues (A~;k)k_>~. By the minimax principle (cf. [1]) the following a priori estimates hold for any 1- 1: (i) (ii)

x" ' n : j<- -x" j ;(°). l

,

2

- - ~(o) ,,j;,

(iii) n ,', (°) "rn;j 2 L 2(g} ) -< Aj;1

"

(2.14)

In view of (2.14) (i), (ii) and (iii) we can assume without loss of generality (by taking a subsequence if necessary) that for 1 ~ j -~ N + 1 the following limits exist:

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(i') Aj:=!imA

j;

(ii') ~ : = !irn ~O.;j weaklyin

H~(a) ;

(iii') [[~[Ir2~,) =lirn [[g,,;~l[L2(a,) = 0.

(2.15)

Combining (2.15) (ii') and (iii') we conclude that !ira g,~;j = 0

weakly in H~(g2').

(2.16)

This shows that q,j is in H~(J2) with supp ~ _C ~*:l~JN+lOk" Therefore H'o(a

(2.17)

) .

On the other hand for any 1 - < j - < N + I ,

1-
and hj <- - "~0) " 1;j

(2.19)

'

The statements (2.16)-(2.19) imply that Ai = •x(0) ,1;/, ~0j [ai = 4~j;1 and ~.[a, = 0 for k # j . z ~ are linearly Noting that q6 . . . . . ~0N+~ have disjoint supports we conclude that ~02,..., ~0N+ independent. Observe that ~0~ = lim,_~= q~];j strongly in L 1(12). Since det((g,~, la, ) II--
Let J C [~ be a subset with # J = N and define Eu; ~ := {q E E N : dqAN: F]----~~N is a linear isomorphism}. An alternative definition of E u j is the following. Denote by Aq,] the N × N matrix whose columns are given by dqaN(%, ) with j E J = {/1
d qAN(~%) ) .

T. Kappeler, S. Kuksin / Physica D 86 (1995) 349-362

357

ENj = { q E E N : det(Aq,j) ¢: 0}. Note that det(Aqj) is a real analytic function on E u. Therefore {q C E N" det(Aq.~)= 0} is a real analytic variety.

Lemma 2.4. Let N E ~. Then there exists a subset JN C_~ with #JN = N such that (1) EN'~EN:jN is a real analytic variety of at least codimension 1. (2) EN;~u is open and dense in F. (3) p.r((F~E;JN ) D K(r)) = O. Proof. By Lemma 2.3 there exists q E E N such that dqA N is onto. Therefore the restriction of dqA N to a subspace Fj with # J sufficiently large but finite is already onto. As a consequence, there exists a subset JN c N with ~JN = N so that dqA N [F~Nis a linear isomorphism. Therefore, {q E E N : det(Aq;JN ) ~ 0} is not empty and as det(AqjN) is a real analytic function on E N, statements (1) and (2) follow. Statement (3) follows from Lemma 2.1 and Lemma 2.2 (3). []

3. Proof of Theorem 2

The proof is presented in a number of lemmas. Introduce for N-> 1 and o~ @ 77 N the sets E ~ ) ( a ) : = {q E E N : A u ( q ) ' a # 0 } ,

(3,1)

E ~ ) ( j ; a) := {q ~ EI1,NIu{j} " AN(q)" a -- hi(q) ¢=0}.

(3,2)

(])

Lemma 3.1. (1) For ~ E EN\{0} the set E N \ E N (o~) is a real analytic variety of at least codimension 1 and txr(K(r ) D (ENkE(NI)(ot))) = 0 for any r > 0. (2) For any j>-N + 1, a ~.~_U the set ED,N]U{j}kE(NZ)(j" , or) is a real analytic variety of at least codimension 1 a n d / x r (K(r) D (Etl,uluu)kE(uZ)(j; a))) = 0 for any r > 0. Proof. (1) As A u ( q ) . a is a real analytic function, and in view of Lemma 2.2, EukE~)(a) is an analytic variety. Note that, due to Lemma 2.4, Au(q).a does not vanish identically. This implies that E u ~ ) ( a ) is at least of codimension 1. The fact that K(r)D Eu\(E~)(ot)) has zero/~r-measure follows then from Lemma 2.1. (2) According to Lemma 2.2, for j -> N + 1, EI1,NIu{j } = {q U_E N : Aj_l(q) < A j ( q ) < Aj+l(q) }

is open and dense. Since both As and Aj are real analytic functions on E[1,NlU{j}, for any a G Z N, the set {q E EB,Nlu{j} : aN(q).a -- At(q) = 0} is a real analytic variety. Arguing as in (1) (with N replaced by N + 1) one sees that AN(q). a -- At(q) does not vanish identically. Therefore E[1,Nlu{j}kEN t2)(J,.. or) is at least of codimension 1. Apply again L e m m a 2.1 to complete the proof of statement (2). [] Let q E EN;1N D K(r) for some r > 0. Let Ju C [~ with :~JN = N. Choose M -> 1 so that ,IN C [1, M] and define J : = [1, M]kl u.

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Definition. A (Ju, M)-cylindrical neighbourhood of q in K(r), •-U,q:jN;M'=BjN + Bj + K(r)~\[l.gl

(3.3)

is called admissible if there exist R E Q, R > 0, and C =

C(q, N, M) > 0 so that

~--q;JN;MC_EN;jN fq K(r) ,

(3.4)

BjN= BjN(R ) is a disc of radius R with center ~rgNq in FjNN K(r),

(3.5)

Bj = Bj(R) is a disc of radius R with center ~rjq in Fj N K(r),

(3.6)

and so that, for any embedding with

P2EBj+K(r)~\U.M ] the

I]aN;p211cl <--C

II(AN;p2)-']]c~ <~C .

and

m a p AN;pz:BjN----)~ N,

p ~ A N ( p t + p 2 ) is a C ~(3.7)

Note that for any q E EN;JNA K(r), the admissible neighbourhoods of q form a basis of neighbourhoods of q in K(r). In the rest of this section we will only use admissible ones. For convenience we write p ~Eq;JN:M as p = P l +P2 where Pl = 7r,u(P) ;

P2 = YrN\JN(P)"

(3.8)

Lemma 3.2. For 0 < p -< 1 and a E 7/N\{0} let Uo;~(q) "= { p E ~q~#,;M : ]AN(p) . a] <--]-~} .

(3.9)

Then /x, ( <9< 0

(,_J

1 af~zN\{o}

Uo;~(q))=limg,( p~0

~

aCz~ \{0}

Uo;~(q))=O.

Proof. To prove (3.10) it suffices, due to Fubini's theorem, to show that for any P2 fixed in lim

~]

p---~OaEzN\{o}

~.Lr,JN(Uo;a;p2) = 0,

(3.10) K(r)~\j N

(3.11)

uniformly with respect to Pz, where

Up;,;p 2 := {Pl: P = P l +P2 E Uo:~} and /x~,ju is the measure introduced in Section 2. The identity (3.11) follows from the following estimate: There exists C = Cu. q > 0 independent of p and a E zN\{0} such that for any 0 < p --< 1, a E YN\{0},
(3.12)

~* ~rN\jN(--q;JN;M ) the map

AN;pz: {Pl: P = P l +P2E~--'q;JN;M} ----~N ,

PI"--~AN(Pl + P 2 ) ,

is a diffeomorphism which satisfies the estimates (3.7). Therefore there exists a constant C = independent of p and a such that, with 3' denoting the Lebesgue measure,

Cu, q > 0

T. Kappeler, S. Kuksin / Physica D 86 (1995) 349-362

]J~rJN(Up.a.p2)
({ x ~ g :

IXl < Cl,

x

~

<

lx,l<- -i l +l, Ixjl

})

P

c , , 2 <--j ~< N

2Co

-I IN+,

359

[]

•

For ~ ~ 7/N, 0 < P -< 1, j -> 1, p @ EN;JN, consider the function fo;.:j(p) : = au(P), t~ -

)tj(p)

.

Write p = P l +P2 where Pl = 7rjNP, P2 = 7r~\JNP" Recall that if Aj(p) is a simple eigenvalue, then Aj(p) is differentiable at p and its L2-gradient is given by 3Aj op(x) -

j(x, p)2

where ~oj is an L2-normalized eigenfunction corresponding to Aj. Using Sobolev's embedding theorem and the fact that ~0j is L2-normalized it follows that there exists a constant C = C(q, N) independent of p and j such that Lip(BsN--, R, p, ~Ai(p, +P2)) -< C , i.e. Pl--->A j ( P l + P 2 ) is Lipschitz continuous uniformly with respect to p and j. By (3.7), AN;p: defines a Cl-diffeomorphism and therefore the map TI v---->A j ( A ; I p 2 ( . Q )

+

P2)

defined for 7/ in AN;p2(BsN) C ~s, is Lipschitz continuous, uniformly in ,), P2 and j. I.e., there exists G = Ca(q, N, M) > 0 such that

Lip(AN;p2(B6,)--> ~, ~ ~ Aj(A~Vlp2(~) +P2)) < Ca.

(3.13')

We summarize these results in the following way which is convenient for later use.

Lemma 3.3. (1) There exists l = l(q, N, M) such that for any ol E E N with ]a] >-l, for any j E N and each p = p~ + P2 E ="q:JN:M ½1a I -> Lip(ANw2(BJ N) ~ N, r / ~ Aj(A~Ip2(rl) + P2 )).

(3.13)

(2) There exits J0 E N, J0 >- - M + 1, so that for any a E 7/N, j ->Jo and any p = Pl + P2 E ~=" q ;JN;M with estimate (3.13) holds.

I~;~u(P)J- 1 the

Proof. The first assertion follows from the estimate (3.13') by choosing l >-2ca. To show (2), notice that Ifo;~;j(P)[ -< 1 implies that 1-->-IAN(p).a I+Aj(p)

or

Aj(p)--l+lAN(p)'a

I.

Using the asymptotics of Aj(p) together with (3.7) one concludes that there exists J0 E N such that the

360

T. Kappeler, S. Kuksin / Physica D 86 (1995) 3 4 9 - 3 6 2

estimate hi(p)_< 1 + statement (1).

implies that I~1 >-Z with l given as in (1). Therefore we can apply []

laN(p)-~1

Lemma 3.4. For j E ~kJ N with j -> N + 1 and a E ZN\{0} define Vo;~;j( q) : = { P E,~q;JN;M : IAN(p) " ct -- Aj(p)] <-- ] - ~ } • i f L > 3~d, then there exist 1 - 1 and J0 >- M + 1 such that -

-

]Zr( 0JOl~l , U->or V°;~;J(q)) =lim/xr{ p--,0 \lJ>-J°lotl u >-1 or V°;~;J(q)) = 0 "

(3.14)

Proof. To prove (3.14) it suffices, again due to Fubini's theorem, to show that, with

one has lim ~

p--~O J>--Joor

/.t,,~N(Vp;~;j;p2) = 0

(3.15)

I~l->t uniformly in P2. To prove (3.15) we use Lemma 3.4. Recall that fp;a;j(P) can be written as follows: for 71 E au;p2(Bju ) and Pl = a~vlp2(n),

f# ;,~;j(Pl + P2) = ~7" a

2tj(aNlpz('q) + P2) "

-

(3.16)

If j - j 0 or [a]->l, the intersection of AN;e2(Vp;~;j;p2 ) with a line of the form {-% + t a/[a[:tEO~} is contained in an interval of length 4pla[ -L-1. Indeed if no + tl a/]a] and no + t2 a/]a[ are two points in this intersection, t 2 >-tl, then, with

,(

p~(tj) = AN;p2 no +

(j = 1, 2),

one obtains

(t 2 -tl)-(-dT.a a Therefore It2 - t,I-<

- ½1~l(t2

4p/l~l

TM

-

tl ) <--[fp;~;j(Pl(t,)+Pz)--fp;~;j(p,(t2)+P2)l

2pL <- i~1

as claimed and

P ~r(v~;o;j;p:) _< c +i~1, -~c

(3.17)

for some constant C = C(q, N, M). Notice that for a E 7/N\{0} fixed, ~ ( j e ~ ; v~;~;j;p= ~ ~) <- c l ~ l ~'2 .

(3.18)

The estimate (3.18) follows easily from Weyl's estimate of the eigenvalues Aj(p): Aj(p) = Cdj 2/d + o(j2/d) ,

(3.19)

T. Kappeler, S. Kuksin / Physica D 86 (1995) 349-362

361

where d = dim O, Cd is a constant independent of p and the error term is uniform for p G Eq;JN;M' This implies that, for j ---j~ with ja sufficiently large,

1 >-IAN(p)'a -- aj(p)l- I,~j(p)l - fill->

cj 2'" -

fill.

(3.20)

As a consequence of (3.20) we have for j->j~ and some positive constant C = CN;q;j

j <__C[a] a'2 "

(3.21)

Clearly, (3.21) remains valid for j--3d,

E

J-~j0o,

]-£r,JN(Vp;a;j;p2) ~ C E

o,0

10/]d/2

P

P

IL+--------7--i~ c o,0E i~1,+,

I~t_>t

which leads to (3.15).

[]

Proof of Theorem 2. For any q E K(r) (q ENj N of the form Zje J 0j~, J C_N finite, and 0j E Q Vj E J, choose all admissible (JN, M)-cylindrical neighbourhoods --Wq;~N;MC_K(r) 71 ENj ~. Then O=lxr({P@~qju;M:forany P > 0 there exists a vector a @ YN\{0} uc. t.at

<

P

according to Lemma 3.2; O : t,Zr({ p

E ~,--~q;jN;M:for any p > 0 there exists J>-Jo o r e ~ Z N, with lal _>1

suchthatlAN(p).a-Aj(p)[<_[--~})

(3.23)

according to Lemma 3.4, and 0 =/*r({ P E ~q;Ju;M: there exists 1 <--j --
(3.24)

according to Lemma 3.1. The admissible neighbourhoods ,,Wq;~u;Mwith q as specified above define a countable cover of ENj u Cl K(r). Indeed if q E ENj N A K(r), then there exists an admissible neighbourhood =--qJN;M= BjN(R) x Bj(R) x K(r)~\b,Ml with discs of radius R E Q to which Lemma 3.2 and Lemma 3.4 apply. There exists 4 = Zje J Oj~0j with ] finite and 4j E Q such that

4 ~ ~'~'q;JN;M = B,N(R) × B,(R) x K(r)~,I,,MI with I1%(~?- q)ll < g / 4 and [~rj(4 - q)ll < R/4. Then -w#jN;M with discs of radius R/2 is also admissible and clearly q E ~'~:JN;M" This proves Theorem 2. []

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T. Kappeler, S. Kuksin / Physica D 86 (1995) 349-362

Acknowledgement B o t h a u t h o r s wish to t h a n k for t h e k i n d h o s p i t a l i t y at I H E S .

References [1] R. Courant and D. Hilbert, Methods of Modern Mathematical Physics, Vol. 1 (New York, 1962). [2] V.I. Arnold, V.V. Kozlov and A.I. Neistadt, Mathematical Aspects of Classical and Celestial Mechanics, Encycl. of Mathem. Sci., Vol. 3 (Springer, 1989). [3] Y. Colin de Verdi~re, Construction de Laplaciens dont une partie finie du spectre est donn6e, Ann. Sci. ENS. 20 (1987) 599-615. [4] T. Kappeler, Multiplicities of the eigenvalues of the Schr6dinger equation in any dimension, J. Func. Anal. 77 (1988) 346-351. [5] T. Kappeler and B. Ruf, On the nodal line of the second eigenfunction of elliptic operators, J. Reine Angew. Math. 396 (1989) 1-13. [6] S.B. Kuksin, An averaging theorem for distributed conservative systems and its applications to the Von Karman equations, Prikl. Matem. Mekhan. 53:2 (1988) 196-205 [English translation in P.M.M. USSR 53:2 (1989) 150-157]. [7] S.B. Kuksin, Nearly integrable infinite-dimensionalHamiltonian systems, Lecture Notes in Mathematics Vol. 1556 (Springer, 1993). [8] J.P. Ramis, Sous-ensembles analytiques d'une vari6t6 Banachique complexe, Ergeb. Math. Grenzgeb. 53 (Springer, 1970).