Diophantine tori and spectral asymptotics for non selfadjoint operators

Michael Hitrik

Description

Niveau: Supérieur
Diophantine tori and spectral asymptotics for non-selfadjoint operators Michael Hitrik Department of Mathematics University of California Los Angeles CA 90095-1555, USA Johannes Sjostrand Centre de Mathematiques Laurent Schwartz Ecole Polytechnique FR–91128 Palaiseau France and UMR 7640 CNRS San Vu˜ Ngo.c Institut Fourier UMR CNRS-UJF 5582 BP 74, 38402 Saint-Martin d'Heres France Prepublication de l'Institut Fourier no 665 (2005) Abstract We study spectral asymptotics for small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori enjoying a Diophantine property. We get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases: in the first case, we assume that the strength of the perturbation is O(h?) for some ? > 0 and is bounded from below by a fixed positive power of h. In the second case, is assumed to be sufficiently small but independent 1

• spectral instability


• fourier integral operator


• selfadjoint


• recently there has


• original selfadjoint


• been equipped


• there


• weyl quantization

Sujets

• Schwartz
• École polytechnique
• Sjöstrand
• Fourier integral operator
• Self-adjoint
• Weyl quantization

Informations

• Publié par: chaeh
• Nombre de lectures: 7
• Langue: English

Extrait

Diophantine tori and spectral asymptotics non-selfadjoint operators MichaelHitrikJohannesSj¨ostrand DepartmentofMathematicsCentredeMathe´matiques University of California Laurent Schwartz Los Angeles Ecole Polytechnique CA 90095-1555, USA FR–91128 Palaiseau [e-mail] France and UMR 7640 CNRS [e-mail]

SanV˜uNgo . c Institut Fourier UMR CNRS-UJF 5582 BP 74, 38402

Saint-Martind’H`eres France [e-mail]

for

Prepublication de l’Institut Fourier 665 (2005) ´ http://www-fourier.ujf-grenoble.fr/prepublications.html

Abstract

We study spectral asymptotics for small non-selfadjoint perturbations of selfadjointh-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori enjoying a Diophantine property. We get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different

cases: in the first case, we assume that the strengthof the perturbation is O(h δ ) for someδ >0 and is bounded from below by a fixed positive power of h. In the second case,is assumed to be sufficiently small but independent

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ofh, and we describe the eigenvalues completely in a fixedh-independent domain in the complex spectral plane.

Keywords and Phrases:Non-selfadjoint, eigenvalue, spectral asymptotics, La- grangian torus, Diophantine condition, completely integrable, KAM Mathematics Subject Classification 2000 35P20, 37J35, 37J40, 53D22,: 35P15, 58J37, 58J40, 70H08

1 Introduction and

statement of main results

Recently there has been a large number of new developments for non-selfadjoint problems. These include semiclassical spectral asymptotics for non-selfadjoint ope- rators in low dimensions [22], [29], [28], [40], [35], resolvent estimates and pseudospec- tral behavior [15], [13], [6], spectral instability questions [19], [34], and evolution problems and decay to equilibrium for the Fokker-Planck operator [20]. The pur- pose of this work is to continue a line of development initiated in [28], which opened up the possibility of carrying out a spectral analysis for non-selfadjoint operators in dimension two, that is as precise as the corresponding analysis for selfadjoint prob- lems in dimension one. In [28], it was established that for a wide and stable class of non-selfadjoint operators in dimension two, it is possible to describe all eigenvalues individually in a fixed domain in the complex plane, by means of a Bohr-Sommerfeld quantization condition. The underlying reason for this result is a version of the KAM theorem without small divisors, in a complex domain. The work [28] has been continued in a series of papers [41], [23], [24], [25], all of them done in the context of small non-selfadjoint perturbations of selfadjoint opera- tors, with the important additional assumption that the classical flow of the leading symbol of the unperturbed part should be periodic in some energy shell. While the case of a periodic classical flow is very special indeed, in the aforementioned works, we have already given some applications of the general results to spectral asymp- totics for damped wave equations on analytic Zoll surfaces [39], [21], while barrier topresonancesforsemiclassicalSchr¨odingeroperatorshavebeentreatedin[26]. Now a classical Hamiltonian with a periodic flow can be naturally viewed as a degenerate case of a completely integrable symbol, and an even more general and much more interesting dynamical situation occurs when considering a symbol that is merely close to a completely integrable one. Continuing our previous works, in this case it seems to be of interest to study the spectrum of non-selfadjoint operators that are small perturbations of a selfadjoint operator, whose classical flow is close to a completely integrable one. The present work is the first one where we begin

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to study this problem, and when doing so, as our starting point, we shall take a general assumption that the real energy surface of the unperturbed leading symbol contains several flow invariant Lagrangian tori satisfying a Diophantine condition. According to a classical theorem of Kolmogorov [3], the existence of such invariant tori is guaranteed when the unperturbed symbol in question is close to a completely integrable non-degenerate one. We shall begin by describing the general assumptions on our operators, which will be the same as in [23], [24], and [25]. LetMdenoteR 2 or a compact real- f analytic manifold of dimension 2. We shall letMstand for a complexification ofM, f so thatM=C 2 in the Euclidean case, and in the manifold case,Mis a Grauert tube ofM. WhenM=R 2 , let P  =P(x, hD x , ;h) (1.1) be the Weyl quantization onR 2 of a symbolP(x, ξ, ;h) depending smoothly on ∈neigh (0,Rvalues in the space of holomorphic functions of () with x, ξ) in a tubular neighborhood ofR 4 inC 4 , with

|P(x, ξ, ;h)| ≤ O(1)m(Re (x, ξ)) (1.2) there. Heremis assumed to be an order function onR 4 , in the sense thatm >0 and m(X)≤C 0 hX−Yi N 0 m(Y), X, Y∈R 4 , C 0 , N 0 >0.(1.3) We also assume that m≥1.(1.4) We further assume that ∞ P(x, ξ, ;h)∼ X p j, (x, ξ)h j , h→0,(1.5) j=0

in the space of such functions. We make the ellipticity assumption |p 0, (x, ξ)| ≥ C 1 m(Re (x, ξ)),|(x, ξ)| ≥C,(1.6) for someC >0. WhenMis a compact manifold, we letP  be a differential operator onM, such that for every choice of local coordinates, centered at some point ofM, it takes the form P  = X a α, (x;h)(hD x ) α ,(1.7) |α|≤m

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wherea α, (x;h) is a smooth function ofwith values in the space of bounded holomorphic functions in a complex neighborhood ofx= 0. We further assume that ∞ a α, (x;h)∼ X a α,,j (x)h j , h→0,(1.8) j=0

in the space of such functions. The semi-classical principal symbolp 0, , defined on T ∗ M, takes the form p 0, (x, ξ) = X a α,,0 (x)ξ α ,(1.9) if (x, ξ) are canonical coordinates onT ∗ M, and we make the ellipticity assumption

|p 0, (x, ξ)| ≥ 1 C hξi m ,(x, ξ)∈T ∗ M,|ξ| ≥C,

(1.10)

for some largeC > we assume that0. (HereMhas been equipped with some Riemannian metric, so that|ξ|andhξi= (1 +|ξ| 2 ) 1/2 are well-defined.) Sometimes, we writep  forp 0, and simplypforp 0,0 ssum . A e P =0 is formally selfadjoint. (1.11) In the case whenMis compact, we let the underlying Hilbert space beL 2 (M, µ(dx)) for some positive real-analytic densityµ(dx) onM. Under these assumptions,P  will have discrete spectrum in some fixed neighbor- hood of 0∈C, whenh >0, ≥0 are sufficiently small, and the spectrum in this region will be contained in a band|Imz| ≤ O(). Assume for simplicity that (withp=p =0 ) p −1 (0)∩T ∗ M (1.12)is connected,

and let us also assume that the energy levelE= 0 is non-critical, so thatdp6= 0 alongp −1 (0)∩T ∗ M. LetH p =p 0 ξ ∙ ∂x −p 0 x ∙ ∂ξ be the Hamilton field ofp. We introduce the following hypothesis, assumed to hold throughout this work: The setp −1 (0)∩T ∗ Mcontains finitely many analyticH-invariant (1.13) p Lagrangian tori Λ j ,1≤j≤L,such that each Λ j carries real analytic coordinatesx 1 , x 2 identifying Λ j withT 2 so that along Λ j ,we have, H p =a 1 ∂ x 1 +a 2 ∂ x 2 ,(1.14)

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wherea 1 ,a 2 ∈Rsatisfy the Diophantine condition,

∙k| ≥ 1 | C a 0 |k| N 0 ,06=k∈Z 2 ,(1.15) for some fixedC 0 ,N 0 >0. HereT 2 =R 2 /2πZ 2 is the standard 2-torus. We write out the first few terms in a Taylor expansion ofp  in a neighborhood ofp −1 (0)∩T ∗ M, p  =p+iq+O( 2 ).(1.16) When 0≤K∈C 0 ∞ (R) is such that R K(t)dt= 1 andT >0, we introduce a “smoothed out” flow average ofq, hqi T ,K,p =hqi T ,K = Z K T (−t)q◦exp (tH p )dt, K T (t) = T 1 K  T t  ,(1.17)

defined nearp −1 (0)∩T ∗ M. The standard flow average corresponds to takingK= 1 [−1,0] , and we shall then writehqi T ,K =hqi T . LetG T be an analytic function, defined in a neighborhood ofp −1 (0)∩T ∗ M, such that H p G T =q− hqi T ,K . This is a convolution equation along theH p –trajectories, and as in [39] and [24], we solve it by setting G T = Z T J T (−t)q◦exp (tH p )dt, J T (t) = T 1 J  t T  ,(1.18)

where the functionJis compactly supported, smooth away from 0, with J 0 (t) =δ(t)−K(t).

Composing the principal symbol (1.16) with the holomorphic canonical transfor- mation exp (iH G T ), and conjugating the operatorP  by means of the corresponding Fourier integral operatorU  =e h G T (x,hD x ) , defined microlocally nearp −1 (0)∩T ∗ M, we may reduce our operator to a new one, still denoted byP  , which has the principal symbol p  ◦exp (iH G T ) =p+ihqi T ,K +O T ( 2 ). Moreover, it is still true thatP =0 is the original selfadjoint operator. Repeating anargument,explainedforexamplein[39],whichmakesuseofthesharpGa˚rding

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inequality, we obtain a first localization of the spectrum ofP  : ifz∈ spectrum ofP  is such that|Rez| ≤δ, then as, δ, h→0, Im  z ∈ " T li → m ∞ p i − n 1 ( f 0) Rehqi T ,K −o(1), T li → m ∞ p s −1 u ( p 0) Rehqi T ,K +o(1) # .

Cin the

(1.19)

This estimate remains valid forK= 1 [−1,0]  us also notice that along the. Let diophantine torus Λ j , 1≤j≤L, we have uniformly, asT→ ∞, hqi T =F j +O  T 1  .(1.20) HereF j is the mean value ofqover Λ j , computed with respect to the natural smooth flow-invariant measure on Λ j , with respect to which theH p -flow on Λ j is ergodic. b In the case whenK∈C 0 ∞ (R), using the rapid decay ofK, it is easy to see that (1.20) improves to hqi T ,K =F j +O  T 1 ∞  . We shall assume from now on that

F 1 =F 2 =. . .=F L is independent ofj,(1.21) and we write thenF j =F, 1≤j≤L. As will be explained in section 2, for eachj, there exists a smooth canonical transformation

κ ∞,j : neigh(Λ j , T ∗ M)→neigh(ξ= 0, T ∗ T 2 ), mapping Λ j toξ= 0, such that p  ◦κ ∞ −1 ,j =p ∞,j (ξ) +iq j (x, ξ) +O( 2 ) +O(ξ ∞ ),(1.22) andp ∞,j (ξ) =a∙ξ+O(ξ 2 ), withasatisfying (1.15). We furthermore may assume that the energy surfacep ∞ −1 ,j (0) has the formξ 2 =f j (ξ 1 ), for some smooth function f j , withf j (0) = 0,f j 0 (0)6 the coordinates Using= 0.ξ 1 ,ξ 2 , we define hq j i(ξ 1 (2 )= π) 2 Z q j (x, ξ)dx.(1.23) For each smallδ >0, we use the coordinate functionsξ 1 ◦κ ∞,j andξ 2 ◦κ ∞,j near Λ j to decompose the real energy surface as follows, p −1 (0) = Ω − (δ)∪Λ δ ∪Ω + (δ),

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where L Λ δ =p −1 (0)∩ [ (ξ 1 ◦κ ∞,j ) −1 ((−δ, δ)). j=1 Here the sets Ω ± (δdisjoint, compact, with finitely many connected components,) are while in general they are not invariant under theH p –flow. Recall thatFstands for the common value of the average ofqover the tori Λ j , 1≤j≤L. We introduce the following global assumption:

There existN 1 , N 2 ∈N\{0}and a sequenceδ=δ j →0 such that (1.24) Ω i + n ( f δ)  hReqi δ −N 1 ,K −ReF  ≥δ N 2 , Ω su ( p δ)  hReqi δ −N 1 ,K −ReF  ≤ −δ N 2 . − Here 0≤K∈C 0 ∞ is as in (1.17), and we adopt the convention that whenN 2 = 1, ±δ N 2 in the right hand side of (1.24) should be replaced by±δ/C 1 for someC 1 >0. Theorem 1.1Letα 1j andα 2,j be the fundamental cycles inΛ j ,1≤j≤L, defined , by κ ∞,j (α k,j ) ={x∈T 2 ;x k = 0}, k= 1,2. We write thenS j = (S 1,j , S 2,j )andk j = (k(α 1,j ), k(α 2,j ))for the values of the actions and the Maslov indices of the cycles, respectively. Let us make the global dynamical assumption(1.24), and assume that the differentials of the functionsp ∞,j andRehq j i, defined in(1.22)and(1.23)are linearly independent whenξ= 0,1≤ j≤L furthermore that. Assume=O(h δ ),δ >0, satisfies≥h K , for someK fixed but arbitrarily large. LetC >0be sufficiently large. Then the eigenvalues of P  in the rectangle |Rez| C < h δ ,|Imz−ReF| h C < δ (1.25) are given by P j (∞)  h  k− k 4 j  − 2 S π j , ;h  +O(h ∞ ), k∈Z 2 ,1≤j≤L. HereP j (∞) (ξ, ;h)is smooth inξ∈neigh(0,R 2 )and∈neigh(0,R), real-valued for = 0, and has an asymptotic expansion in the space of such functions, ∞ P j (∞) (ξ, ;h)∼ X h l p ( j, ∞ l ) (ξ, ),1≤j≤L. l=0 We have p (∞) (ξ, ) =p(ξ) +ihqi(ξ) +O( 2 ).

j,0 ∞,j j

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Our next result treats the case when the strength of the perturbationis suf- ficiently small but independent ofh. In this case, we obtain a complete spectral result in a fixedh-independent domain.

Theorem 1.2Let us continue to writeS j andk j for the actions and Maslov indices of the fundamental cycles inΛ j ,1≤j≤L that. Assumeh 1/3−δ < ≤ 0 1, for someδ >0. As in Theorem1.1, we make the assumption(1.24)and assume ifferenti fp ∞,j (ξ)andRehq j i(ξ)are linearly independent forξ= 0, that the d als o 1≤j≤L. LetC >0be large enough. the eigenvalues of ThenP  in e e N |Rez| ≤  1 C /N ,  Imz  −ReF  ≤  1 C / (1.26)

are given by z(j, k)∼ n X∞ =0 h n pe ( j, ∞ n )  h  k− k 4 j  − 2 S j , π   , k∈Z 2 ,1≤j≤L,

with pe (∞) e j,n (ξ, ) =O( −2(n−1) + −n/N ), n= 0,1,2, . . . ,1≤j≤L, e holomorphic forξ=O( 1/N ), and pe ( j, ∞) 0 (ξ, ) =p j (ξ) +iq j (ξ, ) +O   N e −1  . Herep j is real on the real domain and the differentials ofp j (ξ)andReq j (ξ, )are linearly independent whenξ== 0,1≤j≤L have. We p j (ξ) +iq j (ξ, ) =a∙ξ+iF+O((ξ, ) 2 ), a=a j . e e The parametersNandN/Ncan be taken arbitrarily large.

Remark. As we shall see in section 2, the assumption (1.24) implies the existence of a suitable weight function, which allows us to microlocalize the spectral problem forP  in a rectangle of the form (1.25) or (1.26) to a small neighborhood of the union of the tori Λ j , 1≤j≤L Indeed, it is the existence—see Proposition 2.3. of the weight function that allows us to carry out the complete spectral analysis in such rectangles. The purpose of the condition (1.24) is to provide an explicit criterion of a purely dynamical nature, which suffices for the construction of the global weight. As will also be seen in section 2, in the case when theH p -flow is

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