Spectra for semiclassical operators with periodic bicharacteristics in dimension two
aa r X i v : . [ m a t h . SP ] J a n Spectra for semiclassical operators with periodicbicharacteristics in dimension two
Michael A. HallDepartment of MathematicsUniversity of CaliforniaLos AngelesCA 90095-1555, USA [email protected]
Michael HitrikDepartment of MathematicsUniversity of CaliforniaLos AngelesCA 90095-1555, USA [email protected]
Johannes Sj¨ostrandIMB, Universit´e de Bourgogne9, Av. A. Savary, BP 47870FR–21078 Dijon, Franceand UMR 5584 CNRS [email protected]
Abstract
We study the distribution of eigenvalues for selfadjoint h –pseudodifferentialoperators in dimension two, arising as perturbations of selfadjoint operatorswith a periodic classical flow. When the strength ε of the perturbation is ≪ h , the spectrum displays a cluster structure, and assuming that ε ≫ h (or sometimes ≫ h N , for N > Contents Clustering of eigenvalues and averaging reduction 93 Normal form near a Lagrangian torus 14 x dependence . . . . . . . . . . . . . . . . . . . . . . . 153.2 Removing the x dependence . . . . . . . . . . . . . . . . . . . . . . . 17 The spectral theory of selfadjoint partial differential operators, whose associatedclassical flow is periodic, has a long and distinguished tradition, starting with theclassical works of J. J. Duistermaat-V. Guillemin [9] and A. Weinstein [27], in thehigh energy limit, in the case of compact manifolds. Subsequently, many importantcontributions to the theory were given, [5], [12], [1], [28], and the case of semiclassicalpseudodifferential operators was treated in [15], [8], [21]. In particular, assumingthat the Hamilton flow is periodic in some energy shell, the cluster structure of thespectrum has been established in [15]. That work also contains some precise resultsconcerning the semiclassical asymptotics for the counting function of eigenvaluesin the clusters, with the celebrated Bohr-Sommerfeld quantization rule obtained asa special case in dimension one, see also [7]. Let us also remark that apart fromtheir intrinsic interest in spectral theory, starting from the work [9], operators withperiodic classical flow have frequently served as a source of examples of situationswhere various spectral estimates become optimal — see [2] for a recent manifestationof this in the context of uniform L p resolvent estimates for the Laplacian on acompact Zoll manifold.The purpose of this paper is to show how the microlocal techniques of [16], [17],[18], developed in the context of analytic non-selfadjoint perturbations of selfadjointoperators with periodic classical flow in dimension two, apply to a class of selfadjoint operators of the form P ε = P ( x, hD x ) + εQ ( x, hD x ). Here P = P ( x, hD x ) and Q = Q ( x, hD x ) are selfadjoint, with P elliptic at infinity and with the classicalflow of P periodic in a band of energies around 0. It is then well-known, and willbe recalled below, that the spectrum of P near 0 exhibits a cluster structure, eachcluster being of size ≤ O ( h ), and with the separation between the adjacent clusters2f size h . The parameter ε ∈ [0 , ε ), ε >
0, measures the strength of the selfadjointperturbation, and in order to have the clustering for the spectrum of P ε , one shouldhave ε ≪ h . The general problem is then to understand the internal structure ofthe spectral clusters of the perturbed operator P ε in some detail, in the semiclassicallimit h →
0. In this work, assuming that ε ≫ h N , where N ≥ P , we obtain semiclassical complete asymptotic expansionsfor the individual eigenvalues of P ε in subclusters, corresponding to regular valuesof the leading symbol of the perturbation Q , averaged along the classical flow. Weremark that contrary to [16], [17], [18], no analyticity assumptions are needed here,and the spectral analysis is carried out within the framework of the standard L –spaces.Let M stand for R or a smooth compact 2–dimensional Riemannian manifold.When M = R , we let P ε = P w ( x, hD x , ε ; h ) , < h ≪ , (1.1)be the h –Weyl quantization on R of a symbol P ( x, ξ, ε ; h ) depending smoothly on ε ∈ neigh(0 , R ) taking values in the symbol class S ( m ). Here m is assumed to bean order function on R in the sense that m > ∃ C ≥ , N > , m ( X ) ≤ C h X − Y i N m ( Y ) , X, Y ∈ R . (1.2)The symbol class S ( m ) is given by S ( m ) = (cid:8) a ∈ C ∞ ( R ); ∀ α ∈ N , ∃ C α > , ∀ X ∈ R , | ∂ αX a ( X ) | ≤ C α m ( X ) (cid:9) . We shall assume throughout that m ≥ . (1.3)Assume furthermore that P ( x, ξ, ε ; h ) ∼ ∞ X j =0 h j p j ( x, ξ, ε ) , h → , (1.4)in the space S ( m ). We make the ellipticity assumption, | p ( x, ξ, ε ) | ≥ C m ( x, ξ ) , | ( x, ξ ) | ≥ C, (1.5)for some C >
0. 3hen M is a compact manifold, we first recall the standard class of semiclassicalsymbols on T ∗ M , S m ( T ∗ M ) = n a ( x, ξ ; h ) ∈ C ∞ ( T ∗ M × (0 , (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ a ( x, ξ ; h ) (cid:12)(cid:12)(cid:12) ≤ C αβ h ξ i m −| β | o . Associated to S m ( T ∗ M ) is the corresponding class of semiclassical pseudodifferentialoperators denoted by L m ( M ). Let P ε be a C ∞ function of ε ∈ neigh(0 , R ) withvalues in L m ( M ), m >
0. Let f M ⊂ M be a coordinate chart identified with a convexbounded domain in R n in such a way that the Riemannian volume element µ ( dx )reduces to the Lebesgue measure in f M . We then have on f M , for every u ∈ C ∞ ( f M ), P ε u ( x ) = 1(2 πh ) Z Z e ih ( x − y ) · ξ p (cid:18) x + y , ξ, ε ; h (cid:19) u ( y ) dy dξ + Ru ( x ) . (1.6)Here p ( x, ξ, ε ; h ) is a smooth function of ε with values in S m loc ( f M × R ), and R is neg-ligible in the sense that its Schwartz kernel R ( x, y ) satisfies ∂ αx ∂ βy R ( x, y ) = O ( h ∞ ),for all α , β . We further assume that the symbol p ( x, ξ, ε ; h ) has an asymptoticexpansion in S m loc ( f M × R ), as h → p ( x, ξ, ε ; h ) ∼ ∞ X j =0 h j p j ( x, ξ, ε ) , p j ∈ S m − j loc . (1.7)The semiclassical principal symbol of P ε in this case is given by p ( x, ξ, ε ), and wemake the ellipticity assumption, | p ( x, ξ, ε ) | ≥ C h ξ i m , ( x, ξ ) ∈ T ∗ M, | ξ | ≥ C, for some C >
0. Here we recall that since M has been equipped with some Rie-mannian metric, | ξ | and h ξ i = (1 + | ξ | ) / are well defined. Let us also recallfrom [24] that while the complete symbol p in (1.7) depends on the choice of localcoordinates, the principal symbol p and the subprincipal symbol p are invariantlydefined, provided that we use local coordinates in (1.6) for which the Riemannianvolume density becomes equal to the Lebesgue measure.In what follows, we shall write p ε for the principal symbol p ( x, ξ, ε ) of P ε , andsimply p for p ( x, ξ, ε = 0). We shall assume that for ε ∈ neigh(0 , R ), P ε is formally selfadjoint . (1.8)In the case when M is compact, we let the underlying Hilbert space be L ( M, µ ( dx )).4or h > H ( m ), the naturallydefined Sobolev space associated with the order function m (so that in the compactcase, H ( m ) is the standard semiclassical Sobolev space H m ( M )), P ε becomes a well-defined selfadjoint operator on L ( M ). Moreover, the assumptions above imply thatthe spectrum of P ε in a fixed neighborhood of 0 is discrete, for h > ε ≥ p − (0) ⊂ T ∗ M is connected and that dp = 0 along p − (0). Let H p = p ′ ξ · ∂∂x − p ′ x · ∂∂ξ be the Hamilton vector field of p . Weintroduce the following basic assumption, assumed to hold throughout this work:for E ∈ neigh(0 , R ),The H p -flow is periodic on p − ( E ) with minimal (1.9)period T ( E ) > E. Let g : neigh(0 , R ) → R be the smooth function defined by g ′ ( E ) = T ( E )2 π , g (0) = 0 , (1.10)so that g ◦ p has a 2 π -periodic Hamilton flow. Set f = g − . We then have thefollowing well known result, due to [15], following the earlier works [27], [5]. Theorem 1.1
Assume that the subprincipal symbol of P vanishes. Then the spec-trum of P near is contained in the union of the intervals of the form I k = f ( h ( k − θ )) + [ −O ( h ) , O ( h )] , k ∈ Z , (1.11) pairwise disjoint for h > small enough. Here θ = α / S / πh , where α ∈ Z and S ∈ R are the Maslov index and the classical action, respectively, computedalong a closed H p -trajectory ⊂ p − (0) , of period T (0) .Remark . We refer to [22] for a self-contained discussion of Maslov indices of loopsof Lagrangian subspaces and closed Hamiltonian trajectories. Remark . Let us observe that up to a constant, the function g in (1.10) is equal to1 / π times the classical action along a closed H p –orbit in p − ( E ), E ∈ neigh(0 , R ),see [7].Let us write p ε = p + εq + O ( ε m ) , (1.12)5 ( h ) ∼ h Figure 1: Spectral clusters for the unperturbed operator P . The size of each clusteris O ( h ), with the separation between adjacent clusters being of order h .in the case M = R , and p ε = p + εq + O ( ε h ξ i m ) in the compact case. Here q issmooth and real-valued on T ∗ M . Let h q i = 1 T ( E ) Z T ( E ) / − T ( E ) / q ◦ exp ( tH p ) dt on p − ( E ) , (1.13)and notice that H p h q i = 0. In view of (1.9), the space of closed H p –orbits in p − (0),Σ = p − (0) / exp ( R H p ) , is a 2-dimensional symplectic manifold, and h q i can naturally be viewed as a functionon Σ.The following is the main result of this work. Theorem 1.2
Let us assume that (1.9) holds and that the subprincipal symbol of P vanishes. Let F be a regular value of h q i , considered as a function on Σ . Assumethat h q i − ( F ) ⊂ Σ is a connected closed curve, and let us introduce the correspondingLagrangian torus, Λ ,F = { ρ ∈ T ∗ M ; p ( ρ ) = 0 , h q i ( ρ ) = F } . When γ and γ are the fundamental cycles in Λ ,F with γ being given by a closed H p –trajectory of minimal period, we write S = ( S , S ) and α = ( α , α ) for theactions and the Maslov indices of the cycles, respectively. Assume next that thespectrum of P near clusters into bands of size O (1) h N , for some N ≥ . Let usassume that h N ≪ ε ≪ h. Let
C > be large enough. There exists a smooth function f ( ξ ; h ) = f ( ξ ) + N − X j =2 h j f j ( ξ ) , ξ ∈ neigh(0 , R ) , (1.14)6 uch that for each k ∈ Z , with h ( k − α / − S / π small enough, the eigenvaluesof P ε in the set (cid:12)(cid:12)(cid:12)(cid:12) z − f (cid:18) h ( k − α − S π ; h (cid:19) − εF (cid:12)(cid:12)(cid:12)(cid:12) < εC (1.15) are given by b P (cid:18) h ( k − α − S π , h ( ℓ − α − S π , ε, h N ε ; h (cid:19) + O ( h ∞ ) , ℓ ∈ Z . (1.16) Here b P ( ξ, ε, h N /ε ; h ) is smooth in ξ ∈ neigh(0 , R ) , smooth in ε, h N ε ∈ neigh(0 , R ) ,and has a complete asymptotic expansion in the space of such functions, as h → , b P (cid:18) ξ, ε, h N ε ; h (cid:19) ∼ f ( ξ ; h ) + ε (cid:18) r (cid:18) ξ, ε, h N ε (cid:19) + hr (cid:18) ξ, ε, h N ε (cid:19) + . . . (cid:19) . We have r ( ξ ) = h q i ( ξ ) + O (cid:18) ε + h N ε (cid:19) , r j = O (1) , j ≥ , corresponding to the action-angle coordinates near the Lagrangian torus Λ ,F . O ( ε ) f ( h ( k − θ )) f ( h ( k − θ )) + εF ∼ εh ( )Figure 2: Spectral asymptotics in a subcluster of the k th spectral cluster of theoperator P ε , corresponding to the regular value F of the leading symbol of theperturbation, averaged along the classical flow. Here h ≪ ε ≪ h . The red crossesrepresent the eigenvalues of P ε in (1.15), given by (1.16). Remark . In the case when the compact manifold Λ ,F has several connected com-ponents, the result of Theorem 1.2 may be extended by showing that the set ofeigenvalues z in (1.15) agrees with the union of the spectral contributions comingfrom each of the connected components, modulo O ( h ∞ ), each contribution havingthe description as in the theorem. See also the discussion in Section 3 below. Example . Let M be a compact symmetric surface of rank one, and let P = − h ∆ − M . The assumption (1.9) then holds and from [12] we know that the spectrum7f P clusters into bands of diameter 0 and separation of order h . Furthermore,the eigenvalues of P + 1 depend quadratically on h , and we may conclude that thefunctions f j in (1.14) vanish, for j >
2, while f is a constant. Taking 2 < N ∈ N tobe any fixed integer, we see that the result of Theorem 1.2 applies to the Schr¨odingeroperator P ε = P + εq = h ( − ∆ + q ) −
1, where ε = h and q ∈ C ∞ ( M ) is real-valued, cf. with [27], [6], [10]. Let now A be a smooth real-valued 1-form on M andconsider the magnetic Schr¨odinger operator on M given by P ε = h d ∗ λA d λA + h q − . (1.17)Here d A u = du + iA ∧ u , and d ∗ A is the Riemannian adjoint of d A . Theorem 1.2applies to the operator P ε in (1.17) when ε = λh , provided that 0 < λ ≪ q ( x, ξ ) = q A ( x, ξ ) = 2 h ξ, A ♯ i , where A ♯ isthe vector field associated to A by means of the Riemannian metric. We may alsoremark if B = dA is the magnetic field and dA = d e A , then, since H ( M ) = 0, wehave e A = A + dϕ , where ϕ ∈ C ∞ ( M ) is real-valued. It follows that q e A − q A = H p ϕ ,where p = ξ is the leading symbol of P , and therefore the flow average h q e A i = h q A i depends on the magnetic 2-form B = dA only. See also Section 6 below. Remark . It seems quite likely that the result of Theorem 1.2 can be extended tothe case when F is a non-degenerate critical value of h q i , cf. with [6], [10]. Wewould also like to mention that the result of Theorem 1.2 can be viewed as a Bohr-Sommerfeld quantization condition in the spectral clusters, corresponding to theregular values for a reduced one-dimensional operator, and here there are somedirect links with [4], [3], and the theory of Toeplitz operators on reduced compactsymplectic spaces such as Σ. See also [14].The plan of the paper is as follows. In Section 2, after re-deriving the clustering of thespectrum of P , we carry out an averaging reduction of P ε , microlocally in an energyshell. In Section 3, we microlocalize further to a suitable Lagrangian torus andconstruct a quantum Birkhoff normal form for P ε near the torus, very much followingthe approach of [16]. In Section 4 we solve a suitable global Grushin problem for P ε and identify the spectrum in the subclusters precisely, thereby completing themain part of the proof of Theorem 1.2. In Section 5, we complete the discussionby addressing the case when the spectral clusters of P are of size O ( h N ), N > ε in Theorem 1.2 in this case. In Section6, we finally give an application to the magnetic Schr¨odinger operator on R in theresonant case. Acknowledgements . The first author was supported by the UCLA DissertationYear Fellowship. The third author was supported by the grant NOSEVOL ANR2011 BS 01019 01. 8
Clustering of eigenvalues and averaging reduc-tion
For future reference, it will be convenient and natural for us to start by recalling aproof of Theorem 1.1 — see also Proposition 2.1 of [17]. When z ∈ neigh(0 , R ), letus consider the equation ( P − z ) u = v, u ∈ H ( m ) . (2.1)Let χ ∈ C ∞ ( T ∗ M ; [0 , χ = 1 near p − (0). Semiclassical ellipticregularity gives, with the L norms throughout, that || (1 − χ ) u || ≤ O (1) || v || + O ( h ∞ ) || u || , (2.2)where χ = Op wh ( χ ) is the corresponding quantization. Here and in what follows,when M = R , we use the h –Weyl quantization, while when M is compact, we fixthe choice of the quantization map Op wh : S m ( T ∗ M ) → L m ( M ), given by the Weylquantization in special local coordinates as recalled in the introduction, with theassociated symbol map: L m ( M ) → S m ( T ∗ M ) /h S m − ( T ∗ M ).Turning the attention to a neighborhood of p − (0), let γ ⊂ p − (0) be a closed H p -orbit, where we know that T (0) is the minimal period of γ . From Section 3 of [16],we recall the following result. Proposition 2.1
There exists a smooth real-valued canonical transformation κ : neigh( γ, T ∗ M ) → neigh( τ = x = ξ = 0 , T ∗ ( S t × R x )) , (2.3) mapping γ onto { τ = x = ξ = 0 } , such that p ◦ κ − = f ( τ ) . Here f has been definedin (1.10) . Following [16], we recall that the canonical transformation κ can be implementedby a multi-valued microlocally unitary h -Fourier integral operator U = O (1) : L ( M ) → L f ( S × R ), so that the improved Egorov property holds — see thediscussion in Section 2 of [16]. Here L f ( S × R ) is the space of functions definedmicrolocally near τ = x = ξ = 0 in T ∗ ( S × R ), which satisfy the Floquet-Blochperiodicity condition, u ( t − π, x ) = e πiθ u ( t, x ) , θ = S πh + α . (2.4)9s explained in [16], the multi-valuedness of U is a reflection of the fact that thedomain of definition of the canonical transformation κ is not simply connected, thecorresponding first homotopy group being generated by the closed trajectory γ .It follows that there exists a selfadjoint operator e P with the leading symbol f ( τ )near { τ = x = ξ = 0 } and with vanishing subprincipal symbol, so that e P U = U P microlocally near γ , so that( e P U − U P )Op wh ( χ ) = O ( h ∞ ) : L ( M ) → L ( M ) , (2.5)and χ w ( x, hD x )( e P U − U P ) = O ( h ∞ ) : L f ( S × R ) → L f ( S × R ) , for every χ ∈ C ∞ (neigh( γ, T ∗ M )) and for every χ ∈ C ∞ ( T ∗ ( S × R )) supportednear τ = x = ξ = 0. The operator e P acts on the space of functions satisfyingthe Floquet-Bloch condition (2.4), defined microlocally near τ = x = ξ = 0 in T ∗ ( S × R ).Let us remark next that an orthonormal basis for the space L f ( S ) of functions u ∈ L ( R ) satisfying a Floquet-Bloch periodicity condition analogous to (2.4),with the x -variable suppressed, consists of the functions e k ( t ) = 1 √ π exp ( i ( k − θ ) t ) , θ = S πh + α , k ∈ Z , (2.6)which satisfy f ( hD t ) e k ( t ) = f ( h ( k − θ )) e k ( t ). It follows that if z ∈ neigh(0 , R ) issuch that | z − f ( h ( k − θ )) | ≥ Ch , k ∈ Z , for C > e P − z = f ( hD t ) + h R − z, R = O (1) : L f ( S × R ) → L f ( S × R ) , is invertible, microlocally near τ = x = ξ = 0, with the norm of the inverse being O ( h − ).Let us now take finitely many closed H p –trajectories γ , . . . , γ N ⊂ p − (0) and smallopen neighborhoods Ω j of γ j , with Ω j invariant under the H p -flow, 1 ≤ j ≤ N ,such that p − (0) ⊆ ∪ Ω j . Associated to this open cover, we take cutoff functions0 ≤ χ j ∈ C ∞ (Ω j ) such that H p χ j = 0 and P χ j = 1 near p − (0). Let U j denotea multi-valued microlocally unitary h -Fourier integral operator associated to thecanonical transformation near γ j , as in Proposition 2.1.10or each j , 1 ≤ j ≤ N , using (2.1), we see that( P − z ) χ j u + [ χ j , P ] u = χ j v, (2.7)and therefore, U j ( P − z ) χ j u = ( e P − z ) U j χ j u + O ( h ∞ ) u = U j χ j v + U j [ P , χ j ] u. (2.8)When z ∈ neigh(0 , R ) avoids the intervals I k in (1.11), we just saw that the operator e P − z = f ( hD t ) + h R j − z, possesses a microlocal inverse of norm O (1 /h ). Using (2.8), we conclude that || χ j u || ≤ O (cid:18) h (cid:19) || v || + O (cid:18) h (cid:19) || [ P, χ j ] u || + O ( h ∞ ) || u || . (2.9)Since the subprincipal symbols of P and χ j both vanish, we have in the operatorsense, [ P , χ j ] = O ( h ) — see also [11] for composition rules for the subprincipalsymbols. Using (2.9) and summing over j we get || N X j =1 χ j u || ≤ O (cid:18) h (cid:19) || v || + O ( h ) || u || . (2.10)Combining (2.2) and (2.10), we obtain || u || ≤ O (cid:18) h (cid:19) || v || + O ( h ) || u || . (2.11)Taking h small enough, we conclude that ( P − z ) − exists and satisfies ( P − z ) − = O (1 /h ) : L → L , when z ∈ neigh(0 , R ) avoids the intervals I k , k ∈ Z .Having recalled a proof of Theorem 1.1, let us now proceed to carry out an averagingreduction for the selfadjoint operator P ε , replacing the leading symbol q of theperturbation by its average along closed orbits of the H p –flow. We notice that sucha reduction has a very long tradition [27], [13], and the following discussion will betherefore somewhat brief.Let G ∈ C ∞ in a neighborhood of p − (0) be real-valued and such that H p G = q − h q i , (2.12)11here h q i is the flow average, defined in (1.13). As recalled in [16], we may take G = 1 T ( E ) Z T ( E ) / − T ( E ) / (cid:20) R − ( t ) (cid:18) t + 12 T ( E ) (cid:19) + 1 R + ( t ) (cid:18) t − T ( E ) (cid:19)(cid:21) q ◦ exp ( tH p ) dt, on p − ( E ). By a Taylor expansion, we then get p ε ◦ exp ( εH G ) = p + ε ( q − H p G ) + O ( ε ) = p + ε h q i + O ( ε ) . Similarly, with G , G , . . . denoting a sequence of smooth real-valued functions tobe determined, and G ∼ P ∞ j =0 ε j G j , if we expand p ε ◦ exp ( εH G ) asymptotically, weclaim that we can iteratively solve for G j so that p ε ◦ exp ( εH G ) = p + ε h q i + O ( ε ) , where the O ( ε ) error term is real-valued and Poisson commutes with p , modulo O ( ε ∞ ). Explicitly, if G ≤ N = G + εG + ε G + . . . + ε N G N satisfies p ε ◦ exp ( εH G ≤ N ) = p + ε h q i + ε q + . . . + ε N +1 q N +1 + ε N +2 r N +2 + O ( ε N +3 ) , where q j are real-valued and H p q j = 0, 2 ≤ j ≤ N + 1, then for G ≤ N +1 = G ≤ N + ε N +1 G N +1 , with G N +1 ∈ C ∞ to be determined, we have by a variation on theCampbell-Hausdorff formula [20],exp ( εH G ≤ N +1 ) = exp ( εH G ≤ N + ε N +2 H G N +1 )= exp ( εH G ≤ N )exp ( ε N +2 H G N +1 )(1 + O ( ε N +3 )) , (2.13)where the O ( ε N +3 )–bound is in the C ∞ -sense. This implies that p ε ◦ exp ( εH G ≤ N +1 )= p + ε h q i + ε q + . . . + ε N +1 q N +1 + ( r N +2 − H p G N +1 ) ε N +2 + O ( ε N +3 ) . (2.14)As above, we may find a smooth real-valued solution of H p G N +1 = r N +2 − h r N +2 i ,defined near p − (0).The functions G j , j ≥
0, are defined in a fixed neighborhood of p − (0), and by Borel’slemma we may choose G ( x, ξ, ε ) ∈ C ∞ near p − (0), smooth in ε ∈ neigh(0 , R ), whichis given by G ∼ ∞ X j =0 ε j G j , (2.15)12symptotically in the C ∞ –sense. We have then achieved that p ε ◦ exp ( εH G ) is ininvolution with p modulo O ( ε ∞ ), in a fixed neighborhood of p − (0), as desired.Now an application of Cartan’s formula shows that the canonical transformationexp ( εH G ) is exact in the sense that the 1-form (exp ( εH G )) ∗ λ − λ is exact, where λ is the fundamental 1-form on T ∗ M . By Egorov’s theorem, we may therefore quantizethe real-valued smooth canonical transformation exp ( εH G ) by a (single-valued) h -Fourier integral operator U ε = O (1) : L ( M ) → L ( M ), which is microlocallyunitary near p − (0). Then we have that the selfadjoint operator e P ε := U − ε P ε U ε ,defined microlocally near p − (0), is such that its leading symbol is of the form p + ε h q i + O ( ε ), where the O ( ε ) term is in involution with p , modulo O ( ε ∞ ).Furthermore, by the results of Section 2 of [16], we know that if we choose theprincipal symbol of the microlocally unitary Fourier integral operator U ε to be ofconstant argument, then U ε enjoys the improved Egorov property, so that on thelevel of symbols we have e P ε = P ε ◦ exp ( εH G ) + O ( h ). A natural choice of U ε istherefore given by U ε = e − iεG/h , since then the principal symbol of U ε solves a realtransport equation, using also that the subprincipal symbol of G vanishes.We summarize the discussion above in the following result. Proposition 2.2
There exists G ( x, ξ, ε ) ∈ C ∞ (neigh( p − (0) , T ∗ M )) real-valued,depending smoothly on ε ∈ neigh(0 , R ) , with the asymptotic expansion (2.15) inthe space of real-valued smooth functions in a fixed neighborhood of p − (0) , suchthat the microlocally defined selfadjoint operator e P ε = e iεG/h P ε e − iεG/h depends on ε in a C ∞ -fashion and has the leading symbol of the form p + ε h q i + O ( ε ) ,where the O ( ε ) –term Poisson commutes with p modulo O ( ε ∞ ) . The subprincipalsymbol of e P ε is O ( ε ) . Assume furthermore that ε ≪ h , so that the spectrum of P ε near retains a cluster structure, being confined to the union of intervals I k ( ε ) = f ( h ( k − θ )) + [ −O ( h + ε ) , O ( h + ε )] , k ∈ Z . If z ∈ Spec( P ε ) ∩ neigh(0 , R ) is such that z ∈ I k ( ε ) , for some k , then we have ε min neigh ( p − (0)) h q i − O ( ε + h ) ≤ z − f ( h ( k − θ )) ≤ ε max neigh ( p − (0)) h q i + O ( ε + h ) . Here the last estimate follows by an application of sharp G˚arding’s inequality.13
Normal form near a Lagrangian torus
In Proposition 2.2, we have reduced ourselves to a microlocally defined selfadjointoperator e P ǫ , acting on L ( M ), with the leading symbol of the form p + ε h q i + O ( ε ) , where the O ( ε )–term Poisson commutes with p , modulo O ( ε ∞ ). The subprincipalsymbol of e P ε is O ( ε ). In what follows, when working with the operator e P ε , tosimplify the notation, we shall drop the tilde and write P ε instead.Let F ∈ R be such that min p − (0) h q i < F < max p − (0) h q i and assume that F isa regular value of h q i , viewed as a function on the space of closed orbits Σ. Afterreplacing q by q − F we may assume that F = 0, and let us consider the H p –flowinvariant set Λ , = { ρ ∈ T ∗ M ; p ( ρ ) = 0 , h q i ( ρ ) = 0 } . We know that T (0) is the minimal period of all closed H p –trajectories in Λ , andsince dp, d h q i are linearly independent at each point of Λ , , we see that Λ , is aLagrangian manifold which is a union of finitely many tori. Assume for simplicitythat Λ , is connected so that it is equal to a single Lagrangian torus. Since thefunctions p, h q i are in involution, they form a completely integrable system in aneighborhood of Λ , . We have action-angle coordinates near Λ , [19], given by asmooth real-valued canonical transformation κ : neigh( ξ = 0 , T ∗ T ) → neigh(Λ , , T ∗ M ) , T = R / π Z , (3.1)mapping the zero section in T ∗ T onto Λ , , and such that p ◦ κ = p ( ξ ), h q i ◦ κ = h q i ( ξ ). Here we make the identification T ∗ T ∼ = ( R / π Z ) x × R ξ . Since the classicalflow of p is periodic with minimal period T (0) in Λ , , we may and will choose κ so that in fact p ◦ κ = p ( ξ ), by letting ξ be the normalized action of a closed H p -trajectory of minimal period — see the discussion in Section 4 of [16]. The linearindependence of the differentials of p and h q i implies that p ′ (0) = 0, ∂ ξ h q i (0) = 0.We may also remark that when expressed in terms of the action coordinate ξ , thefunction p becomes p ( ξ ) = f ( ξ ), where the smooth function f has been introducedafter (1.10).Implementing the real canonical transformation κ in (3.1) by means of a multi-valuedmicrolocally unitary h -Fourier integral operator U : L f ( T ) → L ( M ), which alsohas the improved Egorov property [16], we get a new selfadjoint operator U − P ε U ,which for simplicity, will still be denoted by P ε , P ε : L f ( T ) → L f ( T ) . (3.2)14ere the operator P ε is defined microlocally near ξ = 0 in T ∗ T , with the full (Weyl)symbol of the form P ε ∼ ∞ X j =0 h j p j ( x, ξ, ε ) , (3.3)the principal symbol being p ( x, ξ, ε ) = p ( ξ ) + ε h q i ( ξ ) + O ( ε ) (3.4)with the O ( ε ) error term independent of x modulo O ( ε ∞ ). Furthermore, p ( x, ξ, ε ) = O ( ε ) , and all the terms in the expansion (3.3) are smooth and real-valued. The dependenceon ε ∈ [0 , ε ) in (3.3) is still C ∞ . The space L f ( T ) here stands for the subspace of L ( R ) consisting of Floquet periodic functions u ( x ), satisfying u ( x − ν ) = e iν · Θ u ( x ) , ν ∈ (2 π Z ) , Θ = S πh + α . Here S = ( S , S ) with S j being the action of the generator γ j of the first homotopygroup of Λ , , with γ being given by a closed H p –trajectory of minimal period, and α = ( α , α ) is the corresponding Maslov index. x dependence Our first goal is to eliminate the x -dependence in p j , j ≥
1, in (3.3). Let A = A ( x, ξ, ε ) ∈ C ∞ be real-valued, x ∈ T , ξ ∈ neigh(0 , R ), smooth in ε ≥
0, andlet us consider the conjugation of the selfadjoint operator P ε by the unitary h -pseudodifferential operator e iA w . We have e − iA w P ε e iA w = e − i ad A w P ε = ∞ X k =0 ( − i ad A w ) k k ! P ε , (ad A w ) P ε = [ A w , P ε ] . Identifying the symbols with the corresponding h -Weyl quantizations, we obtainthat e − iA w P ε e iA w = P ε + e − iA w [ P ε , e iA w ]= p ( x, ξ, ε ) + h ( p ( x, ξ, ε ) + H p A ( x, ξ, ε )) + O ( h ) .
15e shall now show that A can be chosen real-valued smooth, so that p + H p A becomes independent of x , modulo O ( ε ∞ ). In doing so, we shall construct the C ∞ –symbol A as a formal power series in ε . Introducing the Taylor expansions, p ( x, ξ, ε ) ∼ ∞ X ℓ =0 ε ℓ p ,ℓ ( x, ξ ) , p ( x, ξ, ε ) ∼ ∞ X ℓ =1 ε ℓ p ,ℓ ( x, ξ ) , and writing A ∼ ∞ X ℓ =1 ε ℓ a ℓ ( x, ξ ) , we compute the power series expansion of the Poisson bracket, H p A ∼ X k ≥ ,ℓ ≥ ε k + ℓ { p ,k , a ℓ } = ∞ X m =1 ε m f m , where f m = X k + ℓ = m,k ≥ ,ℓ ≥ { p ,k , a ℓ } . We would like to choose the coefficients a ℓ , ℓ ≥
1, so that p ,ℓ + f ℓ is independentof x , for all ℓ ≥
1. When ℓ = 1, we have p , + f = p , + ∂ ξ p ∂ x a , and since ∂ ξ p (0) = 0, we can determine a real-valued by solving the transport equation, p , + ∂ ξ p ∂ x a = h p , i x , the right hand side standing for the average of p , with respect to x . Arguinginductively, assume that the smooth real-valued functions a , . . . a m have alreadybeen determined. The term p ,m +1 + f m +1 is of the form p ,m +1 + ∂ ξ p ∂ x a m +1 + X k + ℓ = m +1 , ℓ 0) = 0.Continuing to argue in the spirit of [16], [17], we shall now look for an additionalconjugation by means of unitary h –Fourier integral operators, which eliminates the x -dependence in the full symbol in (3.5). Following [16], to that end it will beconvenient to construct the conjugating operator by viewing ε and h /ε as twoindependent small parameters, provided that ε is not too small.On the level of symbols, we write, using that e p ( x , ξ, ε ) = εq ( x , ξ, ε ), where q isreal-valued and C ∞ in all variables, e P ε = p ( ξ ) + ε (cid:18) h q i ( ξ ) + O ( ε ) + hq ( x , ξ, ε ) + h ε e p + h h ε e p + . . . (cid:19) = p ( ξ ) + ε (cid:18) r ( x , ξ, ε, h ε ) + hr ( x , ξ, ε, h ε ) + h r + . . . (cid:19) , (3.6)with r ( x , ξ, ε, h ε ) = h q i ( ξ ) + O ( ε ) + h ε e p , (3.7) r ( x , ξ, ε, h ε ) = q ( x , ξ, ε ) + h ε e p ,r j ( x , ξ, ε, h ε ) = h ε e p j +2 , j ≥ . When eliminating the variable x , let us introduce the basic assumption that ε = O ( h δ ) , δ > , (3.8)and also, assume that h ε ≤ δ , (3.9)for some δ > h . Replacing first (3.9) by thestrengthened hypothesis, h ε ≤ O ( h δ ) , δ > , (3.10)let us describe the construction of a unitary conjugation eliminating the x –depen-dence in e P ε .When b = b ( x , ξ, ε, h ε ) is real-valued and smooth for ξ ∈ neigh(0 , R ), ε , h /ε ∈ [0 , ε ), and is such that b = O ( ε + h /ε ) in the C ∞ –sense, we consider the selfadjointoperator e ih B e P ǫ e − ih B , B = b w ( x , hD x , ε, h /ε ) . (3.11)18ince the commutator [ B , p ( hD x )] = 0, we see that the full symbol of the conju-gated operator (3.11) is real-valued and of the form p ( ξ ) + ε ( b r + h b r + . . . ) , where by Egorov’s theorem, b r = r ◦ exp ( H b ) = ∞ X k =0 k ! H kb r , while b r j = O (1) for j ≥ 1. Since the canonical transformation exp ( H b ) is exact, wesee that the conjugated operator still acts on the space L f ( T ) of Floquet periodicfunctions.It follows from (3.7) that b r = h q i ( ξ ) + O (cid:18) ε + h ε (cid:19) − ∂ ξ h q i ∂ x b + O (cid:18) ε, h ε (cid:19) ! , and using that ∂ ξ h q i 6 = 0, it becomes clear how to construct a real-valued smoothsymbol b = O ( ε + h /ε ), defined near ξ = 0 in T ∗ T , smooth in ε , h /ε ∈ neigh(0 , R ), as a formal Taylor series in ε , h /ε , so that b r = h q i + O ( ε + h /ε )is independent of x , modulo O ( h ∞ ), in view of (3.8), (3.10).Dropping the assumption (3.10), we now come to discuss the construction of aconjugating Fourier integral operator when only (3.9) is valid. Following [16], weconsider the eikonal equation for ϕ = ϕ ( x , ξ, ε, h /ε ), r (cid:18) x , ξ , ξ + ∂ x ϕ, ε, h ε (cid:19) = h r ( · , ξ, ε, h ε ) i , (3.12)where h·i in the right hand side stands for the average with respect to x . Since ∂ ξ h q i 6 = 0, by Hamilton-Jacobi theory, (3.12) has a smooth real-valued solutionwith ∂ x ϕ single-valued and ∂ x ϕ = O (cid:18) ε + h ε (cid:19) . Taylor expanding (3.12) and using that ∂ ξ r (cid:18) x , ξ, ε, h ε (cid:19) = ∂ ξ h q i ( ξ ) + O (cid:18) ε + h ε (cid:19) , 19e get ϕ = ϕ per + x ζ , where ϕ per = O ( ε + h /ε ) is periodic in x and ζ = ζ ( ξ, ε, h /ε ) = O (( ε, h /ε ) ).Let us set η = η (cid:18) ξ, ε, h ε (cid:19) = ( ξ , ξ + ζ ) , and ψ (cid:18) x , η, ε, h ε (cid:19) = ϕ per + x · η, where ϕ per is viewed as a function of η rather than ξ . Associated to the function ψ is the real-valued smooth canonical transformation κ : ( ψ ′ η , η ) → ( x, ψ ′ x ) , (3.13)which is an O ( ε + h /ε )–perturbation of the identity in the C ∞ –sense, and such thatif ( x, ξ ) = κ ( y, η ), then ξ = η . We have by construction,( r ◦ κ ) ( y, η, ε, h ε ) = r ( x, ψ ′ x , ε, h /ε ) = h r ( · , ξ, ε, h ε ) i = h r ( · , η, ε, h ε ) i + O (cid:16)(cid:0) ε, h /ε (cid:1) (cid:17) , which is a function of ( y, η ), independent of y .We can quantize the canonical transformation κ in (3.13) by a microlocally unitaryFourier integral operator, and after conjugation by this operator, we obtain a newoperator, still denoted by e P ε , which is of the form (3.6), where r = h q i ( ξ ) + O (cid:18) ε + h ε (cid:19) is independent of x , and r j = O (1) in the C ∞ –sense, for j ≥ 1. Furthermore, asexplained in Section 4 of [16], the conjugated operator e P ε still acts on the space L f ( T ) of Floquet periodic functions.Let us consider therefore an operator of the form e P ε = p ( ξ ) + ε (cid:18) r ( ξ, ε, h ε ) + hr ( x , ξ, ε, h ε ) + . . . (cid:19) , r = h q i ( ξ ) + O ( ε + h /ε ) is independent of x , and r j = O (1), j ≥ r j are real-valued, smooth, and depend smoothly on ε , h /ε ∈ neigh(0 , R ). To eliminate the x –dependence in the lower order terms r j , j ≥ 1, we could argue as in the previous step, making the terms r j independentof x one at a time, but here we would like to describe a slightly different method,which has the merit of being more direct. Let us look for a conjugation by an ellipticunitary pseudodifferential operator of the form e iB/h , where B ( x , ξ, ε, h ε ; h ) = ∞ X ν =1 h ν b ν ( x , ξ, ε, h ε ) . Here b ν are real-valued smooth and depend smoothly on ε , h /ε ∈ neigh(0 , R ). Theconjugated operator e ih B e P ε e − ih B = e ih ad B e P ǫ = ∞ X k =0 ( i ad B ) k h k k ! e P ǫ is selfadjoint and can be expanded as follows, p ( ξ ) + ε ∞ X k =0 ∞ X j =1 ... ∞ X j k =1 ∞ X ℓ =0 h ℓ + j + .. + j k k ! (cid:18) ih ad b j (cid:19) .. (cid:18) ih ad b j k (cid:19) r ℓ = p ( ξ ) + ε ∞ X n =0 h n b r n . (3.14)Here b r n is equal to the sum of all the coefficients for h n coming from the expressions h ℓ + j + .. + j k k ! (cid:18) ih ad b j (cid:19) .. (cid:18) ih ad b j k (cid:19) r ℓ , (3.15)with ℓ + j + .. + j k ≤ n and j ν ≥ 1. In particular, we see that b r n are all real-valued, thanks to the observation that if A , B are selfadjoint, then so is the operator i [ A, B ] = ( i ad A ) B . Then b r = r , b r = r + H b r = r − H r b ,.., b r n = r n − H r b n + s n ,where s n only depends on b , ..., b n − and is the sum of all coefficients of h n arisingin the expressions (3.15) with ℓ + j + .. + j k ≤ n , j , .., j k , ℓ < n , j ν ≥ b , b , . . . real-valued smooth, successively, with b j = O (1), such that all the coefficients b r j in (3.14) are independent of x and= O (1).The discussion in this section may be summarized in the following theorem.21 heorem 3.1 Let us make all the general assumptions of Section and let F ∈ R be a regular value of h q i , viewed as a function on the space of closed orbits Σ . Assumethat the Lagrangian manifold Λ ,F : p = 0 , h q i = F is connected. When γ and γ are the fundamental cycles in Λ ,F with γ corre-sponding to a closed H p –trajectory of minimal period, we write S = ( S , S ) and α = ( α , α ) for the actions and the Maslov indices of the cycles, respectively. As-sume furthermore that ε = O ( h δ ) , δ > , is such that h /ε ≤ δ , for some δ > sufficiently small but fixed. There exists a smooth Lagrangian torus b Λ ,F ⊂ T ∗ M ,which is an O ( ε + h /ε ) –perturbation of Λ ,F in the C ∞ –sense, such that when ρ ∈ T ∗ M is away from a small neighborhood of b Λ ,F and | p ( ρ ) | ≤ /C , for C > sufficiently large, we have |h q i ( ρ ) − F | ≥ O (1) . There exists a C ∞ real-valued canonical transformation κ : neigh( b Λ ,F , T ∗ M ) → neigh( ξ = 0 , T ∗ T ) , mapping to b Λ ,F to ξ = 0 , and a corresponding uniformly bounded h -Fourier integraloperator U = O (1) : L ( M ) → L f ( T ) , which has the following properties:1. The operator U is microlocally unitary near b Λ ,F : if U ∗ = O (1) : L f ( T ) → L ( M ) is the complex adjoint, then for every χ ∈ C ∞ (neigh( b Λ ,F , T ∗ M )) , wehave ( U ∗ U − 1) Op wh ( χ ) = O ( h ∞ ) : L ( M ) → L ( M ) . (3.16) For every χ ∈ C ∞ (neigh( ξ = 0 , T ∗ T )) , we have ( U U ∗ − χ w ( x, hD x ) = O ( h ∞ ) : L f ( T ) → L f ( T ) . 2. We have a normal form for P ε : Acting on L f ( T ) , there exists a selfadjointoperator b P (cid:16) hD x , ε, h ε ; h (cid:17) with the symbol b P (cid:18) ξ, ε, h ε ; h (cid:19) ∼ p ( ξ ) + ε ∞ X j =0 h j r j (cid:18) ξ, ε, h ε (cid:19) , | ξ | ≤ O (1) , mooth in ξ ∈ neigh(0 , R ) , and smooth in ε , h /ε ∈ neigh(0 , R ) , such that r = h q i ( ξ ) + O (cid:18) ε + h ε (cid:19) , and r j = O (1) , j ≥ , and such that b P U = U P ε microlocally near b Λ ,F , i.e. (cid:16) b P U − U P ε (cid:17) Op wh ( χ ) = O ( h ∞ ) , χ w ( x, hD x ) (cid:16) b P U − U P ǫ (cid:17) = O ( h ∞ ) , in the operator sense, for every χ , χ as in . Throughout this section, we shall assume that ε ≪ h and that the lower bound h /ε ≤ δ ≪ P ε near 0 is confined to the union of intervals, I k ( ε ) = f ( h ( k − θ )) + [ −O ( ε ) , O ( ε )] , k ∈ Z , θ = S πh + α , disjoint for all h > z ∈ neigh(0 , R )is such that | z − f ( h ( k − θ )) − εF | ≤ εC , C ≫ , (4.1)for some k ∈ Z , and z avoids the union of the pairwise disjoint open intervals J ℓ ( h )of length εh/ O (1), that are centered at the quasi–eigenvalues b P (cid:18) h ( k − α − S π , h ( ℓ − α − S π , ε, h ε ; h (cid:19) , (4.2)for ℓ ∈ Z , then the operator P ε − z : H ( m ) → L ( M )is bijective. 23o that end, consider a partition of unity on T ∗ M ,1 = χ + ψ , + + ψ , − + ψ , + + ψ , − . (4.3)Here χ ∈ C ∞ ( T ∗ M ) is supported in a small flow invariant neighborhood of b Λ ,F where the operator U of Theorem 3.1 is defined and unitary, and where P ε is inter-twined with b P , and χ = 1 near b Λ ,F . Thanks to Theorem 3.1, we also assume, aswe may, that on the operator level,[ P ε , χ ] = O ( h ∞ ) : L → L . (4.4)Furthermore, the functions ψ , ± ∈ C ∞ ( T ∗ M ) are supported in flow invariant regionsΩ ± , such that ± ( h q i − F ) ≥ / O (1) in Ω ± , respectively. Moreover, we can arrangeso that ψ , ± are in involution with p , the principal symbol of P ε =0 . Finally, ψ , ± ∈ C ∞ b ( T ∗ M ) are such that ± p > / O (1) in the support of ψ , ± .Let us consider the equation,( P ε − z ) u = v, u ∈ H ( m ) , when z ∈ neigh(0 , R ) satisfies (4.1) for some k ∈ Z . We then claim that, with thenorms taken in L , || (1 − χ ) u || ≤ O (cid:18) ε (cid:19) || v || + O ( h ∞ ) || u || . (4.5)When establishing (4.5), we only have to prove this bound with ψ , ± in place of 1 − χ ,as the estimate involving ψ , ± follows from the semiclassical elliptic regularity.Let γ ⊂ p − (0) be a closed H p -orbit away from b Λ ,F , and assume, to fix the ideas,that h q i ≥ F + 1 /C near γ . Let ψ , e ψ ∈ C ∞ be supported in a small flow-invariantneighborhood of γ and assume that H p ψ = H p e ψ = 0 and that e ψ = 1 near supp ψ .In view of a standard iteration argument [16], it suffices to prove that || ψ u || ≤ O (cid:18) ε (cid:19) || v || + O ( h ) || e ψ u || + O ( h ∞ ) || u || . (4.6)In doing so, we shall use the normal form for P ε near γ , recalled in the proof ofTheorem 1.1 in Section 2. We have( P ε − z ) ψ u = ψ v + [ P ε , ψ ] u. P ε , ψ ] = O ( h + ǫh ) = O ( εh ) as an operator on L , since h ≤ ε and thesubprincipal symbols of P and ψ vanish. Applying the Fourier integral operator U introduced in the proof of Theorem 1.1 and using Egorov’s theorem, we obtain,modulo an error term of norm O ( h ∞ ) || u || , (cid:0) f ( hD t ) + ε h q i ( hD t , x, hD x ) + O ( ε + h ) − z (cid:1) U ψ u = U ( ψ v + [ P ε , ψ ] u ) . (4.7)Let us now check that the operator f ( hD t )+ ε h q i ( hD t , x, hD x ) − z , acting on L f ( S × R ), is invertible, microlocally near τ = x = ξ = 0, with the norm of the inversebeing O (1 /ε ), provided that z ∈ neigh(0 , R ) is such that (4.1) holds. To that end,we consider a direct sum orthogonal decomposition, f ( hD t ) + ε h q i ( hD t , x, hD x ) − z = M k ′ ∈ Z ( f ( h ( k ′ − θ )) + ε h q i ( h ( k ′ − θ ) , x, hD x ) − z ) , (4.8)where it is understood that we only consider the values of k ′ ∈ Z for which h ( k ′ − θ )is small enough. Using that ε ≪ h , we see that for each k ′ = k , with k given in (4.1),the corresponding direct summand in (4.8) is invertible, microlocally near x = ξ = 0with a norm of the inverse being O ( h − ). When verifying the microlocal invertibilityin the case k ′ = k , we write z = f ( h ( k − θ )) + εw , where | w − F | ≤ /C , C ≫ h q i ( τ, x, ξ ) − F ∼ 1, for τ , x , ξ ≈ 0, and the operator f ( h ( k − θ )) + ε h q i ( h ( k − θ ) , x, hD x ) − z = ε ( h q i ( h ( k − θ ) , x, hD x ) − w )is therefore invertible, microlocally near x = ξ = 0, with the O ( ε − ) bound for thenorm of the inverse.From (4.7) we therefore infer that || ψ u || ≤ O (cid:18) ε (cid:19) ( || v || + || [ P ε , ψ ] u || ) + O ( h ∞ ) || u || , (4.9)and using also that || [ P ε , ψ ] u || ≤ O ( εh ) || e ψ u || + O ( h ∞ ) || u || , we obtain the bounds (4.6) and then (4.5).Relying upon (4.5), we shall now complete the proof of the fact that the spectrumof P ε in the region (4.1) is contained in the union of the intervals J ℓ ( h ) centered atthe quasi-eigenvalues (4.2). Let us write( P ε − z ) χ u = χ v + [ P ε , χ ] u, O ( h ∞ ) || u || . Applying the unitary Fourier integral operator U of Theorem 3.1, weget, modulo an error term of norm O ( h ∞ ) || u || , (cid:16) b P − z (cid:17) U χ u = U ( χ v + [ P ε , χ ] u ) . Now an expansion in a Fourier series shows that the operator b P − z is invertible,microlocally near ξ = 0, with a microlocal inverse of the norm O (( εh ) − ), providedthat z in the set (4.1) avoids the intervals J ℓ ( h ). We get || χu || ≤ O (cid:18) εh (cid:19) || v || + O ( h ∞ ) || u || , and combining this estimate together with (4.5) we infer that the operator P ε − z : H ( m ) → L ( M ) is injective, hence bijective, since it is a Fredholm operator of indexzero by general arguments, for h > z in (4.1) varies in an interval J ℓ ( h ) centered around the quasi-eigenvalue in(4.2), contained in the set in (4.1), for some ℓ ∈ Z , we may follow Section 6 of [16]and set up a globally well posed Grushin problem for the operator P ε − z . Since thecorresponding discussion here is even simpler than that of [16], we shall only recallthe main steps. Let us define the rank one operators R + : L ( M ) → C , R − : C → L ( M ) , given by R + u = ( U χu, e kℓ ) , R − u − = u − U ∗ e kℓ . Here e kℓ ( x ) = 12 π e ih ( h ( k − θ ) x + h ( ℓ − θ ) x ) , θ j = α j S j πh , j = 1 , , the scalar product in the definition of R + is taken in the space L f ( T ), and U ∗ isthe complex adjoint of U . The arguments of Section 6 of [16] can now be applied asthey stand to show that for every ( v, v + ) ∈ L ( M ) × C , the Grushin problem( P ε − z ) u + R − u − = v, R + u = v + , has a unique solution ( u, u − ) ∈ H ( m ) × C . We have the corresponding estimate εh || u || + | u − | ≤ O (1) ( || v || + εh | v + | ) , u = Ev + E + v + , u − = E − v + E − + v + , then repeating the arguments of [16], we find that E − + ( z ) = z − b P (cid:18) h ( k − θ ) , h ( ℓ − θ ) , ε, h ε ; h (cid:19) + O ( h ∞ ) . Since the eigenvalues of P ε in the interval J ℓ ( h ) are precisely the values of z forwhich E − + ( z ) vanishes [25], we see that we have established Theorem 1.2, in thegeneral case when the clusters of P are of size O ( h ), and when ε is in the range h ≪ ε ≪ h . Remark . The number of the eigenvalues of P ε in the subcluster (4.1) is ∼ h − , whichis of the same order of magnitude as the total number of eigenvalues of P ε in the k th spectral cluster f ( h ( k − θ )) + [ −O ( ε ) , O ( ε )]. See also Chapter 15 of [7]. In this section, following [18], we shall extend the range of ε in Theorem 1.2, in thecase when the spectrum of P near 0 clusters into bands of size O (1) h N , N > P ε , ε ∈ neigh(0 , R ), be a smooth family of selfadjoint operators, such the as-sumptions of the introduction are satisfied. As we saw in Section 3, microlocally nearthe Lagrangian torus Λ ,F , the operator P can be reduced by successive averagingprocedures to an operator of the form P ∼ ∞ X j =0 h j p j ( x , ξ ) , (5.1)defined near ξ = 0 in T ∗ T , and such that p = p ( ξ ), p = 0. We then have thefollowing result. Proposition 5.1 Assume that the subprincipal symbol of P vanishes and that thespectrum of P clusters into intervals of size ≤ O ( h N ) , for some integer N > .Then the terms p j ( x , ξ ) = p j ( ξ ) in (5.1) are independent of ( x , ξ ) when ≤ j ≤ N − . P near a closed H p –trajectory was considered. This minordifference does not affect the validity of the result, and the proof of Proposition 5.1is essentially the same as that of Proposition 12.1 in [18], making use of a suitablefamily of O ( h / )–Gaussian quasimodes on the one-dimensional torus.An application of the discussion in Section 3 together with Proposition 5.1 allowsus to conclude that when 0 = ε ∈ neigh(0 , R ), microlocally near the torus Λ ,F , theoperator P ε can be reduced to the following form, P ε = ∞ X j =0 h j p j ( x , ξ, ε ) , ( x, ξ ) ∈ T ∗ T , where p ( x , ξ, ε ) = p ( ξ ) + ε h q i + O ( ε )is independent of x , and and p ( x , ξ, ε ) = εq ( x , ξ, ε ) ,p j ( x , ξ, ε ) = p j ( ξ ) + εq j ( x , ξ, ε ) , ≤ j ≤ N − . It follows that we can write, P ε = p ( ξ ; h ) + ε (cid:18) r (cid:18) x , ξ, ε, h N ε (cid:19) + hr (cid:18) x , ξ, ε, h N ε (cid:19) + h r . . . (cid:19) , (5.2)where p ( ξ ; h ) = p ( ξ ) + N − X j =2 h j p j ( ξ ) ,r (cid:18) x , ξ, ε, h N ε (cid:19) = h q i ( ξ ) + O ( ε ) + h N ε p N ( x , ξ, ε ) ,r (cid:18) x , ξ, ε, h N ε (cid:19) = q ( x , ξ, ε ) + h N ε p N +1 ( x , ξ, ε ) , and more generally, r j (cid:18) x , ξ, ε, h N ε (cid:19) = q j ( x , ξ, ε ) + h N ε p N + j ( x , ξ, ε ) , ≤ j ≤ N − ,r j (cid:18) x , ξ, ε, h N ε (cid:19) = h N ε p j + N ( x , ξ, ε ) , j ≥ N . h N ε ≤ δ ≪ , and we see that a natural analog of Theorem 3.1 is valid, with the small parameter h /ε replaced by h N /ε . The arguments of Section 4 can therefore also be applied,with minor modifications, and we obtain the full statement of Theorem 1.2, for ε inthe range h N ≪ ε ≪ h . Let us consider the magnetic Schr¨odinger operator on R , P = X j =1 ( hD x j + A j ( x )) + V ( x ) . (6.1)Here the magnetic and electric potentials A = ( A , A ) and V are assumed to besmooth and real-valued, with ∂ α A , ∂ α V ∈ L ∞ ( R ), for all α ∈ N . It is then wellknown that P is essentially selfadjoint on L ( R ), starting from C ∞ ( R ).Let us assume that V ≥ V ′′ (0) > 0. We furtherassume that lim inf | x |→∞ V ( x ) > . The spectrum of the selfadjoint nonnegative operator P is then discrete in a neigh-borhood of 0.Associated to P in (6.1) is the Weyl symbol given by p ( x, ξ ) = X j =1 ( ξ j + A j ( x )) + V ( x ) , x, ξ ∈ R . (6.2)Assume that near 0, for j = 1 , 2, we have A j ( x ) = O ( x m − ) , (6.3)for some m ≥ 3, and that V ( x ) = 12 V ′′ (0) x · x + O ( x m ) . (6.4)29fter a linear symplectic change of coordinates, we obtain, as ( x, ξ ) → p ( x, ξ ) = p ( x, ξ ) + X j =1 A j,m − ( x ) ξ j + p m ( x ) + O (( x, ξ ) m +1 ) . (6.5)Here p ( x, ξ ) = X j =1 λ j x j + ξ j ) , λ j > ,A j,m − is a homogeneous polynomial of degree m − 1, and p m ( x ) is a homogeneouspolynomial of degree m . In what follows, in order to fix the ideas, we shall considerthe case m = 4. Assume also, for simplicity, that the electrical potential V satisfies V ( − x ) = V ( x ) and the magnetic potential A satisfies A ( − x ) = − A ( x ). We canthen rewrite (6.5) as follows, p ( x, ξ ) = p ( x, ξ ) + X j =1 A j, ( x ) ξ j + p ( x ) + O (( x, ξ ) ) . (6.6)We assume that λ = ( λ , λ ) fulfills the resonant condition, λ · k = 0 , (6.7)for some 0 = k ∈ Z . We shall then be interested in eigenvalues E of P with E ∼ ε , where h δ < ε ≪ 1, 0 < δ < / 2. The general arguments of [23] imply thatthe corresponding eigenfunctions are microlocally concentrated in the region where( x, ξ ) = O ( ε / ), and we introduce therefore the change of variables x = ε / y . Then1 ε P ( x, hD x ) = 1 ε P ( ε / ( y, e hD y )) , e h = hε ≪ . It follows from (6.6) that the symbol of the corresponding e h -pseudodifferential op-erator is 1 ε p ( ε / ( y, η )) = p ( y, η ) + εq ( y, η ) + O ( ε ) , to be considered in the region where | ( y, η ) | = O (1). Here q ( y, η ) = X j =1 A j, ( y ) η j + p ( y ) . (6.8)30he resonant assumption (6.7) implies that the H p –flow is periodic on p − ( E ), for E ∈ neigh(1 , R ), with period T > E , and we shallassume that T is the minimal period for the H p –flow. We may therefore applyTheorem 1.2 to discuss the invertibility of P ( x, hD x ) − ε (1 + z ) = ε (cid:18) ε P ( x, hD x ) − − z (cid:19) , z ∈ neigh(0 , R ) , in the range of energies E = ε (1 + z ), given by h N / ( N +1) ≪ E ≪ h / , for all N = 2 , , . . . . Notice also that since the eigenvalues of p w ( x, e hD x ) dependlinearly on e h , the functions f j , j ≥ 2, occurring in Theorem 1.2, all vanish. Weobtain the following result. Proposition 6.1 Assume that (6.7) holds and that the H p –flow has a minimalperiod T > on p − (1) . Let h q i stand for the average of the homogeneous function q in (6.8) along the trajectories of the Hamilton vector field of p , and assume that h q i is not identically zero. Let F ∈ R be a regular value of h q i restricted to p − (1) .Let ε satisfy h N /N +1 ≪ ε ≪ h / , for some N ≥ fixed. Then for z ∈ neigh(0 , R ) in the set (cid:12)(cid:12)(cid:12)(cid:12) z − f (cid:18)e h ( k − α − S π (cid:19) − εF (cid:12)(cid:12)(cid:12)(cid:12) < ε O (1) , f ( E ) = 2 πT E, e h = hε , the eigenvalues of P of the form ε (1 + z ) are given by z = b P e h ( k − α − S π , e h ( ℓ − α − S π , ε, e h N ε ; e h ! + O ( h ∞ ) , ℓ ∈ Z . Here b P ( ξ, ε, e h N /ε ; e h ) has an expansion, as e h → , b P ξ, ε, e h N ε ; e h ! ∼ f ( ξ ) + ε ∞ X n =0 e h n r n ξ, ε, e h N ε ! , where r ( ξ ) = h q i ( ξ ) + O ε + e h N ε ! , r j = O (1) , j ≥ . he coordinates ξ = ξ ( E ) and ξ = ξ ( E, F ) are the normalized actions of theLagrangian tori Λ E,F : p = E, h q i = F, for E ∈ neigh(1 , R ) , F ∈ neigh( F , R ) , given by ξ j = 12 π Z γ j ( E,F ) η dy − Z γ j (1 ,F ) η dy ! , j = 1 , , with γ j ( E, F ) being fundamental cycles in Λ E,F , such that γ ( E, F ) corresponds toa closed H p –trajectory of minimal period T . Furthermore, S j = Z γ j (1 ,F ) η dy, and α j ∈ Z is fixed, j = 1 , . We shall finish this section by providing an explicit example, illustrating Proposition6.1 in the case when λ = (1 , T = 2 π is the minimal period for the H p –flow, and our task becomes computing the flow average h q i and determining itscritical values, viewed as a function on the compact symplectic manifold Σ. In thiscase, as we saw in [18], the manifold Σ can naturally be identified with the complexprojective line C P ∼ = S .Continuing to follow [18], let us recall first how to compute the trajectory averageof a monomial x α ξ β with | α | + | β | = m , for some m ∈ { , , , ... } . To this end, it isconvenient to introduce z j = x j + iξ j ∈ C , j = 1 , , (6.9)and we then notice that along a H p -trajectory we get in the z , z coordinates: z j ( t ) = e − iλ j t z j (0) . (6.10)Then we write x j ( t ) = Re z j ( t ), ξ j ( t ) = Im z j ( t ), so that x ( t ) α ξ ( t ) β = Y j =1 ((Re z j ( t )) α j (Im z j ( t )) β j )= 12 | α | + | β | i | β | Y j =1 (( z j (0) e − iλ j t + z j (0) e iλ j t ) α j ( z j (0) e − iλ j t − z j (0) e iλ j t ) β j ) . (6.11)32xpanding the product by means of the binomial theorem, we see that the timeaverage is equal to the time-independent term, and since this average is constantalong each trajectory we shall replace the symbols z j (0) simply by z j .For simplicity, we shall assume that p = 0 in (6.8), and then we write A j, ( x ) = X k =0 a j,k x k x − k , j = 1 , . (6.12)The associated magnetic field B ( x ) = ∂A , ∂x − ∂A , ∂x is given by B ( x ) = b x + b x x + b x , (6.13)where b = 3 a , − a , , b = 2( a , − a , ) , b = a , − a , . (6.14)Using (6.11), we get h ξ x i = 0 , h ξ x x i = 116 i ( z z z − z z z ) = − ρ / ρ / sin( θ − θ ) , h ξ x x i = 116 i ( z z − z z ) = − ρ ρ sin 2( θ − θ ) , h ξ x i = 316 i ( z z z − z z z ) = − ρ / ρ / sin( θ − θ ) , h ξ x i = 316 i ( z z z − z z z ) = 32 ρ / ρ / sin( θ − θ ) , h ξ x x i = 116 i ( z z − z z ) = 12 ρ ρ sin 2( θ − θ ) , h ξ x x i = 116 i ( z z z − z z z ) = 12 ρ / ρ / sin( θ − θ ) , h ξ x i = 0 . 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