6 pages, 1 figureInternational audienceWe consider tunneling between symmetric wells for a 2-D semi-classical Schrödinger operator for energies close to the quadratic minimum of the potential V in two cases: (1) excitations of the lowest frequency in the harmonic oscillator approximation of V; (2) more general excited states from Diophantine tori with comparable quantum numbers
τ -quasiperiodic : (P(τ ), X (τ )) = (P(ωτ + ϕ 0 )), X (ωτ + ϕ 0 )). Here ω is a vector of periods and (P(τ ), X (τ )) are 2π-periodic functions with respect to each phase.This occurs when H is completely integrable, and Λ are compact, connected leaves of maximal dimension d (i.e. lagrangian) in the fibration of T * M by the momentum map m, that we assume to be non singular at Λ = Λ 0 . Then Arnold-Mineur-Liouville theorem states [Ar2] that in a suitable action-angle coordinate system, Λ are diffeomorphic to the torus T d = R d /2πZ d , and parametrized by the action variables I ∈ R d + . The non-criticality of m implies that all components I j of I are non zero in a neighborhood of Λ 0 .In general the new hamiltonian H has no reason to be integrable. Nevertheless, in a neighborhood of Λ 0 , there are action-angle coordinates (J, ϕ) giving a full family of tori Λ J which are almost invariant for the hamiltonian vector field X H . Under some diophantine condition, it it possible to improve these tori, so that they become invariant within an arbitrary accuracy. This is actually the case, for using a generalization of Birkhoff transformations (better known near a stationary point of X H , ) in the symplectic coordinates given by Darboux-Weinstein theorem, H can be brought to a normal form, we call the Birkhoff normal form (BNF). This was carried out by M.Hitrik, J.Sjöstrand and S.Vu-Ngoc [HiSjVu].An alternative, and more pragmatic way to our opinion, was devised in [BeDoMa], and consists in providing directly suitable action-angle coordinates (J, ϕ), without resorting to Darboux-Weinstein theorem. We believe it has the advantage of being easily implemented numerically in some concrete applications.Once the classical Hamiltonian is taken to BNF, we may address the problem of finding quasi-modes supported on the Λ J , with a suitable accuracy. A necessary condition for Λ J is to satisfy Maslov quantization condition. A similar problem arises in an attempt of quantizing the so-called KAM tori. Quoting J.J. Duistermaat [Du1,p.231], while slightly changing notations : ". . . there is no reason why the tori on which [Maslov quantization condition] holds for some h > 0 should persist. For most [invariant embedded lagrangian manifolds ι : Λ → T * M ] one can expect that h → 1 h · θ [the cohomology class of the closed 1-form ι * (pdx)] is dense in H 1 (Λ; R)/H 1 (Λ; Z). Extracting a subsequence 1 h = τ n k , from τ n = τ 0 + nσ 0 , n ∈ Z [which ensures Maslov quantization condition to hold], such that τ n · θ converges to −α/4 [Maslov class] in H 1 (Λ; R)/H 1 (Λ; Z), one can still obtain some asymptotic estimates for the spectrum of H but in general the convergence of τ n ·θ to −α/4 is expected to be slow and the asymptotic estimates for the spectrum are correspondingly weak."This observation was made more precise by Y.Colin de Verdière [CdV,Prop.7.5] : namely, if H 1 (Λ; R) is of dimension 2 (i.e. for a 2-d torus Λ), then the extraction above can be made, Proof: Consider first y ′ J j , J y ′ J k . Differentiating (1.5) w.r...
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