Semiclassical Approximation for Schrödinger Operators at High Energy

Abstract

Let H _ћ = −ћ ² ∂ ² _φ /2+U(φ) be a Schrödinger operator acting in L ²(T) with T an ℓ-dimensional torus and U an analytic periodic function on T. Approximate semiclassical expansions for the eigenfunctions and eigenvalues of H _ћ are developed which are asymptotic in inverse powers of the classical action variables for the corresponding classical Hamiltonian. The leading term in the eigenvalue expansion is the energy associated with a KAM torus; the fact that KAM tori are abundant at high energy is exploited to show that the rank of the approximate eigenfunctions with energy ≤ E approaches the rank of the true eigenvalues of H _ћ , for E large.