Invariant Tori, Effective Stability, and Quasimodes with Exponentially Small Error Terms II –¶Birkhoff Normal Forms

Abstract

The aim of this paper (part I and II) is to explore the relationship between the effective (Nekhoroshev) stability for near-integrable Hamiltonian systems and the semi-classical asymptotics for Schrodinger operators with exponentially small error terms. Given a real analytic Hamiltonian H close to a completely integrable one and a suitable Cantor set \( \Theta \) defined by a Diophantine condition, we are going to find a family \( \Lambda_{\omega}, \omega \in \Theta \), of KAM invariant tori of H with frequencies \( \omega \in \Theta\) which is Gevrey smooth in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union \( \Lambda \) of the invariant tori which can be viewed as a simultaneous Birkhoff normal form of H around all invariant tori \( \Lambda_{\omega} \). This leads to effective stability of the quasiperiodic motion near \( \Lambda \). As an application we obtain in part II (semiclassical) quasimodes with exponentially small error terms which are associated with a Gevrey family of KAM tori for its principal symbol H. To do this we construct a quantum Birkhoff normal form of the Schrodinger operator around \( \Lambda \) in suitable Gevrey classes starting from the Birkhoff normal form of H.