KAM Tori and Quantum Birkhoff Normal Forms

Popov, Georgi¹

¹ Département de Mathématiques, UMR 6629, Université de Nantes - CNRS, B.P. 92208, 44322 Nantes-Cedex 03, France

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1999-2000), Exposé no. 19, 13 p.

Résumé

This talk is concerned with the Kolmogorov-Arnold-Moser (KAM) theorem in Gevrey classes for analytic hamiltonians, the effective stability around the corresponding KAM tori, and the semi-classical asymptotics for Schrödinger 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 find a family ${\Lambda }_{\omega },\omega \in \Theta$, of KAM invariant tori of $H$ with frequencies $\omega \in \Theta$ which is Gevrey smooth with respect to $\omega$ in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union $\Lambda$ of the KAM tori which can be viewed as a Birkhoff normal form (BNF) of $H$ around $\Lambda$. This leads to effective stability of the quasiperiodic motion near $\Lambda$. We investigate the semi-classical asymptotics of a Schrödinger type operator with a principal symbol $H$. We obtain semiclassical quasimodes with exponentially small error terms which are associated with the Gevrey family of KAM tori ${\Lambda }_{\omega },\omega \in \Theta$. To do this we construct a quantum Birkhoff normal form (QBNF) of the Schrödinger operator around $\Lambda$ in suitable Gevrey classes starting from the BNF of $H$. As an application, we obtain a sharp lower bound for the counting function of the resonances which are exponentially close to a suitable compact subinterval of the real axis.