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},\phantom{\rule{4pt}{0ex}}\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},\phantom{\rule{4pt}{0ex}}\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.

@article{SEDP_1999-2000____A19_0, author = {Popov, Georgi}, title = {KAM Tori and Quantum Birkhoff Normal Forms}, journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)}, publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {1999-2000}, note = {talk:19}, mrnumber = {1813182}, zbl = {1056.37078}, language = {en}, url = {http://www.numdam.org/item/SEDP_1999-2000____A19_0} }

Popov, Georgi. KAM Tori and Quantum Birkhoff Normal Forms. Séminaire Équations aux dérivées partielles (Polytechnique) (1999-2000), Talk no. 19, 13 p. http://www.numdam.org/item/SEDP_1999-2000____A19_0/

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