KAM Tori and Quantum Birkhoff Normal Forms
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1999-2000), Exposé no. 19, 13 p.

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) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:19},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {1999-2000},
mrnumber = {1813182},
zbl = {1056.37078},
language = {en},
url = {http://archive.numdam.org/item/SEDP_1999-2000____A19_0/}
}
Popov, Georgi. KAM Tori and Quantum Birkhoff Normal Forms. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1999-2000), Exposé no. 19, 13 p. http://archive.numdam.org/item/SEDP_1999-2000____A19_0/

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