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 Θ defined by a Diophantine condition, we find a family Λ ω ,ωΘ, of KAM invariant tori of H with frequencies ωΘ which is Gevrey smooth with respect to ω in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union Λ of the KAM tori which can be viewed as a Birkhoff normal form (BNF) of H around Λ. This leads to effective stability of the quasiperiodic motion near Λ. 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 Λ ω ,ωΘ. To do this we construct a quantum Birkhoff normal form (QBNF) of the Schrödinger operator around Λ 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.

Popov, Georgi 1

1 Département de Mathématiques, UMR 6629, Université de Nantes - CNRS, B.P. 92208, 44322 Nantes-Cedex 03, France
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     title = {KAM {Tori} and {Quantum} {Birkhoff} {Normal} {Forms}},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
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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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