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 close to a completely integrable one and a suitable Cantor set defined by a Diophantine condition, we find a family , of KAM invariant tori of with frequencies which is Gevrey smooth with respect to in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union of the KAM tori which can be viewed as a Birkhoff normal form (BNF) of around . This leads to effective stability of the quasiperiodic motion near . We investigate the semi-classical asymptotics of a Schrödinger type operator with a principal symbol . We obtain semiclassical quasimodes with exponentially small error terms which are associated with the Gevrey family of KAM tori . To do this we construct a quantum Birkhoff normal form (QBNF) of the Schrödinger operator around in suitable Gevrey classes starting from the BNF of . 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}, pages = {1--13}, publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {1999-2000}, zbl = {1056.37078}, mrnumber = {1813182}, language = {en}, url = {http://archive.numdam.org/item/SEDP_1999-2000____A19_0/} }
TY - JOUR AU - Popov, Georgi TI - KAM Tori and Quantum Birkhoff Normal Forms JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:19 PY - 1999-2000 SP - 1 EP - 13 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://archive.numdam.org/item/SEDP_1999-2000____A19_0/ LA - en ID - SEDP_1999-2000____A19_0 ER -
%0 Journal Article %A Popov, Georgi %T KAM Tori and Quantum Birkhoff Normal Forms %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:19 %D 1999-2000 %P 1-13 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://archive.numdam.org/item/SEDP_1999-2000____A19_0/ %G en %F 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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