Annales Henri Poincaré

, Volume 1, Issue 2, pp 249-279

First online:

Invariant Tori, Effective Stability, and Quasimodes with Exponentially Small Error Terms II –¶Birkhoff Normal Forms

  • G. PopovAffiliated withDépartement de Mathématiques, UMR 6629, Université de Nantes, CNRS, B.P. 92208, F-44322 Nantes-Cedex 03, France, e-mail:

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The aim of this paper is to obtain quasimodes for a Schrödinger type operator P h in a semi-classical limit \( (h \searrow 0) \) with exponentially small error terms which are associated with Gevrey families of KAM tori of its principal symbol H. To do this we construct a Gevrey quantum Birkhoff normal form of P h around the union \( \Lambda \) of the KAM tori starting from a suitable Birkhoff normal form of H around \( \Lambda \). As an application we prove sharp lower bounds for the number of resonances of P h defined by complex scaling which are exponentially close to the real axis. Applications to the discrete spectrum are also obtained.