, Volume 1, Issue 2, pp 249-279

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

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Abstract.

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.

Submitted 15/09/98, accepted 06/01/99