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

. The aim of this paper is to obtain quasimodes for a Schrödinger type operator P_h in a semi-classical limit (h \searrow 0) (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 L \Lambda of the KAM tori starting from a suitable Birkhoff normal form of H around L \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.