Invariant tori, effective stability, and quasimodes with exponentially small error terms. I: Birkoff normal forms (Q1588817)

A near--integrable Hamiltonian system with a non--degenerate integrable part is considered. A family of invariant tori \(\Lambda_{\omega}\) is found, when the frequencies \(\omega\) belong to a suitable Cantor set \(\Theta\), which is defined by a Diophantine condition. Let \(\Lambda\) be the union of the invariant tori of the system, which can be viewed as a simultaneous Birkhoff normal form of the Hamiltonian around all invariant tori of the family \(\Lambda_{\omega}\). A symplectic Gevrey normal form is obtained in a neighborhood of \(\Lambda\), where the effective stability in the Nekhoroshev sense of the quasiperiodic motions is shown. In the second part of this paper, the relation of this result to the semiclassical asymptotics for Schrödinger operators with exponentially small error terms is investigated.