Annales Henri Poincaré

, Volume 1, Issue 2, pp 249–279

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

  • G. Popov

DOI: 10.1007/PL00001005

Cite this article as:
Popov, G. Ann. Henri Poincaré (2000) 1: 249. doi:10.1007/PL00001005

Abstract.

The aim of this paper is to obtain quasimodes for a Schrödinger type operator Ph 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 Ph 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 Ph defined by complex scaling which are exponentially close to the real axis. Applications to the discrete spectrum are also obtained.

Copyright information

© Birkhäuser Verlag Basel, 2000

Authors and Affiliations

  • G. Popov
    • 1
  1. 1.Département de Mathématiques, UMR 6629, Université de Nantes, CNRS, B.P. 92208, F-44322 Nantes-Cedex 03, France, e-mail: popov@math.univ-nantes.frFR