Annales Henri Poincaré

, Volume 1, Issue 2, pp 249–279 | Cite as

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

  • G. Popov


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.


Lower Bound Error Term Normal Form Real Axis Small Error 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations