Diophantine tori and spectral asymptotics for non selfadjoint operators

Niveau: Supérieur
Diophantine tori and spectral asymptotics for non-selfadjoint operators Michael Hitrik Department of Mathematics University of California Los Angeles CA 90095-1555, USA Johannes Sjostrand Centre de Mathematiques Laurent Schwartz Ecole Polytechnique FR–91128 Palaiseau France and UMR 7640 CNRS San Vu˜ Ngo.c Institut Fourier UMR CNRS-UJF 5582 BP 74, 38402 Saint-Martin d'Heres France Prepublication de l'Institut Fourier no 665 (2005) Abstract We study spectral asymptotics for small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori enjoying a Diophantine property. We get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases: in the first case, we assume that the strength of the perturbation is O(h?) for some ? > 0 and is bounded from below by a fixed positive power of h. In the second case, is assumed to be sufficiently small but independent 1

• spectral instability


• fourier integral operator


• selfadjoint


• recently there has


• original selfadjoint


• been equipped


• there


• weyl quantization