Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2. (English) Zbl 1061.35186

Melin, Anders; Sjöstrand, Johannes

Lebeau, Gilles (ed.), On microlocal analysis. Volume dedicated to Jean-Michel Bony on the occasion of his 60th birthday. Paris: Société Mathématique de France (ISBN 2-85629-132-5/pbk). Astérisque 284, 181-244 (2003).

The authors determine all eigenvalues in an \(h\)-independent domain in the complex plane for a class of non-selfadjoint \(h\)-pseudodifferential operators in 2 dimensions. They show that these eigenvalues are given by a Bohr-Sommerfeld quantization condition. No complete integrability is assumed and as a geometrical step in their proof a KAM-type theorem is obtained.
For the entire collection see [Zbl 1014.00039].

Reviewer: Niels Jacob (Swansea)

MSC:

35S05	Pseudodifferential operators as generalizations of partial differential operators
81S10	Geometry and quantization, symplectic methods
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
35P05	General topics in linear spectral theory for PDEs
37J40	Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
58J40	Pseudodifferential and Fourier integral operators on manifolds

Keywords:

\(h\)-pseudodifferential operators; eigenvalues; KAM-theory