Diophantine tori and Weyl laws for non-selfadjoint operators in dimension two

TL;DR

This paper derives a Weyl law for eigenvalue distribution of non-selfadjoint perturbations of integrable selfadjoint operators in two dimensions, linking eigenvalue counts to classical flow averages.

Contribution

It establishes a Weyl law for eigenvalues in non-selfadjoint perturbations, connecting spectral distribution to classical integrable flow and Diophantine tori.

Findings

Weyl law for eigenvalues in spectral bands bounded by Diophantine tori.

Eigenvalue count asymptotics expressed via long-time averages of perturbations.

Results applicable to semiclassical analytic pseudodifferential operators in two dimensions.

Abstract

We study the distribution of eigenvalues for non-selfadjoint perturbations of selfadjoint semiclassical analytic pseudodifferential operators in dimension two, assuming that the classical flow of the unperturbed part is completely integrable. An asymptotic formula of Weyl type for the number of eigenvalues in a spectral band, bounded from above and from below by levels corresponding to Diophantine invariant Lagrangian tori, is established. The Weyl law is given in terms of the long time averages of the leading non-selfadjoint perturbation along the classical flow of the unperturbed part.