Rational invariant tori and band edge spectra for non-selfadjoint operators

TL;DR

This paper develops semiclassical asymptotics for eigenvalues of non-selfadjoint perturbations of integrable selfadjoint operators in two dimensions, focusing on spectral edges associated with rational invariant tori.

Contribution

It provides complete asymptotic expansions for eigenvalues near spectral band edges related to rational flow-invariant tori in a non-selfadjoint setting.

Findings

Asymptotic expansions for eigenvalues near spectral edges

Analysis of spectra associated with rational invariant tori

Extension of semiclassical methods to non-selfadjoint operators

Abstract

We study semiclassical asymptotics for spectra of non-selfadjoint perturbations of selfadjoint analytic $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Complete asymptotic expansions are established for all individual eigenvalues in suitable regions of the complex spectral plane, near the edges of the spectral band, coming from rational flow-invariant Lagrangian tori.