Perturbations of selfadjoint operators with periodic classical flow

TL;DR

This paper analyzes how small non-selfadjoint perturbations affect the eigenvalues of a self-adjoint operator with periodic classical flow in two dimensions, providing a detailed asymptotic description of the eigenvalues.

Contribution

It offers a complete asymptotic characterization of eigenvalues under specific perturbation strengths when the classical flow is periodic, including cases where perturbation size is independent of Planck's constant.

Findings

Eigenvalues are asymptotically described in certain complex rectangles.

The analysis covers perturbations with strength independent of Planck's constant.

The results apply to a class of non-selfadjoint perturbations with periodic classical flow.

Abstract

We consider non-selfadjoint perturbations of a self-adjoint $h$-pseudodifferential operator in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength $\epsilon $ of the perturbation satisfies $h^{\delta_0} <\epsilon \le \epsilon_0$ for some $\delta_0\in ]0,1/2[$ and a sufficiently small $\epsilon _0>0$. We get a complete asymptotic description of all eigenvalues in certain rectangles $[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$. In particular we are able to treat the case when $\epsilon >0$ is small but independent of $h$.