Bohr-Sommerfeld quantization conditions for non-selfadjoint perturbations of selfadjoint operators in dimension one

TL;DR

This paper describes the spectrum of certain non-selfadjoint operators in one dimension using Bohr-Sommerfeld quantization, extending previous work and applying it to PT-symmetric operators.

Contribution

It provides a spectral description for non-selfadjoint perturbations of selfadjoint operators in one dimension using quantization conditions, building on prior research.

Findings

Spectrum characterized by Bohr-Sommerfeld conditions

Extension of quantization methods to non-selfadjoint operators

Application to PT-symmetric pseudo-differential operators

Abstract

In this paper, we give a description of the spectrum of a class of non-selfadjoint perturbations of selfadjoint operators in dimension one and we show that it is given by Bohr-Sommerfeld quantization conditions. To achieve this, we make use of previous work by Michael Hitrik, Anders Melin and Johannes Sj{\"o}strand. We also give an application of our result in the case of PT-symmetric pseudo-differential operators.