Complex WKB Analysis of a PT Symmetric Eigenvalue Problem

TL;DR

This paper uses complex WKB analysis to explain the spectral properties of PT symmetric eigenvalue problems, revealing quantisation conditions that relate to eigenvalue reality, positivity, and degeneracies.

Contribution

It introduces a complex WKB approach to analyze PT symmetric spectra, providing new insights into eigenvalue structure and degeneracies.

Findings

Spectral properties explained via quantisation conditions

Eigenvalues' reality and positivity linked to spectral structure

Pattern of eigenvalue degeneracies further examined

Abstract

The spectra of a particular class of PT symmetric eigenvalue problems has previously been studied, and found to have an extremely rich structure. In this paper we present an explanation for these spectral properties in terms of quantisation conditions obtained from the complex WKB method. In particular, we consider the relation of the quantisation conditions to the reality and positivity properties of the eigenvalues. The methods are also used to examine further the pattern of eigenvalue degeneracies observed by Dorey et al. in [1,2].