Spectra of PT-Symmetric Operators and Perturbation Theory

TL;DR

This paper develops criteria for the existence of complex eigenvalues in PT-symmetric operators, focusing on spectral properties and perturbation effects in non-self-adjoint operators within Hilbert spaces.

Contribution

It introduces new criteria for eigenvalue existence and non-existence in PT-symmetric operators, advancing understanding of their spectral behavior.

Findings

Criteria for complex eigenvalues in PT-symmetric operators

Conditions for eigenvalue non-existence

Application to Schrödinger operators with PT symmetry

Abstract

Criteria are formulated both for the existence and for the non-existence of complex eigenvalues for a class of non self-adjoint operators in Hilbert space invarariant under a particular discrete symmetry. Applications to the PT-symmetric Schr\"odinger operators are discussed.