Eigenvalue Dynamics of a PT-symmetric Sturm-Liouville Operator. Criteria of the Similarity to a Self-adjoint or Normal Operator

TL;DR

This paper studies the eigenvalue behavior of PT-symmetric Sturm-Liouville operators, identifying conditions under which they are similar to self-adjoint operators, with explicit analysis of a complex Airy model.

Contribution

It provides a detailed analysis of eigenvalue dynamics for PT-symmetric Sturm-Liouville operators and explicitly determines critical parameters for real spectra and similarity to self-adjoint operators.

Findings

Eigenvalues move in a predictable manner as parameters vary.

Explicit critical parameter values for real spectra are derived.

The complex Airy operator serves as an exactly solvable model.

Abstract

The goal of the paper is to investigate the dynamics of the eigenvalues of the Sturm-Liouville operator with summable PT-symmetric potential on the finite interval. It turns out that the case of a complex Airy operator presents an exactly solvable model which allows us to trace the dynamics of the movement of the eigenvalues in all details and to find explicitly the critical parameter values, in particular, to specify precisely the number $\varepsilon_1$ such that for $0<\varepsilon<\varepsilon_1$ the operator has a real spectrum and is similar to a self-adjoint operator.