WKB Analysis of PT-Symmetric Sturm-Liouville problems

TL;DR

This paper uses WKB analysis to study PT-symmetric Sturm-Liouville problems on finite domains, revealing eigenvalue behaviors, critical points, and the accuracy of asymptotic predictions even at low energies.

Contribution

It introduces a WKB-based method to analyze eigenvalues and critical points of PT-symmetric Sturm-Liouville problems on finite domains, highlighting novel spectral phenomena.

Findings

Eigenvalues grow like n^2 for large n

Critical points where eigenvalues become complex are identified

WKB analysis accurately predicts eigenvalues and critical points

Abstract

Most studies of PT-symmetric quantum-mechanical Hamiltonians have considered the Schroedinger eigenvalue problem on an infinite domain. This paper examines the consequences of imposing the boundary conditions on a finite domain. As is the case with regular Hermitian Sturm-Liouville problems, the eigenvalues of the PT-symmetric Sturm-Liouville problem grow like $n^2$ for large $n$. However, the novelty is that a PT eigenvalue problem on a finite domain typically exhibits a sequence of critical points at which pairs of eigenvalues cease to be real and become complex conjugates of one another. For the potentials considered here this sequence of critical points is associated with a turning point on the imaginary axis in the complex plane. WKB analysis is used to calculate the asymptotic behaviors of the real eigenvalues and the locations of the critical points. The method turns out to be surprisingly accurate even at low energies.