Comparison of two semiclassical expansions for a family of PT-symmetric oscillators

TL;DR

This paper compares two semiclassical methods for calculating eigenvalues of PT-symmetric oscillators, highlighting differences due to path choices in complex plane integrals and deriving a unified WKB expression.

Contribution

It introduces a general WKB expression that unifies previous semiclassical expansions for PT-symmetric oscillators, clarifying discrepancies caused by path choices.

Findings

Different semiclassical expansions depend on complex path choices.

A unified WKB expression encompasses earlier methods.

Path selection affects eigenvalue calculations in PT-symmetric systems.

Abstract

We show that a recently developed semiclassical expansion for the eigenvalues of PT-symmetric oscillators of the form $V(x)=(ix)^{2N+1}+bix$ does not agree with an earlier WKB expression for $V(x)=-(ix)^{2N+1}$ the case $b=0$. The reason is due to the choice of different paths in the complex plane for the calculation of the WKB integrals. We compare the Stokes and anti-Stokes lines that apply to each case for the quintic oscillator and derive a general WKB expression that contains the two earlier ones.