WKB Analysis of PT-Symmetric Sturm-Liouville problems. II

TL;DR

This paper extends WKB analysis to accurately approximate the spectrum of PT-symmetric Sturm-Liouville problems with potentials like $igx^{2N+1}$, especially where previous methods failed, such as the cubic potential.

Contribution

It introduces an extended WKB method that accounts for paths through pairs of turning points, improving spectral approximations for complex odd-power potentials.

Findings

Accurately approximates spectra of $V=igx^3$ and similar potentials.

Works well for potentials with half-integer powers of $x$ on specific Riemann sheets.

Extends previous one-turning-point analysis to include pair of turning points.

Abstract

In a previous paper it was shown that a one-turning-point WKB approximation gives an accurate picture of the spectrum of certain non-Hermitian PT-symmetric Hamiltonians on a finite interval with Dirichlet boundary conditions. Potentials to which this analysis applies include the linear potential $V=igx$ and the sinusoidal potential $V=ig\sin(\alpha x)$. However, the one-turning-point analysis fails to give the full structure of the spectrum for the cubic potential $V=igx^3$, and in particular it fails to reproduce the critical points at which two real eigenvalues merge and become a complex-conjugate pair. The present paper extends the method to cases where the WKB path goes through a {\it pair} of turning points. The extended method gives an extremely accurate approximation to the spectrum of $V=igx^3$, and more generally it works for potentials of the form $V=igx^{2N+1}$. When applied to potentials with half-integral powers of $x$, the method again works well for one sign of the coupling, namely that for which the turning points lie on the first sheet in the lower-half plane.