Conjecture on the analyticity of PT-symmetric potentials and the reality of their spectra

TL;DR

This paper explores the conjecture that PT-symmetric complex potentials must be analytic to have real spectra, demonstrating this with a specific class of potentials and discussing the conditions for spectral reality.

Contribution

It provides evidence supporting the conjecture that non-analytic PT-symmetric potentials generally do not have real spectra, using a specific potential form as a case study.

Findings

Non-analytic PT-symmetric potentials tend to have complex spectra.

Analyticity appears necessary for PT-symmetric potentials to have real spectra.

The potential V(x)=(ix)^a|x|^b exemplifies the conjecture.

Abstract

The spectrum of the Hermitian Hamiltonian $H=p^2+V(x)$ is real and discrete if the potential $V(x)\to\infty$ as $x\to\pm\infty$. However, if $V(x)$ is complex and PT-symmetric, it is conjectured that, except in rare special cases, $V(x)$ must be analytic in order to have a real spectrum. This conjecture is demonstrated by using the potential $V(x)=(ix)^a|x|^b$, where $a,b$ are real.