Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics

TL;DR

This paper explores spectral equivalences, supersymmetry, and reality properties in PT-symmetric quantum mechanics, revealing new relationships between differential equations and integrable models, and proving a conjecture on spectral reality.

Contribution

It uncovers novel spectral equivalences, generalizes supersymmetry transformations at quasi-exactly solvable points, and proves a conjecture on the reality of spectra in PT-symmetric systems.

Findings

Discovered spectral equivalences with second- and third-order differential equations.

Proved a conjecture on the reality of spectra in PT-symmetric quantum systems.

Developed an efficient numerical method for analyzing these problems.

Abstract

The one-dimensional Schrodinger equation for the potential $x^6+\alpha x^2 +l(l+1)/x^2$ has many interesting properties. For certain values of the parameters l and alpha the equation is in turn supersymmetric (Witten), quasi-exactly solvable (Turbiner), and it also appears in Lipatov's approach to high energy QCD. In this paper we signal some further curious features of these theories, namely novel spectral equivalences with particular second- and third-order differential equations. These relationships are obtained via a recently-observed connection between the theories of ordinary differential equations and integrable models. Generalised supersymmetry transformations acting at the quasi-exactly solvable points are also pointed out, and an efficient numerical procedure for the study of these and related problems is described. Finally we generalise slightly and then prove a conjecture due to Bessis, Zinn-Justin, Bender and Boettcher, concerning the reality of the spectra of certain PT-symmetric quantum-mechanical systems.