Publication details for Patrick DoreyDorey, Patrick, Dunning, Clare & Tateo, Roberto (2001). Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics. Journal of Physics A: Mathematical and General 34(28): 5679-5704.
- Publication type: Journal Article
- ISSN/ISBN: 0305-4470, 1361-6447
- DOI: 10.1088/0305-4470/34/28/305
- Further publication details on publisher web site
- Durham Research Online (DRO) - may include full text
Author(s) from Durham
The one-dimensional Schrödinger equation for the potential x6 + αx2 + l(l + 1)/x2 has many interesting properties. For certain values of the parameters l and α the equation is in turn supersymmetric (Witten) and 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. Generalized 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 generalize slightly and then prove a conjecture due to Bessis, Zinn-Justin, Bender and Boettcher, concerning the reality of the spectra of certain -symmetric quantum mechanical systems.