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

Dorey, Patrick; Dunning, Clare; Tateo, Roberto

Abstract

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.

Citation

Dorey, P., Dunning, C., & Tateo, R. (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. https://doi.org/10.1088/0305-4470/34/28/305

• Journal Article Type: Article
• Publication Date: Jul 1, 2001
• Deposit Date: Feb 26, 2008
• Journal: Journal of Physics A: Mathematical and General
• Print ISSN: 0305-4470
• Electronic ISSN: 1361-6447
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 34
• Issue: 28
• Pages: 5679-5704
• DOI: https://doi.org/10.1088/0305-4470/34/28/305