Spectral equivalences, Bethe ansatz equations, and reality properties in 𝒫𝒯-symmetric quantum mechanics
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Open Access
- 6 July 2001
- journal article
- Published by IOP Publishing in Journal of Physics A: General Physics
- Vol. 34 (28), 5679-5704
- https://doi.org/10.1088/0305-4470/34/28/305
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.Keywords
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