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

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dc.creatorDorey, Patrick
dc.creatorDunning, Clare
dc.creatorTateo, Roberto
dc.date2001-03-07
dc.date2001-07-11
dc.date.accessioned2026-07-07T10:53:31Z
dc.date.available2026-07-07T10:53:31Z
dc.descriptionThe one-dimensional Schrodinger equation for the potential $x^6+αx^2 +l(l+1)/x^2$ 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.
dc.description32 pages, 12 figures, Latex2e, amssymb, cite and graphicx. v2: References, and a new section with two further dualities, added; typos corrected. v3: Ref [8] corrected. v4: 5 more references,and more typos corrected v5: New title, note added at end and 3 new refs
dc.identifierhttps://arxiv.org/abs/hep-th/0103051
dc.identifierhttp://arxiv.org/abs/hep-th/0103051
dc.identifierJ.Phys.A34:5679-5704,2001
dc.identifierdoi:10.1088/0305-4470/34/28/305
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/185511
dc.subjectHigh Energy Physics - Theory
dc.subjectCondensed Matter
dc.subjectMathematical Physics
dc.subjectExactly Solvable and Integrable Systems
dc.subjectQuantum Physics
dc.titleSpectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
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