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A Recurrence Relation Approach to Higher Order Quantum Superintegrability

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dc.contributor.author Kalnins, E.G.
dc.contributor.author Kress, J.M.
dc.contributor.author Miller Jr., W.
dc.date.accessioned 2019-02-11T15:45:22Z
dc.date.available 2019-02-11T15:45:22Z
dc.date.issued 2011
dc.identifier.citation A Recurrence Relation Approach to Higher Order Quantum Superintegrability / E.G Kalnins, J.M. Kress, W. Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 30 назв. — англ. uk_UA
dc.identifier.issn 1815-0659
dc.identifier.other 2010 Mathematics Subject Classification: 20C99; 20C35; 22E70
dc.identifier.other DOI:10.3842/SIGMA.2011.031
dc.identifier.uri http://dspace.nbuv.gov.ua/handle/123456789/146806
dc.description.abstract We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates. We show that the method provides rigorous proofs of superintegrability and explicit constructions of higher order generators for the symmetry algebra. We apply the method to 5 families of systems, each depending on a parameter k, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system, and we give proofs of quantum superintegrability for all rational values of k, new for 4 of these systems. In addition, we show that the explicit information supplied by the special function recurrence relations allows us to prove, for the first time in 4 cases, that the symmetry algebra generated by our lowest order symmetries closes and to determine the associated structure equations of the algebras for each k. We have no proof that our generating symmetries are of lowest possible order, but we have no counterexamples, and we are confident we can can always find any missing generators from our raising and lowering operator recurrences. We also get for free, one variable models of the action of the symmetry algebra in terms of difference operators. We describe how the Stäckel transform acts and show that it preserves the structure equations. uk_UA
dc.description.sponsorship This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html. uk_UA
dc.language.iso en uk_UA
dc.publisher Інститут математики НАН України uk_UA
dc.relation.ispartof Symmetry, Integrability and Geometry: Methods and Applications
dc.title A Recurrence Relation Approach to Higher Order Quantum Superintegrability uk_UA
dc.type Article uk_UA
dc.status published earlier uk_UA

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