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      Models for quadratic algebras associated with second order superintegrable systems in 2D

      Kalnins, Ernie G.; Miller, W., Jr.; Post, Sarah
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      Kalnins SIGMA.pdf
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      DOI
       10.3842/SIGMA.2008.008
      Link
       www.imath.kiev.ua
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      Kalnins, E. G., Miller, W., Jr. & Post, S. (2008). Models for quadratic algebras associated with second order superintegrable systems in 2D. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 4, 1- 21.
      Permanent Research Commons link: https://hdl.handle.net/10289/1566
      Abstract
      There are 13 equivalence classes of 2D second order quantum and classical superintegrable systems with nontrivial potential, each associated with a quadratic algebra of hidden symmetries. We study the finite and infinite irreducible representations of the quantum quadratic algebras though the construction of models in which the symmetries act on spaces of functions of a single complex variable via either differential operators or difference operators. In another paper we have already carried out parts of this analysis for the generic nondegenerate superintegrable system on the complex 2-sphere. Here we carry it out for a degenerate superintegrable system on the 2-sphere. We point out the connection between our results and a position dependent mass Hamiltonian studied by Quesne. We also show how to derive simple models of the classical quadratic algebras for superintegrable systems and then obtain the quantum models from the classical models, even though the classical and quantum quadratic algebras are distinct.
      Date
      2008-01
      Type
      Journal Article
      Publisher
      SIGMA
      Rights
      This article has been published in the journal: Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). Used with permission.
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      • Computing and Mathematical Sciences Papers [1455]
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