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      Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

      Kalnins, Ernie G.; Kress, Jonathan M.; Miller, W., Jr.
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      DOI
       10.1063/1.2817821
      Link
       scitation.aip.org
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      Kalnins, E G, Kress, J M & Miller, W. J.(2007). Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties. Journal of Mathematical Physics, 48(11), 1-26.
      Permanent Research Commons link: https://hdl.handle.net/10289/1791
      Abstract
      A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n−1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multiseparability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions, and with quasiexactly solvable systems. Here, we announce a complete classification of nondegenerate (i.e., four-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly ten nondegenerate potentials. ©2007 American Institute of Physics
      Date
      2007
      Type
      Journal Article
      Publisher
      American Institute of Physics
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      This article has been published in Journal of Mathematical Physics. ©2007 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.
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      • Computing and Mathematical Sciences Papers [1455]
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