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SIGMA 4 (2008), 008, 21 pages arXiv:0801.2848
https://doi.org/10.3842/SIGMA.2008.008
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics
Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D
Ernest G. Kalnins a, Willard Miller Jr. b and Sarah Post b
a) Department of Mathematics, University of Waikato, Hamilton, New Zealand
b) School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA
Received October 25, 2007, in final form January 15, 2008; Published online January 18, 2008
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.
Key words:
superintegrability; quadratic algebras; Wilson polynomials.
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