Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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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