Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D

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.

This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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• Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere
• Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
• Tools for Verifying Classical and Quantum Superintegrability
• Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials
• On realizations of polynomial algebras with three generators via deformed oscillator algebras
• Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials
• Quadratic algebra for superintegrable monopole system in a Taub-NUT space
• Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
• Bôcher Contractions of Conformally Superintegrable Laplace Equations
• Classical and Quantum Superintegrability of Stäckel Systems
• Superintegrable systems, polynomial algebra structures and exact derivations of spectra
• Dynamical symmetry algebra of two superintegrable two-dimensional systems
• Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras
• Generalized quadratic commutator algebras of PBW-type
• Laplace equations, conformal superintegrability and Bôcher contractions