Kalnins, E. G., Miller, W., Jr., & Post, S. (2010). Models for the 3D Singular Isotropic Oscillator Quadratic Algebra. Physics of Atomic Nuclei, 73(2), 359–366. http://doi.org/10.1134/S1063778810020249
Permanent Research Commons link: http://hdl.handle.net/10289/10148
We give the first explicit construction of the quadratic algebra for a 3D quantum superintegrable system with nondegenerate (4-parameter) potential together with realizations of irreducible representations of the quadratic algebra in terms of differential—differential or differential—difference and difference—difference operators in two variables. The example is the singular isotropic oscillator. We point out that the quantum models arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras for superintegrable systems in n dimensions and are closely related to Hecke algebras and multivariable orthogonal polynomials.
This is an author’s accepted version of an article published in the journal: Physics of Atomic Nuclei. © Pleiades Publishing, Ltd., 2010.