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

TL;DR

This paper constructs models for quadratic algebras linked to 2D superintegrable systems, analyzing their representations and connecting classical and quantum models, with applications to systems on the 2-sphere.

Contribution

It provides explicit models for the quadratic algebras of degenerate superintegrable systems on the 2-sphere, extending previous work and linking classical and quantum algebra representations.

Findings

Constructed models for quantum quadratic algebras using differential and difference operators.

Derived classical quadratic algebra models and related them to quantum models.

Connected results to a position dependent mass Hamiltonian studied by Quesne.

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.