Models of Quadratic Algebras Generated by Superintegrable Systems in 2D

TL;DR

This paper constructs operator models for quadratic algebras from 2D superintegrable systems, enabling the determination of energy quantization and eigenvalues through algebraic methods.

Contribution

It provides explicit operator realizations for quadratic algebras of superintegrable systems, linking algebraic models to spectral properties and separation of variables.

Findings

Operator realizations for all Stäckel equivalent systems

Models determine energy quantization and eigenvalues

Applicable to both degenerate and nondegenerate systems

Abstract

In this paper, we consider operator realizations of quadratic algebras generated by second-order superintegrable systems in 2D. At least one such realization is given for each set of St\"ackel equivalent systems for both degenerate and nondegenerate systems. In almost all cases, the models can be used to determine the quantization of energy and eigenvalues for integrals associated with separation of variables in the original system.