Deformed oscillator algebra approach of some quantum superintegrable Lissajous systems on the sphere and of their rational extensions

TL;DR

This paper develops a new algebraic framework for 2D superintegrable systems on the sphere, involving rational extensions and exceptional orthogonal polynomials, extending previous Cartesian coordinate results.

Contribution

It introduces a novel algebraic approach using deformed oscillator algebras for superintegrable systems on the sphere with rational extensions, connecting to exceptional orthogonal polynomials.

Findings

Derived energy spectra algebraically for Lissajous systems

Connected polynomial algebras with generalized Heisenberg algebras

Extended superintegrability to rationally extended models

Abstract

We extend the construction of 2D superintegrable Hamiltonians with separation of variables in spherical coordinates using combinations of shift, ladder, and supercharge operators to models involving rational extensions of the two-parameter Lissajous systems on the sphere. These new families of superintegrable systems with integrals of arbitrary order are connected with Jacobi exceptional orthogonal polynomials (EOP) of type I (or II) and supersymmetric quantum mechanics (SUSYQM). Moreover, we present an algebraic derivation of the degenerate energy spectrum for the one- and two-parameter Lissajous systems and the rationally extended models. These results are based on finitely generated polynomial algebras, Casimir operators, realizations as deformed oscillator algebras and finite-dimensional unitary representations. Such results have only been established so far for 2D superintegrable systems separable in Cartesian coordinates, which are related to a class of polynomial algebras that display a simpler structure. We also point out how the structure function of these deformed oscillator algebras is directly related with the generalized Heisenberg algebras (GHA) spanned by the nonpolynomial integrals.