Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

TL;DR

This paper develops polynomial algebras for higher order superintegrable systems in Cartesian coordinates, linking them to polynomial Heisenberg algebras, with applications to deformations of oscillators and exceptional orthogonal polynomials.

Contribution

It introduces a general framework for polynomial algebras in superintegrable systems, connecting them to Lie algebra structures and exploring new models involving exceptional orthogonal polynomials.

Findings

Constructed polynomial algebras preserving aspects of rak{gl}(n)

Identified deformations of harmonic oscillators related to exceptional polynomials

Determined degeneracies via finite-dimensional representations

Abstract

We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining higher order ladder operators. One feature of these algebras is that they preserve by construction some aspects of the structure of the $\mathfrak{gl}(n)$ Lie algebra. Among the classes of Hamiltonians arising in this framework are various deformations of harmonic oscillator and singular oscillator related to exceptional orthogonal polynomials and even Painlev\'e and higher order Painlev\'e analogs. As an explicit example, we investigate a new three-dimensional superintegrable system related to Hermite exceptional orthogonal polynomials of type III. Among the main results is the determination of the degeneracies of the model in terms of the finite-dimensional irreducible representations of the polynomial algebra.