New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems

TL;DR

This paper introduces new ladder operators for a rational extension of the harmonic oscillator linked to exceptional orthogonal polynomials, enabling algebraic derivation of spectra for certain superintegrable 2D systems.

Contribution

It constructs novel ladder operators for extended harmonic oscillators and uses them to analyze superintegrable systems via polynomial algebras.

Findings

Eigenstates form multiple infinite-dimensional representations

Polynomial algebras allow algebraic spectrum derivation

New higher-order integrals of motion are constructed

Abstract

New ladder operators are constructed for a rational extension of the harmonic oscillator associated with type III Hermite exceptional orthogonal polynomials and characterized by an even integer $m$. The eigenstates of the Hamiltonian separate into $m+1$ infinite-dimensional unitary irreducible representations of the corresponding polynomial Heisenberg algebra. These ladder operators are used to construct a higher-order integral of motion for two superintegrable two-dimensional systems separable in cartesian coordinates. The polynomial algebras of such systems provide for the first time an algebraic derivation of the whole spectrum through their finite-dimensional unitary irreducible representations.