New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies

TL;DR

This paper introduces four new families of two-dimensional quantum superintegrable systems derived from k-step extensions of harmonic and radial oscillators, utilizing exceptional orthogonal polynomials and polynomial algebras to analyze their spectra and degeneracies.

Contribution

It presents novel superintegrable systems based on k-step rational extensions and develops algebraic methods for spectrum and degeneracy analysis using polynomial algebras.

Findings

Construction of four new superintegrable systems from k-step extensions.

Use of exceptional orthogonal polynomials in wavefunctions.

Algebraic derivation of spectra and degeneracies using polynomial algebras.

Abstract

Four new families of two-dimensional quantum superintegrable systems are constructed from k-step extension of the harmonic oscillator and the radial oscillator. Their wavefunctions are related with Hermite and Laguerre exceptional orthogonal polynomials (EOP) of type III. We show that ladder operators obtained from alternative construction based on combinations of supercharges in the Krein-Adler and Darboux Crum ( or state deleting and creating ) approaches can be used to generate a set of integrals of motion and a corresponding polynomial algebra that provides an algebraic derivation of the full spectrum and total number of degeneracies. Such derivation is based on finite dimensional unitary representations (unirreps) and doesn't work for integrals build from standard ladder operators in supersymmetric quantum mechanics (SUSYQM) as they contain singlets isolated from excited states. In this paper, we also rely on a novel approach to obtain the finite dimensional unirreps based on the action of the integrals of motion on the wavefunctions given in terms of these EOP. We compare the results with those obtained from the Daskaloyannis approach and the realizations in terms of deformed oscillator algebras for one of the new families in the case of 1-step extension. This communication is a review of recent works.