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

Ian Marquette

doi:10.1088/1742-6596/597/1/012057

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.

Highlights

Superintegrable systems in classical and quantum mechanics have many properties that make them very interesting models from both mathematics and physics point of view [1]
It was demonstrated in another paper that new type of ladder operators with m + 1 infinite chains of levels can be created in the case of 1-step extension of the harmonic oscillator from which a new set of integrals and a corresponding polynomial algebra provide via finite dimensional unirreps the whole spectrum and total number of degeneracies [17]
We presented four new families of systems involving k-step extension of the harmonic oscillator and radial oscillator related to Hermite and Laguerre exceptional orthogonal polynomials (EOP) of type III

Summary

Introduction

Superintegrable systems in classical and quantum mechanics have many properties that make them very interesting models from both mathematics and physics point of view [1]. It was demonstrated in another paper that new type of ladder operators with m + 1 infinite chains of levels can be created in the case of 1-step extension of the harmonic oscillator from which a new set of integrals and a corresponding polynomial algebra provide via finite dimensional unirreps the whole spectrum and total number of degeneracies [17]. These new ladder operators for these 1D exactly solvable systems related to type III Hermite EOP (Xm1,m2) do not allow to generate integrals from which all the spectrum can be obtained for 2D superintegrable generalisations From these results, we presented a way to construct new ladder from deleting and creating approach of supersymmetric quantum mechanics for k-step extension of the harmonic oscillator and radial oscillator both related to type III EOP ( Hermite and Laguerre ).

Direct approach of constructing unirreps

Conclusion