Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics II:Painleve transcendent potentials

TL;DR

This paper explores a superintegrable quantum system involving Painleve transcendent potentials, constructing a cubic algebra of integrals, and analyzing its spectra and supersymmetric properties.

Contribution

It introduces a new superintegrable quantum potential expressed via Painleve transcendent and develops its algebraic and supersymmetric framework.

Findings

Constructed a cubic algebra of integrals of motion.

Derived energy spectra using Fock type representations.

Obtained ground state wave functions through supersymmetric analysis.

Abstract

We consider a superintegrable quantum potential in two-dimensional Euclidean space with a second and a third order integral of motion. The potential is written in terms of the fourth Painleve transcendent. We construct for this system a cubic algebra of integrals of motion. The algebra is realized in terms of parafermionic operators and we present Fock type representations which yield the corresponding energy spectra. We also discuss this potential from the point of view of higher order supersymmetric quantum mechanics and obtain ground state wave functions.