Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion

TL;DR

This paper demonstrates how supersymmetry can be used to generate new superintegrable quantum systems with higher order integrals of motion, expanding the class of known exactly solvable models.

Contribution

It introduces a method using supersymmetry to construct new superintegrable Hamiltonians with higher order integrals of motion, including systems involving Painleve transcendents.

Findings

Generated new superintegrable potentials with higher order integrals

Constructed systems with integrals of order seven and quadratic integrals

Applied Mielnik's construction in supersymmetric quantum mechanics

Abstract

The main result of this article is that we show that from supersymmetry we can generate new superintegrable Hamiltonians. We consider a particular case with a third order integral and apply the Mielnik's construction in supersymmetric quantum mechanics. We obtain a new superintegrable potential separable in Cartesian coordinates with a quadratic and quintic integrals and also one with a quadratic integral and an integral of order seven. We also construct a superintegrable system written in terms of the fourth Painleve transcendent with a quadratic integral and an integral of order seven.