An infinite family of superintegrable systems from higher order ladder operators and supersymmetry

TL;DR

This paper introduces a new family of quantum superintegrable systems derived from higher order ladder operators and supersymmetry, revealing their algebraic structures and involving Painleve transcendents.

Contribution

It presents a novel method to generate superintegrable Hamiltonians with higher order integrals using ladder operators and supersymmetric quantum mechanics.

Findings

New superintegrable systems involving fifth Painleve transcendent

Construction of fourth order ladder operators from second order supersymmetry

Polynomial algebra structure of the new systems

Abstract

We will discuss how we can obtain new quantum superintegrable Hamiltonians allowing the separation of variables in Cartesian coordinates with higher order integrals of motion from ladder operators. We will discuss also how higher order supersymmetric quantum mechanics can be used to obtain systems with higher order ladder operators and their polynomial Heisenberg algebra. We will present a new family of superintegrable systems involving the fifth Painleve transcendent which possess fourth order ladder operators constructed from second order supersymmetric quantum mechanics. We present the polynomial algebra of this family of superintegrable systems.