A family of fourth-order superintegable systems with rational potentials related to Painlev\'e VI

TL;DR

This paper introduces a family of superintegrable two-dimensional Hamiltonian systems with rational potentials linked to Painlevé VI, expressing wave functions via special polynomials and identifying a fourth-order integral of motion.

Contribution

It presents new rational extensions of the singular oscillator that are superintegrable and connects these systems to Painlevé VI equations, expanding the classification of such systems.

Findings

Systems are superintegrable with a fourth-order integral of motion.

Wave functions involve Laguerre and exceptional Jacobi polynomials.

Potential functions satisfy a nonlinear equation related to Painlevé VI.

Abstract

We discuss a family of Hamiltonians given by particular rational extensions of the singular oscillator in two-dimensions. The wave functions of these Hamiltonians can be expressed in terms of products of Laguerre and exceptional Jacobi polynomials. We show that these systems are superintegrable and admit an integral of motion that is of fourth-order. As such systems have been classified, we see that these potential satisfy a non-linear equation related to Painlev\'e VI. We begin by demonstrating the process with the simpler example of rational extensions of the harmonic oscillator and use the classification of third-order superintegrable systems to connect these families with the known solutions of Painlev\'e IV associated with Hermite polynomials.