A fourth-order superintegrable system with a rational potential related to Painleve VI

TL;DR

This paper explores a superintegrable quantum system with a rational potential linked to Painleve VI, revealing a fourth-order integral of motion, algebraic structure, and special functions involved.

Contribution

It introduces a new superintegrable system with a rational potential connected to Painleve VI and analyzes its algebraic and functional properties.

Findings

Hamiltonian admits a fourth-order integral of motion

Potential related to Painleve VI equation

Integrals form a cubic algebra and relate to deformed oscillators

Abstract

In this paper, we investigate in detail a superintegrable extension of the singular harmonic oscillator whose wave functions can be expressed in terms of exceptional Jacobi polynomials. We show that this Hamiltonian admits a fourth-order integral of motion and use the classification of such systems to show that the potential gives a rational solution associated with the sixth Painlev\'e equation. Additionally, we show that the integrals of the motion close to form a cubic algebra and describe briefly deformed oscillator representations of this algebra.