Superintegrable systems, exceptional polynomials and solutions to Painlevé equations

Abstract

Exceptional orthogonal polynomials are recently discovered complete systemsof polynomial solutions to Sturm-Liouville equations with gaps in their degree sequence. These polynomials can be created by factorization of superintegrable Hamiltonian systems. These systems are descriptions of quantum physical scenarios that have hidden symmetries manifesting as additional integrals of motion, ie. differential operators that must commute with the Hamiltonian. From this fact compatibility conditions are obtained that contain nonlinear differential equations. These include the Painlevé and Chazy equations, a classification of second and higher order differential nonlinear equations. The appearance of these equations allows us to build new solutions to them from the potential of the Hamiltonian. In this work, I will be exploring a family of Hamiltonian systems, and by characterizing the integrals of this superintegrable system, show that a new family of solutions to the nonlinear Chazy- I.a differential equation can be built. This Chazy differential equation is equivalent to Painlevé VI, thus giving a novel connection between the two families of special functions: Jacobi polynomials and the Painlevé VI transcendents. In addition, I determined the algebra generated by these integrals and solved for the energy spectra from the representation of this algebra.