A family of fourth-order superintegrable systems with rational potentials 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 potentials satisfy a nonlinear equation related to Painlevé 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 Painlevé IV associated with Hermite polynomials

Superintegrability and deformed oscillator realizations of quantum TTW Hamiltonians on constant-curvature manifolds and with reflections in a plane

“…Other models in two dimensions, such as the Post-Winternitz systems, could be considered as well as various Darboux deformations which provide isospectral or almost isospectral problems, which lead to exceptional Jacobi polynomial of Type I, II and III [57,58]. In addition, we may also consider the case where the angular part consists of exotic non-elementary functions, since explicit knowledge of the eigenfunctions is not readily obtainable [18,25,26].…”

“…The exceptional properties of superintegrable systems have excited considerable interest in recent years, particularly in their connection with separability and the algebraic approach to solving the Schrödinger equation. The classification of second-order superintegrable systems on conformally flat two-dimensional space is well-understood [1][2][3][4][5][6][7], and current attention is directed towards finding and solving models in higher dimensions [8][9][10][11][12][13][14][15] and of higher orders [16][17][18][19][20][21][22][23][24][25][26].…”