An infinite family of superintegrable Hamiltonians with reflection in the plane

TL;DR

This paper introduces an infinite family of exactly solvable superintegrable quantum systems in the plane involving reflection operators, with wave functions expressed via little -1 Jacobi polynomials and demonstrated superintegrability.

Contribution

It presents a new class of superintegrable Hamiltonians with reflection symmetry, providing explicit solutions, symmetry algebra, and higher-order constants of motion.

Findings

Wave functions expressed in terms of little -1 Jacobi polynomials

Spectra exhibit accidental degeneracies

Superintegrability proved via recurrence relations

Abstract

We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schr\"odinger equations admit separation of variables in polar coordinates and are exactly solvable. The angular part of the wave function is expressed in terms of little -1 Jacobi polynomials. The spectra exhibit "accidental" degeneracies. The superintegrability of the model is proved using the recurrence relation approach. The (higher-order) constants of motion are constructed and the structure equations of the symmetry algebra obtained.