Families of superintegrable Hamiltonians constructed from exceptional polynomials

TL;DR

This paper introduces a new family of exactly solvable two-dimensional quantum Hamiltonians involving exceptional polynomials, demonstrating their superintegrability through higher-order integrals of motion derived from ladder operators.

Contribution

It constructs novel superintegrable Hamiltonians using exceptional orthogonal polynomials, linking quantum properties to classical limits and expanding the class of known superintegrable systems.

Findings

Hamiltonians are exactly solvable with wave functions in Laguerre and exceptional Jacobi polynomials.

Higher-order integrals of motion confirm superintegrability of the quantum systems.

Quantum terms vanish in the classical limit, recovering known superintegrable systems.

Abstract

We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms of Laguerre and exceptional Jacobi polynomials. The Hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems. Additional, higher-order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable.