New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials

TL;DR

This paper introduces new 2D superintegrable quantum systems derived from Hermite and Laguerre exceptional orthogonal polynomials, revealing novel algebraic structures and energy degeneracies linked to EOP holes.

Contribution

It constructs new superintegrable Hamiltonians from Hermite and Laguerre EOP, detailing their polynomial algebras, spectra, and unique degeneracy patterns.

Findings

Derived 2D superintegrable Hamiltonians from EOP

Established polynomial algebra structures and representations

Discovered new energy degeneracy patterns related to EOP holes

Abstract

In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and Laguerre EOP and their related quantum systems new 2D superintegrable Hamiltonians with higher-order integrals of motion and the polynomial algebras generated by their integrals of motion. We obtain the finite-dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum. We also point out a new type of degeneracies of the energy levels of these systems that is associated with holes in sequences of EOP.