Ladder operators for solvable potentials connected with exceptional orthogonal polynomials

TL;DR

This paper develops ladder operators for solvable quantum potentials linked with exceptional orthogonal polynomials, using supersymmetric quantum mechanics and Darboux methods, to analyze their algebraic structure and applications in superintegrable systems.

Contribution

It introduces new constructions of ladder operators for rationally extended potentials, including a novel method combining state-adding and state-deleting approaches.

Findings

Ladder operators close a polynomial Heisenberg algebra for extended potentials.

Extra bound states form isolated singlets in the algebraic structure.

The methods enable building integrals of motion for superintegrable systems.

Abstract

Exceptional orthogonal polynomials constitute the main part of the bound-state wavefunctions of some solvable quantum potentials, which are rational extensions of well-known shape-invariant ones. The former potentials are most easily built from the latter by using higher-order supersymmetric quantum mechanics (SUSYQM) or Darboux method. They may in general belong to three different types (or a mixture of them): types I and II, which are strictly isospectral, and type III, for which k extra bound states are created below the starting potential spectrum. A well-known SUSYQM method enables one to construct ladder operators for the extended potentials by combining the supercharges with the ladder operators of the starting potential. The resulting ladder operators close a polynomial Heisenberg algebra (PHA) with the corresponding Hamiltonian. In the special case of type III extended potentials, for this PHA the k extra bound states form k singlets isolated from the higher excited states. Some alternative constructions of ladder operators are reviewed. Among them, there is one that combines the state-adding and state-deleting approaches to type III extended potentials (or so-called Darboux-Crum and Krein-Adler transformations) and mixes the k extra bound states with the higher excited states. This novel approach can be used for building integrals of motion for two-dimensional superintegrable systems constructed from rationally-extended potentials.