Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry

TL;DR

This paper introduces two new exactly solvable quantum potentials linked to exceptional orthogonal polynomials, extending classical potentials while maintaining shape invariance and isospectrality, thus enriching the set of solvable models in quantum mechanics.

Contribution

The authors construct novel solvable potentials using Laguerre- and Jacobi-type X1 exceptional orthogonal polynomials, expanding the class of shape invariant and isospectral quantum models.

Findings

New potentials are exactly solvable with bound states.

Potentials are shape invariant and isospectral to classical counterparts.

Solutions involve Laguerre- and Jacobi-type X1 exceptional orthogonal polynomials.

Abstract

We construct two new exactly solvable potentials giving rise to bound-state solutions to the Schr\"odinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These potentials, extending either the radial oscillator or the Scarf I potential by the addition of some rational terms, turn out to be translationally shape invariant as their standard counterparts and isospectral to them.