Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials

TL;DR

This paper introduces rational extensions of specific quantum potentials using confluent Darboux transformations, leading to new orthogonal polynomials and an expanded shape invariance property, enriching the mathematical framework of quantum shape invariance.

Contribution

It constructs new rational potentials with enlarged shape invariance and introduces associated novel orthogonal polynomials, expanding the class of exactly solvable models.

Findings

Extended potentials are strictly or quasi-isospectral to original potentials.

New families of orthogonal polynomials depend on a continuous parameter.

Extended potentials exhibit an enlarged shape invariance property.

Abstract

We construct rational extensions of the Darboux-P\"oschl-Teller and isotonic potentials via two-step confluent Darboux transformations. The former are strictly isospectral to the initial potential, whereas the latter are only quasi-isospectral. Both are associated to new families of orthogonal polynomials, which, in the first case, depend on a continuous parameter. We also prove that these extended potentials possess an enlarged shape invariance property.