New rational extensions of solvable potentials with finite bound state spectrum

TL;DR

This paper introduces a novel method using disconjugacy properties of the Schrödinger equation to generate rational extensions of shape invariant potentials with finite spectra, expanding solvable models in quantum mechanics.

Contribution

It develops a new generalized SUSY QM approach that prolongs the dispersion relation to create regular isospectral rational extensions of finite spectrum potentials.

Findings

Constructed new solvable rational extensions of shape invariant potentials.

Expressed spectra of extensions using new orthogonal polynomials.

Analyzed shape invariance properties of the extended potentials.

Abstract

Using the disconjugacy properties of the Schr\"odinger equation, it is possible to develop a new type of generalized SUSY QM partnership which allows to generate new solvable rational extensions for translationally shape invariant potentials having a finite bound state spectrum. For this we prolong the dispersion relation relating the energy to the quantum number out of the physical domain until a disconjugacy sector. The prolonged excited states Riccati-Schr\"odinger (RS) functions are used to build Darboux-B\"acklund transforms which give regular isospectral extensions of the initial potential. We give the spectra of these extensions in terms of new orthogonal polynomials and study their shape invariance properties.