Solvable rational extensions of the Morse and Kepler-Coulomb potentials

TL;DR

This paper demonstrates how to generate infinite solvable rational extensions of certain quantum potentials using Darboux-Bäcklund transformations and symmetry properties, expanding the class of exactly solvable models in quantum mechanics.

Contribution

It introduces a method to produce infinite solvable rational extensions of translationally shape invariant potentials via Darboux-Bäcklund transformations based on unphysical Riccati-Schrödinger functions.

Findings

Infinite set of solvable rational extensions generated

Extensions derived from symmetries and Riccati-Schrödinger functions

Method applicable to Morse and Kepler-Coulomb potentials

Abstract

We show that it is possible to generate an infinite set of solvable rational extensions from every exceptional first category translationally shape invariant potential. This is made by using Darboux-B\"acklund transformations based on unphysical regular Riccati-Schr\"odinger functions which are obtained from specific symmetries associated to the considered family of potentials.