Inter-relations between additive shape invariant superpotentials

TL;DR

This paper develops a method to interconnect all known additive shape invariant superpotentials in quantum mechanics by linking two main classes through transformations, enhancing understanding of their relationships.

Contribution

It introduces a novel approach to generate type I superpotentials from type II, unifying the classification of additive shape invariant superpotentials.

Findings

Established a method to derive type I superpotentials from type II.

Connected two disjoint classes of superpotentials via point canonical transformations.

Provided a pathway to interrelate all known additive shape invariant superpotentials.

Abstract

All known additive shape invariant superpotentials in nonrelativistic quantum mechanics belong to one of two categories: superpotentials that do not explicitly depend on $\hbar$, and their $\hbar$-dependent extensions. The former group themselves into two disjoint classes, depending on whether the corresponding Schr\"odinger equation can be reduced to a hypergeometric equation (type-I) or a confluent hypergeometric equation (type-II). All the superpotentials within each class are connected via point canonical transformations. Previous work showed that type-I superpotentials produce type-II via limiting procedures. In this paper we develop a method to generate a type I superpotential from type II, thus providing a pathway to interconnect all known additive shape invariant superpotentials.