Inter-Relations of Solvable Potentials

TL;DR

This paper explores the relationships among all known solvable potentials in nonrelativistic quantum mechanics, showing how they can be interconnected through transformations and limiting procedures.

Contribution

It establishes a unified framework linking the two classes of Natanzon potentials via transformations and limits, enhancing understanding of their interrelations.

Findings

All solvable potentials are connected through point canonical transformations.

A limiting procedure relates the hypergeometric and confluent hypergeometric classes.

The work unifies previously separate classes of solvable potentials.

Abstract

Solvable Natanzon potentials in nonrelativistic quantum mechanics are known to group into two disjoint classes depending on whether the Schr\"odinger equation can be reduced to a hypergeometric or a confluent hypergeometric equation. All the potentials within each class are connected via point canonical transformations. We establish a connection between the two classes with appropriate limiting procedures and redefinition of parameters, thereby inter-relating all known solvable potentials.