Potentials of the Heun class

TL;DR

This paper reviews methods for generating potentials that transform the one-dimensional Schrödinger equation into hypergeometric and Heun equations, introducing new classes of potentials and comparing with previous studies.

Contribution

It introduces new classes of potentials that transform the Schrödinger equation into Heun and confluent Heun equations, expanding the theoretical framework.

Findings

Derived new classes of potentials for Heun equations

Compared results with previous studies

Extended the analysis to generalized Heun equations

Abstract

We review different methods of generating potentials such that the one-dimensional Schr\"{o}dinger equation (ODSE) can be transformed into the hypergeometric equation. We compare our results with previous studies, and complement the subject with new findings. Our main result is to derive new classes of potentials such that the ODSE can be transformed into the Heun equation and its confluent cases. The generalized Heun equation is also considered.