Mapping Schr\"odinger equation into a Heun-type and identifying the corresponding potential function, energy and wavefunction

TL;DR

This paper transforms the Schrödinger equation into a Heun-type differential equation, enabling the identification of potential functions, energy levels, and wavefunctions, some of which correspond to new integrable quantum systems.

Contribution

It introduces a method to map the Schrödinger equation into a Heun-type form and identifies new integrable quantum systems through this transformation.

Findings

Derived solutions correspond to new integrable quantum systems

Identified potential functions and energy parameters from the Heun-type equation

Established a link between Schrödinger and Heun equations for quantum analysis

Abstract

We transform the Schr\"odinger wave equation to a nine-parameter Heun-type differential equation. Using our solutions of the latter in [J. Math. Phys. 59 (2018) 113507], we are able to identify the associated potential function, energy parameter, and write the corresponding wave function. Some of the solutions obtained correspond to new integrable quantum systems.