Schr\"odinger equation for two quasi-exactly solvable potentials

TL;DR

This paper explores solutions to the Schr"odinger equation with two elliptic potentials using Heun's equation, revealing conditions for finite-series solutions and the existence of bounded infinite-series eigenfunctions.

Contribution

It demonstrates how Heun's equation solutions can be applied to quasi-exactly solvable potentials, identifying parameter conditions for finite and infinite series solutions.

Findings

Finite-series solutions occur when ll is an integer, except ll=-1,-2,-3,-4.

Half odd-integer ll yields hypergeometric finite series.

Infinite-series eigenfunctions are convergent and bounded for all values of the independent variable.

Abstract

We apply solutions of Heun's general equation to the stationary Schr\"odinger equation with two quasi-exactly solvable elliptic potentials which depend on a real parameter $\ell$. We get finite-series solutions from power series expansions for Heun's equation if $\ell$ is an integer, except if $\ell=-1,-2,-3,-4$. If $\ell\neq-5/2$ is half an odd integer, we obtain finite series in terms of hypergeometric functions. The quasi-exact solvability is expressed by the finite series solutions. However, for any value of $\ell$, we find infinite-series eigenfunctions which are convergent and bounded for all values of the independent variable.