On a solution of the Schrodinger equation with a hyperbolic double-well potential

TL;DR

This paper presents an analytical solution to the one-dimensional Schrödinger equation with a hyperbolic double-well potential by transforming it into a confluent Heun equation, identifying conditions for bound state solutions.

Contribution

It introduces a method to solve the Schrödinger equation with a hyperbolic double-well potential using confluent Heun functions, expanding analytical solution techniques.

Findings

Derivation of wavefunctions in terms of confluent Heun functions.

Conditions for reduction to Heun polynomials for bound states.

Analytical expressions for energy eigenvalues.

Abstract

We report a solution of the one-dimensional Schrodinger equation with a hyperbolic double-well confining potential via a transformation to the so-called confluent Heun equation. We discuss the requirements on the parameters of the system in which a reduction to Heun polynomials is possible, representing the wavefunctions of bound states.