On the calculation of bound-state energies supported by hyperbolic double well potentials

TL;DR

This paper calculates bound-state energies and eigenfunctions for a hyperbolic double-well potential using exact solutions and numerical methods, including critical potential strengths and eigenvalue testing.

Contribution

It provides both exact polynomial solutions for specific parameters and numerical energies for general cases of the hyperbolic double-well potential.

Findings

Eigenvalues and eigenfunctions obtained for the potential.

Critical potential strengths identified for bound state support.

Numerical methods validated against exact asymptotic expressions.

Abstract

We obtain eigenvalues and eigenfunctions of the Schr\"{o}dinger equation with a hyperbolic double-well potential. We consider exact polynomial solutions for some particular values of the potential-strength parameter and also numerical energies for arbitrary values of this model parameter. We test the numerical method by means of a suitable exact asymptotic expression for the eigenvalues and also calculate critical values of the strength parameter that are related to the number of bound states supported by the potential.