Exact and approximate solutions of Schr\"odinger's equation with hyperbolic double-well potentials

TL;DR

This paper derives exact and approximate energy solutions for hyperbolic double-well potentials in Schrödinger's equation, introducing polynomial solutions for the confluent Heun equation and employing the asymptotic iteration method for high accuracy.

Contribution

It presents new analytic and approximate solutions for a class of hyperbolic potentials, including polynomial solutions for the confluent Heun equation and applies the asymptotic iteration method for general cases.

Findings

Exact solutions for specific potential parameters.

Polynomial solutions for the confluent Heun equation.

High-accuracy approximate solutions using asymptotic iteration method.

Abstract

Analytic and approximate solutions for the energy eigenvalues generated by the hyperbolic potentials $V_m(x)=-U_0\sinh^{2m}(x/d)/\cosh^{2m+2}(x/d),\,m=0,1,2,\dots$ are constructed. A byproduct of this work is the construction of polynomial solutions for the confluent Heun equation along with necessary and sufficient conditions for the existence of such solutions based on the evaluation of a three-term recurrence relation. Very accurate approximate solutions for the general problem with arbitrary potential parameters are found by use of the {\it asymptotic iteration method}.