Eur. Phys. J. Plus (2016) 131: 277
https://doi.org/10.1140/epjp/i2016-16277-1

Regular Article

Exact and approximate solutions of Schrödinger’s equation with hyperbolic double-well potentials

¹ Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, H3G 1M8, Montréal, Québec, Canada
² School of Mathematical and Computational Sciences, University of Prince Edward Island, 550 University Avenue, C1A 4P3, Charlottetown, PEI, Canada

^* e-mail: nsaad@upei.ca

Received: 22 June 2016
Accepted: 11 July 2016
Published online: 15 August 2016

Abstract

Analytic and approximate solutions for the energy eigenvalues generated by the hyperbolic potentials, V _m(x) = - U ₀sinh^2m(x/d )/cosh^2m+2(x/d ) , m = 0, 1, 2,... , are constructed. A by-product 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 asymptotic iteration method.