Double exponential sinc-collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function

Abstract

We show that the double exponential sinc-collocation method provides an efficient uniformly accurate solution to the one-dimensional time independent Schrödinger equation for a general class of rational potentials of the form V (x) = p(x)/q(x). The derived algorithm is based on the discretization of the Hamiltonian of the Schrödinger equation using sinc expansions. This discretization results in a generalized eigenvalue problem, the eigenvalues of which correspond to approximations of the energy values of the starting Hamiltonian. A systematic numerical study is conducted, beginning with test potentials with known eigenvalues and moving to rational potentials of increasing degree.