An accurate spectral method for solving the Schroedinger equation

Abstract

The solution of the Lippman-Schwinger (L-S) integral equation is equivalent to the the solution of the Schroedinger equation. A new numerical algorithm for solving the L-S equation is described in simple terms, and its high accuracy is confirmed for several physical situations. They are: the scattering of an electron from a static hydrogen atom in the presence of exchange, the scattering of two atoms at ultra low temperatures, and barrier penetration in the presence of a resonance for a Morse potential. A key ingredient of the method is to divide the radial range into partitions, and in each partition expand the solution of the L-S equation into a set of Chebyshev polynomials. The expansion is called "spectral" because it converges rapidly to high accuracy. Properties of the Chebyshev expansion, such as rapid convergence, are illustrated by means of a simple example.