An improved discretization of Schrödinger-like radial equations

Abstract

A new discretization of the radial equations that appear in the solution of separable second order partial differential equations with some rotational symmetry (as the Schrödinger equation in a central potential) is presented. It cures a pathology, related to the singular behavior of the radial function at the origin, that suffers in some cases the discretization of the second derivative with respect to the radial coordinate. This pathology causes an enormous slowing down of the convergence to the continuum limit when the two point boundary value problem posed by the radial equation is solved as a discrete matrix eigenvalue problem. The proposed discretization is a simple solution to that problem. Some illustrative examples are discussed.