An improved discretization of Schrodinger-like radial equations

TL;DR

This paper introduces an improved discretization method for Schrödinger-like radial equations that addresses singularity issues at the origin, enhancing convergence speed in numerical solutions.

Contribution

It presents a novel discretization technique that corrects a known pathology at the origin, improving numerical stability and convergence in solving radial equations.

Findings

Reduces convergence time in numerical solutions

Addresses singularity issues at the origin

Improves stability of discretized 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 Schrodinger equation in a central potential) is presented. It cures a pathology, related to the singular behaviour 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.