High-Order Matrix Numerov for Singular Potentials

TL;DR

This paper enhances the matrix Numerov method for solving the Schrödinger equation with singular potentials by introducing boundary corrections that restore and improve convergence rates, maintaining efficiency.

Contribution

It introduces boundary corrections based on analytic near-origin information to improve the Numerov method's accuracy for singular potentials.

Findings

Restores fourth-order convergence for Coulomb potentials

Achieves higher convergence rates for s- and p-wave energies

Maintains computational efficiency of the original method

Abstract

The matrix Numerov method provides an efficient framework for solving the time-independent Schr\"odinger equation as a matrix eigenvalue problem. However, for singular potentials such as the Coulomb interaction, the expected fourth-order convergence deteriorates for low angular momenta due to the behavior of the potential near the origin. We show that this loss of accuracy originates from an implicit boundary assumption in the standard formulation. By incorporating analytic near-origin information into the discretized Hamiltonian, we derive simple boundary corrections that restore fourth-order convergence and can even produce higher convergence rates for $s$- and $p$-wave energies. The resulting scheme preserves the simplicity and computational efficiency of the original method while significantly improving its accuracy for singular potentials.