A simple and efficient numerical scheme to integrate non-local potentials

TL;DR

This paper introduces a modified, stable numerical scheme for solving the integro-differential Schrödinger equation with non-local potentials, improving upon the trivially equivalent potential method for nuclear physics applications.

Contribution

A slight modification to the trivially equivalent potential method is proposed, removing divergences and enabling stable, precise calculations of non-local potentials in nuclear physics.

Findings

The modified method effectively removes divergences caused by wave function nodes.

It provides a simple, stable, and accurate numerical solution for non-local potentials.

Application to 16O Hartree-Fock calculations demonstrates its practical utility.

Abstract

As nuclear wave functions have to obey the Pauli principle, potentials issued from reaction theory or Hartree-Fock formalism using finite-range interactions contain a non-local part. Written in coordinate space representation, the Schrodinger equation becomes integro-differential, which is difficult to solve, contrary to the case of local potentials, where it is an ordinary differential equation. A simple and powerful method has been proposed several years ago, with the trivially equivalent potential method, where non-local potential is replaced by an equivalent local potential, which is state-dependent and has to be determined iteratively. Its main disadvantage, however, is the appearance of divergences in potentials if the wave functions have nodes, which is generally the case. We will show that divergences can be removed by a slight modification of the trivially equivalent potential method, leading to a very simple, stable and precise numerical technique to deal with non-local potentials. Examples will be provided with the calculation of the Hartree-Fock potential and associated wave functions of 16O using the finite-range N3LO realistic interaction.