Solution of the Schr\"odinger equation containing a Perey-Buck nonlocality

TL;DR

This paper presents two highly accurate numerical methods for solving the radial Schr"odinger equation with nonlocal potentials, specifically applied to Perey-Buck nonlocality, without relying on local equivalent approximations.

Contribution

It introduces Chebyshev polynomial expansion and Sturmian function expansion methods for solving nonlocal Schr"odinger equations with high precision, applicable to general nonlocal kernels.

Findings

Achieved accuracy of 1:10^14 with Chebyshev method.

Achieved accuracy of 1:10^6 after one iteration with Sturmian method.

Applicable to general nonlocality kernels.

Abstract

The solution of a radial Schr\"odinger equation for {\psi}(r) containing a nonlocal potential of the form \int{K(r,r') {\psi}(r') dr'} is obtained to high accuracy by means of two methods. An application to the Perey-Buck nonlocality is presented, without using a local equivalent representation. The first method consists in expanding {\psi} in a set of Chebyshev polynomials, and solving the matrix equation for the expansion coefficients numerically. An accuracy of between 1:10^{6} to 1:10^{14} is obtained, depending on the number of polynomials employed. The second method consists in expanding {\psi} into a set of N Sturmian functions of positive energy, supplemented by an iteration procedure. For N=15 an accuracy of 1:10^{4} is obtained without iterations. After one iteration the accuracy is increased to 1:10^{6}. The method is applicable to a general nonlocality K.