A B-spline Galerkin method for the Dirac equation

TL;DR

This paper develops a B-spline Galerkin method tailored for the Dirac equation, ensuring stability and accuracy without spurious solutions, and validates it through comparison with exact R-matrix calculations across various nuclear charges.

Contribution

It introduces a stable B-spline Galerkin approach for the Dirac equation and demonstrates its effectiveness in accurately computing the R-matrix without spurious solutions.

Findings

No spurious solutions were observed.

Excellent agreement with exact R-matrix results.

Method is effective across a range of nuclear charges and quantum numbers.

Abstract

The B-spline Galerkin method is investigated for the simple eigenvalue problem, $y^{\prime\prime} = -\lambda^2 y$. Special attention is give to boundary conditions. From this analysis, we propose a stable method for the Dirac equation and evaluate its accuracy by comparing the computed and exact R-matrix for a wide range of nuclear charges $Z$ and angular quantum numbers $\kappa$. No spurious solutions were found and excellent agreement was obtained for the R-matrix.