Spectral parameter power series representation for regular solutions of the radial Dirac system

TL;DR

This paper introduces a spectral parameter power series (SPPS) method for solving the radial Dirac system and related spectral problems, providing a new numerical approach with high accuracy and applications to atomic physics.

Contribution

The paper develops a novel SPPS representation for the radial Dirac system and its spectral problem, enabling efficient numerical solutions and applications to physical models.

Findings

The SPPS-based numerical method yields excellent accuracy.

The method is applicable to perturbed Bessel equations.

Application to hydrogen-like atoms demonstrates practical utility.

Abstract

A spectral parameter power series (SPPS) representation for the regular solution of the radial Dirac system with complex coefficients is obtained, as well as a SPPS representation for the (entire) characteristic function of the corresponding spectral problem on a finite interval. Based on the SPPS representation, a numerical method for solving spectral problems is developed. It is shown that the method is also applicable to solving spectral problems for perturbed Bessel equations. We exhibit that the proposed numerical method delivers excellent results. Additionally, an application of the method to find the energy values of hydrogen-like atoms with a finite radius is presented.