Novel method for evaluating the eigenvalues of the Heun differential equation with an application to the Breit equation

TL;DR

This paper introduces a new numerical method based on Green functions and continued fractions to accurately compute eigenvalues of the Heun differential equation derived from the Breit equation, improving precision over previous approaches.

Contribution

The work presents a novel calculation method for eigenvalues of the Heun equation from the Breit equation, achieving high accuracy and consistency with existing results.

Findings

Eigenvalues computed with 25-digit accuracy

Method consistent with previous numerical results

Extension of accuracy beyond prior literature

Abstract

Eigenvalues of the Breit equation, in which only the static Coulomb potential is considered, have been found. Over the past decades several authors have analyzed the Breit equation to obtain numerically or by approximation an estimation of the energy levels. Various approaches have been used and no determination of the energy levels currently exists that is directly based on the second order Heun differential equation derived. The aim of this work is to provide a method of calculation that can be used to numerically calculate the energy levels for various spin states to high accuracy. From the Breit equation, we derive the corresponding second-order Heun differential equation and continued fraction from which the eigenvalues can be determined very accurately. Next, we present a novel method based on the Green function method, which leads to a semi-infinite determinant from which we are able to obtain the numerical values of the eigenvalues by direct calculation. Using suitable numerical methods for the direct calculation of the continued fraction and the semi-infinite determinant, we show that both methods are consistent within 25 digits of accuracy. We show that the correct energy levels for the Dirac equation follow from our results by a suitable mapping of the variables. The results are in total agreement with earlier calculations found in the literature and extend this by several digits of additional accuracy. The condition on the determinant giving the energy levels provides a rich structure that is promising in extending the results of this work.