Coulomb singularities in scattering wave functions of spin-orbit-coupled states

TL;DR

This paper investigates the Coulomb singularity issue in coupled channel scattering with spin-orbit interaction, proposing a matrix-based method to analyze regular solutions near the origin, supported by numerical examples.

Contribution

It introduces a novel matrix approach to handle Coulomb singularities in coupled scattering equations with spin-orbit coupling, extending the Frobenius method.

Findings

Logarithmic divergence identified in solutions near the origin.

Auxiliary functions enable reduction to a first-order system.

Numerical calculations demonstrate the method's effectiveness.

Abstract

We report on our analysis of the Coulomb singularity problem in the frame of the coupled channel scattering theory including spin-orbit interaction. We assume that the coupling between the partial wave components involves orbital angular momenta such that $\Delta l = 0, \pm 2$. In these conditions, the two radial functions, components of a partial wave associated to two values of the angular momentum $l$, satisfy a system of two second-order ordinary differential equations. We examine the difficulties arising in the analysis of the behavior of the regular solutions near the origin because of this coupling. First, we demonstrate that for a singularity of the first kind in the potential, one of the solutions is not amenable to a power series expansion. The use of the Lippmann-Schwinger equations confirms this fact: a logarithmic divergence arises at the second iteration. To overcome this difficulty, we introduce two auxilliary functions which, together with the two radial functions, satisfy a system of four first-order differential equations. The reduction of the order of the differential system enables us to use a matrix-based approach, which generalizes the standard Frobenius method. We illustrate our analysis with numerical calculations of coupled scattering wave functions in a solid-state system.