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

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