自旋轨道耦合态散射波函数中的库仑奇异性
Coulomb singularities in scattering wave functions of spin-orbit-coupled states
摘要：
我们报告我们的库仑奇异性问题的框架内的耦合通道散射理论，包括自旋轨道相互作用的分析。我们假设分波分量之间的耦合涉及轨道角动量，使得$Δl = 0, \pm 2$。在这些条件下，两个径向函数，与角动量$l$的两个值相关联的分波的分量，满足两个二阶常微分方程的系统。我们研究了由于这种耦合而在分析原点附近正则解的行为时产生的困难。首先，我们证明了第一类奇点的潜力，解决方案之一是不服从幂级数展开。李普曼-施温格方程的使用证实了这一事实：在第二次迭代时出现对数发散。为了克服这个困难，我们引入两个辅助功能，连同两个径向函数，满足一个系统的四个一阶微分方程。微分系统的阶数的减少使我们能够使用基于矩阵的方法，该方法推广了标准的Frobenius方法。我们说明我们的分析与耦合散射波函数在固态系统的数值计算。
英文摘要：
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 $Δl = 0, \pm 2$. 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.
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