A time splitting method for the three-dimensional linear Pauli equation

Abstract

We present and analyze a numerical method to solve the time-dependent linear Pauli equation in three space-dimensions. The Pauli equation is a "semi-relativistic" generalization of the Schr\"odinger equation for 2-spinors which accounts both for magnetic fields and for spin, the latter missing in predeeding work on the linear magnetic Schr\"odinger equation. We use a four operator splitting in time, prove stability and convergence of the method and derive error estimates as well as meshing strategies for the case of given time-independent electromagnetic potentials (= "linear" case), thus providing a generalization of previous results for the magnetic Schr\"odinger equation. Some proof of concept examples of numerical simulations are presented.