A semi-Lagrangian time splitting method for the Schrödinger equation with vector potentials∗

In this paper, we present a time splitting scheme for the Schrödinger equation in the presence of electromagnetic eld in the semi-classical regime, where the wave function propagates O(ε) oscillations in space and time. With the operator splitting technique, the time evolution of the Schrödinger equation is divided into three parts: the kinetic step, the convection step and the potential step. The kinetic and the potential steps can be handled by the classical time-splitting spectral method. For the convection step, we propose a semi-Lagrangian method in order to allow large time steps. We prove the unconditional stability conditions with spatially variant external vector potentials, and the error estimate in the l approximation of the wave function. By comparing with the semi-classical limit, the classical Liouville equation in the Wigner framework, we show that this method is able to capture the correct physical observables with time step ∆t ≫ ε. We implement this method numerically for both one dimensional and two dimensional cases to verify that ε−independent time steps can indeed be taken in computing physical observables.