A splitting approach for the magnetic Schr\"odinger equation

TL;DR

This paper introduces a splitting method for the magnetic Schr"odinger equation, demonstrating first-order convergence and effective mass conservation through numerical experiments in multiple dimensions.

Contribution

It develops a novel splitting approach for the magnetic Schr"odinger equation involving unbounded operators and proves its convergence, with practical implementation details.

Findings

The splitting method achieves first-order convergence.

The method conserves mass effectively in numerical simulations.

Numerical examples validate the approach in multiple dimensions.

Abstract

The Schr\"odinger equation in the presence of an external electromagnetic field is an important problem in computational quantum mechanics. It also provides a nice example of a differential equation whose flow can be split with benefit into three parts. After presenting a splitting approach for three operators with two of them being unbounded, we exemplarily prove first-order convergence of Lie splitting in this framework. The result is then applied to the magnetic Schr\"odinger equation, which is split into its potential, kinetic and advective parts. The latter requires special treatment in order not to lose the conservation properties of the scheme. We discuss several options. Numerical examples in one, two and three space dimensions show that the method of characteristics coupled with a nonequispaced fast Fourier transform (NFFT) provides a fast and reliable technique for achieving mass conservation at the discrete level.