The Gaussian wave packets transform for the semi-classical Schr\"odinger equation with vector potentials

TL;DR

This paper introduces a Gaussian wave packets transform for the semi-classical Schrödinger equation with electromagnetic fields, transforming it into a less oscillatory form and proposing efficient, stable numerical methods with high accuracy.

Contribution

The paper develops a novel Gaussian wave packets transform for the semi-classical Schrödinger equation with vector potentials and introduces stable, spectrally accurate numerical schemes for its solution.

Findings

Transform reduces oscillatory Schrödinger equation to a non-oscillatory form.

Proposed methods are unconditionally stable and spectrally accurate in space.

Numerical tests confirm efficiency and accuracy of the methods.

Abstract

In this paper, we reformulate the semi-classical Schr\"odinger equation in the presence of electromagnetic field by the Gaussian wave packets transform. With this approach, the highly oscillatory Schr\"odinger equation is equivalently transformed into another Schr\"odinger type wave equation, the $w$ equation, which is essentially not oscillatory and thus requires much less computational effort. We propose two numerical methods to solve the $w$ equation, where the Hamiltonian is either divided into the kinetic, the potential and the convection part, or into the kinetic and the potential-convection part. The convection, or the potential-convection part is treated by a semi-Lagrangian method, while the kinetic part is solved by the Fourier spectral method. The numerical methods are proved to be unconditionally stable, spectrally accurate in space and second order accurate in time, and in principle they can be extended to higher order schemes in time. Various one dimensional and multidimensional numerical tests are provided to justify the properties of the proposed methods.