Efficient methods for time-dependence in semiclassical Schr\"odinger equations

TL;DR

This paper introduces an efficient, stable numerical method combining Zassenhaus decomposition and Magnus expansion to solve time-dependent semi-classical Schrödinger equations, validated through numerical experiments.

Contribution

It presents a novel unitary method that efficiently handles explicit time dependence in semi-classical Schrödinger equations, improving stability and computational performance.

Findings

Method is stable and unitary.

Numerical experiments demonstrate effectiveness.

Applicable to time-dependent potentials.

Abstract

We build an efficient and unitary (hence stable) method for the solution of the semi-classical Schr\"odinger equation subject with explicitly time-dependent potentials. The method is based on a combination of the Zassenhaus decomposition (Bader, Iserles, Kropielnicka & Singh 2014) with the Magnus expansion of the time-dependent Hamiltonian. We conclude with numerical experiments.