A Bloch decomposition-based stochastic Galerkin method for quantum dynamics with a random external potential

TL;DR

This paper develops a numerical method combining Bloch decomposition and stochastic Galerkin techniques to efficiently solve the Schrödinger equation with random potentials, demonstrating stability, efficiency, and phenomena like Anderson localization.

Contribution

It introduces a novel stochastic Galerkin method based on Bloch decomposition for quantum dynamics with random external potentials, enhancing computational efficiency and stability.

Findings

Method is unconditionally stable.

Numerical results show improved efficiency.

Evidence of Anderson localization phenomena.

Abstract

In this paper, we consider the numerical solution of the one-dimensional Schr\"odinger equation with a periodic lattice potential and a random external potential. This is an important model in solid state physics where the randomness is involved to describe some complicated phenomena that are not exactly known. Here we generalize the Bloch decomposition-based time-splitting pseudospectral method to the stochastic setting using the generalize polynomial chaos with a Galerkin procedure so that the main effects of dispersion and periodic potential are still computed together. We prove that our method is unconditionally stable and numerical examples show that it has other nice properties and is more efficient than the traditional method. Finally, we give some numerical evidence for the well-known phenomenon of Anderson localization.