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

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• 作者：Zhizhang Wu ^{wzz14@mails.tsinghua.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Zhongyi Huang ; ^{zhuang@math.tsinghua.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Schrö ; dinger equation ; Bloch decomposition ; Time-splitting ; Generalized polynomial chaos ; Uncertainty quantification
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：317
• 期：Complete
• 页码：257-275
• 全文大小：765 K

文摘

In this paper, we consider the numerical solution of the one-dimensional Schrödinger equation with a periodic lattice potential and a random external potential. This is an important model in solid state physics where the randomness results from complicated phenomena that are not exactly known. Here we generalize the Bloch decomposition-based time-splitting pseudospectral method to the stochastic setting using the generalized 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.