Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials

Authors:
Shi Jin;Hao Wu;Xu Yang;Zhongyi Huang
Affiliations:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA;Department of Mathematical Sciences, Tsinghua University, Beijing 10084, China;Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA;Department of Mathematical Sciences, Tsinghua University, Beijing 10084, China
Venue:
Journal of Computational Physics
Year:
2010

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Abstract

The linear Schrodinger equation with periodic potentials is an important model in solid state physics. The most efficient direct simulation using a Bloch decomposition-based time-splitting spectral method [18] requires the mesh size to be O(@e) where @e is the scaled semiclassical parameter. In this paper, we generalize the Gaussian beam method introduced in Jin et al. [23] to solve this problem asymptotically. We combine the technique of Bloch decomposition and the Eulerian Gaussian beam method to arrive at an Eulerian computational method that requires mesh size of O(@e). The accuracy of this method is demonstrated via several numerical examples.