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

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The linear Schrödinger 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 ( ϵ ) where ϵ 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 ( ϵ ) . The accuracy of this method is demonstrated via several numerical examples.