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

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 [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.