Semiclassical Models for the Schrödinger Equation with Periodic Potentials and Band Crossings∗

The Bloch decomposition plays a fundamental role in the study of quantum mechanics and wave propagation in periodic media. Most of the homogenization theory developed for the study of high frequency or semiclassical limit for these problems assumes no crossing of the Bloch bands, resulting in classical Liouville equations in the limit along each Bloch band. In this article, we derive semiclassical models for the Schrödinger equation in periodic media that take into account band crossing, which is important to describe quantum transitions between Bloch bands. Our idea is still based on the Wigner transform (on the Bloch eigenfunctions), but in taking the semiclassical approximation, we retain the off-diagonal entries of the Wigner matrix, which cannot be ignored near the point of band crossing. This results in coupled inhomogenious Liouville systems that can suitably describe quantum tunnelling between bands that are not well-separated. We also develop a domain decomposition method that couples these semiclassical models with the classical Liouville equations (valid away from zones of band crossing) for a multiscale computation. Solutions of these models are numerically compared with those of the Schröding equation to justify the validity of these new models for band-crossings.