Some results about the dynamics of an electron in a crystal
We will discuss the asymptotic analysis of a Schrödinger equation modeling the dynamics of an electron in a crystal for small wave-length comparable to the characteristic scale of the crystal. We will be interested in the description of the limit of time averaged energy densities for rather general initial data. We shall decompose these initial data on Bloch bands and authorize leading order contributions on modes that cross. We will explain how conical crossing of Bloch modes do not trap the energy and do not prevent dispersion despite the singularity they create. We will also describe on examples the effects of critical points of the Bloch modes localized at crossing points.