A mathematical analysis of quantum transport in three-dimensional crystals

We present an existence and uniqueness result for a quantum transport model in three dimensional crystals. The model consists of a quantum transport (Wigner) equation posed on the phase space consisting of a discrete position variable and a «continuous» wave vector, which is restricted to a bounded domain in R³ (first Brillouin zone of the crystal). The potential is modeled self-consistently by a discrete Poisson equation (Coulomb interaction). Also we investigate the limits of solutions of this model as the grid spacing tends to zero and show that they converge to the solution of a quantum transport model posed on the «fully continuous» phase space. The transport model derived by this limiting procedure treats the band diagram of the crystal in a semi-classical way and the potential energy term quantum mechanically.