Semiclassical limit for the Schrodinger-Poisson equation in a crystal

We give a mathematically rigorous theory for the limit from a weakly nonlinear Schrodinger equation with both periodic and nonperiodic potential to the semiclassical version of the Vlasov equation. To this end we perform simultaneously a classical limit (vanishing Planck constant) and a homogenization limit of the periodic structure (vanishing lattice length taken proportional to the Planck constant). We introduce a new variant of Wigner transforms, namely the "Wigner-Bloch series" as an adaptation of the Wigner series for density matrices related to two different "energy bands." Another essential tool is estimates on the commutators of the projectors into the Floquet subspaces (''band subspaces") and the multiplicative potential operator that destroy the invariance of these band subspaces under the periodic Hamiltonian. We assume the initial data to be concentrated in isolated bands but allow for band-crossing of the other bands, which is the generic situation in more than one space dimension. The nonperiodic potential is obtained from a coupling to the Poisson equation; i.e., we rake into account the self-consistent Coulomb interaction. Our results also hold for the easier linear case where this potential is given. We hence give the first rigorous derivation of the (nonlinear) "semiclassical equations" of solid stare physics widely used to describe the dynamics of electrons in semiconductors. (C) 2001 John Wiley & Sons, Inc..