Asymptotic Analysis of some Nonlinear Schroedinger Equations for charged particle transport

In the modelling of electrons as quantum mechanical particles several types of NLS arise. The first class are "weakly nonlinear" models such as the Schroedinger-Poisson equation. Relativistic extensions of this model are given by the Dirac-Maxwell system and selfconsistent Pauli equations. Another nonlinearity arises in the context of the Hartree-Fock equations.

It can be approximated by a local function of the density similar to the focusing case of the cubic NLS. We present the models and the open mathematical problems as well as recent progress e.g. on the limit from Schroedinger- Poisson in a crystal to the semiclassical equations of solid state physics using a new variant of Wigner transforms.