In the last decade the method of Wigner transforms has proven to be very succesful in the context of (semi)classical limits and general homogenization limits [1]. The basic idea is to transform a PDE in ``physical'' space into sort of a kinetic equation in ``phase-space''.
For problems with a distinguished scale of oscillations Wigner measures are clearly more convenient than e.g. H-measures.
We present recent results on the adoption of Wigner transforms to homogenization of periodic structures, the so called Wigner series. The outstanding problem of the semiclassical limit of the nonlinear Schrödinger-Poisson problem [2] in a crystal has recently been solved by such Wigner Bloch measures [3], thus giving the first rigorous justification of the ``semiclassical equations'' of solid state physics
| [1] | ``Wigner transforms and Homogenization limits'', P. Gérard, P. A. Markowich, N.J. Mauser and F. Poupaud, Comm.Pure and Appl.Math., 50 (1997) 321-377 |
| [2] | ``Sur les Mesures de Wigner'', P.L. Lions and T. Paul, Revista Mat. Iberoamericana, 9 (1993) 553-618 |
| [3] | ``Semiclassical limit for the Schrödinger-Poisson equation in a crystal'', P. Bechouche, N.J. Mauser and F. Poupaud, Comm.Pure and Appl.Math., XXX (2001) 1-42 |
| E-Mail: | norbert.mauser@univie.ac.at |