The weakly random Schroedinger equation: homogenization and the kinetic limit

We consider a Schroedinger equation with a weakly random time-independent potential. When the correlation function of the potential is, roughly speaking, of the Schwartz class, it has been shown by Spohn (1977), and Erdos and Yau (2001) that the kinetic limit holds -- the expectation of the phase space energy density of the solution converges to the solution of a kinetic equation. We "extend" this result to potentials whose correlation functions satisfy (in some sense) "sharp" conditions, and also prove a parallel homogenization result for slowly varying initial conditions. I will explain the quotation marks above and make some speculations on the genuinely sharp conditions on the random potential that separate various regimes. This talk is a joint work with T. Chen and T. Komorowski