HOMOGENIZATION OF THE SCHRODINGER EQUATION WITH LARGE, RANDOM POTENTIAL

We study the behavior of solutions to a Schrodinger equation with large, rapidly oscillating, mean zero, random potential with Gaussian distribution. We show that in high dimension d > m, where m is the order of the spatial pseudo-differential operator in the Schrodinger equation (with m = 2 for the standard Laplace operator), the solution converges in the L-2 sense uniformly in time over finite intervals to the solution of a deterministic Schrodinger equation as the correlation length epsilon tends to 0. This generalizes to long times the convergence results obtained for short times and for the heat equation in [2]. The result is based on the decomposition of multiple scattering contributions introduced in [6]. In dimension d < m, the random solution converges to the solution of a stochastic partial differential equation; see [1, 13].