Homogenization of the Schroedinger equation with large, random potential
Ningyao Zhang, Guillaume Bal · arXiv (Cornell University) · 2012
We study the behavior of solutions to a Schr{ö}dinger equation with large, rapidly oscillating, mean zero, random potential with Gaussian distribution. We show that in high dimension $d>\mathfrak{m}$, where $\mathfrak{m}$ is the order of the spatial pseudo-differential operator in the Schr{ö}dinger equation (with $\mathfrak{m}=2$ 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 Schr{ö}dinger equation as the correlation length $\varepsilon$ tends to 0. This generalizes to long times the convergence results obtained for short times and for the heat equation. The result is based on a careful decomposition of multiple scattering contributions. In dimension $d