Homogenization of the Schroedinger equation with large, random potential

Details Homogenization of the Schroedinger equation with large, random potential Homogenization of the Schroedinger equation with large, random potential

2012-02-15

arxiv.org/abs/1202.3181v1

Mathematics

Analysis of PDEs

25 pages

Scientific paper

We study the behavior of solutions to a Schr{\"o}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{\"o}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{\"o}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 $d<\mathfrak{m}$, the random solution converges to the solution of a stochastic partial differential equation.

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