HOMOGENIZATION OF THE SCHRÖDINGER EQUATION WITH LARGE, RANDOM POTENTIAL
Ningyao Zhang, Guillaume Bal · Stochastics and Dynamics · 2013
We study the behavior of solutions to a Schrödinger 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 Schrödinger 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 Schrödinger equation as the correlation length ε 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].