  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.
