  We consider the Schr\"odinger equation with a time-independent weakly random
potential of a strength $\epsilon\ll 1$, with Gaussian statistics. We prove
that when the initial condition varies on a scale much larger than the
correlation length of the potential, the compensated wave function converges to
a deterministic limit on the time scale $t\sim\epsilon^{-2}$. This is shown
under the sharp assumption that the correlation function $R(x)$ of the random
potential decays slower than $1/|x|^2$, which ensures that the effective
potential is finite. When $R(x)$ decays slower than $1/|x|^2$ we establish an
anomalous diffusive behavior for the averaged wave function on a time scale
shorter than $\epsilon^{-2}$, as long as the initial condition is "sufficiently
macroscopic". We also consider the kinetic regime when the initial condition
varies on the same scale as the random potential and obtain the limit of the
averaged wave function for potentials with the correlation functions decaying
faster than $1/|x|^2$. We use random potentials of the Schonberg class which
allows us to bypass the oscillatory phase estimates.
