Asymptotics of the phase of the solutions of the random Schrödinger equation

We consider solutions of the Schrödinger equation with a weak time-dependent random potential. It is shown that when the two-point correlation function of the potential is rapidly decaying then the Fourier transform ζ̂ε(t, ξ) of the appropriately scaled solution converges point-wise in ξ to a deterministic limit, exponentially decaying in time. On the other hand, when the two-point correlation function decays slowly, we show that the limit of ζ̂ε(t, ξ) has the form ζ̂0(ξ) exp(iBκ(t, ξ)) where Bκ(t, ξ) is a fractional Brownian motion.