The random Schrödinger equation: slowly decorrelating time-dependent potentials

We analyze the weak-coupling limit of the random Schrödinger equation with low frequency initial data and a slowly decorrelating random potential. For the probing signal with a sufficiently long wavelength, we prove a homogenization result, that is, the properly compensated wave field admits a deterministic limit in the “very low” frequency regime. The limit is “anomalous” in the sense that the solution behaves as exp(−Dt) with s > 1 rather than the “usual” exp(−Dt) homogenized behavior when the random potential is rapidly decorrelating. Unlike in rapidly decorrelating potentials, as we decrease the wavelength of the probing signal, stochasticity appears in the asymptotic limit – there exists a critical scale depending on the random potential which separates the deterministic and stochastic regimes.