  We analyze the weak-coupling limit of the random Schr\"odinger 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 $\exp(-Dt^{s})$ 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.
