The random Schrödinger equation: Slowly decorrelating time-dependent potentials

Abstract

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 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.

Inverse scattering for a random potential

“…We refer to the works in [2,33] for various perspectives on wave propagation (whether classical or quantum) in random media. Let us also mention the papers on random Schrödinger models [3,27], where the potential model involving slowly decaying correlations corresponds closely to the random potential model in the present paper. Notice that our work does not involve assumptions on scaling regimes nor any approximations.…”