The random Schr\"odinger equation: slowly decorrelating time-dependent potentials

TL;DR

This paper studies the behavior of solutions to the random Schr"odinger equation with slowly decorrelating potentials, revealing a transition from deterministic to stochastic regimes depending on the wavelength of the probing signal.

Contribution

It proves a homogenization result for low-frequency initial data in the slowly decorrelating potential regime, identifying an anomalous exponential decay and a critical scale for stochasticity emergence.

Findings

Deterministic limit exists for long-wavelength signals in the low-frequency regime.

The solution exhibits anomalous decay behavior of exp(-Dt^s) with s>1.

A critical wavelength scale separates deterministic and stochastic asymptotic regimes.

Abstract

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.