Communications in Mathematical Sciences

Volume 15 (2017)

Number 2

The random Schrödinger equation: Slowly decorrelating time-dependent potentials

Pages: 359 – 378

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n2.a4

Authors

Yu Gu (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)

Lenya Ryzhik (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)

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 $s \gt 1$ 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.

Keywords

random Schrödinger equation, long range correlation, homogenization, fractional Brownian motion

2010 Mathematics Subject Classification

34E05, 35B27, 35Q41

Published 21 February 2017