The random Schr\"odinger equation: homogenization in time-dependent potentials

TL;DR

This paper studies the behavior of solutions to the Schr"odinger equation with time-dependent random potentials, demonstrating homogenization for low frequencies and persistent randomness at high frequencies, with a kinetic equation describing energy transfer.

Contribution

It proves a homogenization result for low frequency components and characterizes the non-homogenized high frequency dynamics via a kinetic equation.

Findings

Low frequency wave components homogenize under weak random potentials.

High frequency components remain random and do not homogenize.

Energy transfer between frequencies is described by a kinetic equation.

Abstract

We analyze the solutions of the Schr\"odinger equation with the low frequency initial data and a time-dependent weakly random potential. We prove a homogenization result for the low frequency component of the wave field. We also show that the dynamics generates a non-trivial energy in the high frequencies, which do not homogenize -- the high frequency component of the wave field remains random and the evolution of its energy is described by a kinetic equation. The transition from the homogenization of the low frequencies to the random limit of the high frequencies is illustrated by understanding the size of the small random fluctuations of the low frequency component.