The weak coupling limit for the random Schr\"odinger equation: The average wave function

TL;DR

This paper analyzes the weak coupling limit of the random Schrödinger equation with Gaussian potential, showing convergence to a deterministic limit and revealing anomalous diffusive behavior depending on the potential's correlation decay rate.

Contribution

It establishes the convergence of the averaged wave function to a deterministic limit under specific correlation decay conditions and identifies anomalous diffusion phenomena.

Findings

Convergence to a deterministic limit for the wave function when correlation decays slower than 1/|x|^2.

Identification of anomalous diffusive behavior on shorter time scales.

Limit results for the kinetic regime with faster decaying correlations.

Abstract

We consider the Schr\"odinger equation with a time-independent weakly random potential of a strength $\epsilon\ll 1$, with Gaussian statistics. We prove that when the initial condition varies on a scale much larger than the correlation length of the potential, the compensated wave function converges to a deterministic limit on the time scale $t\sim\epsilon^{-2}$. This is shown under the sharp assumption that the correlation function $R(x)$ of the random potential decays slower than $1/|x|^2$, which ensures that the effective potential is finite. When $R(x)$ decays slower than $1/|x|^2$ we establish an anomalous diffusive behavior for the averaged wave function on a time scale shorter than $\epsilon^{-2}$, as long as the initial condition is "sufficiently macroscopic". We also consider the kinetic regime when the initial condition varies on the same scale as the random potential and obtain the limit of the averaged wave function for potentials with the correlation functions decaying faster than $1/|x|^2$. We use random potentials of the Schonberg class which allows us to bypass the oscillatory phase estimates.