Diffusion in the Mean for an Ergodic Schr\"odinger Equation Perturbed by a Fluctuating Potential

TL;DR

This paper proves diffusive behavior and a central limit theorem for a quantum particle in a lattice with a combined static and fluctuating random potential, showing how the diffusion constant depends on the noise strength.

Contribution

It establishes diffusive scaling and a CLT for an ergodic Schrödinger equation with a mixed static and stochastic potential, revealing the impact of noise on localization.

Findings

Diffusive scaling of position moments is proven.

A central limit theorem for the mean position is established.

The diffusion constant scales quadratically with the noise strength.

Abstract

Diffusive scaling of position moments and a central limit theorem are obtained for the mean position of a quantum particle hopping on a cubic lattice and subject to a random potential consisting of a large static part and a small part that fluctuates stochastically in time. If the static random potential is strong enough to induce complete localization in the absence of time dependent noise, then the diffusion constant is shown to go to zero proportional to the square of the strength of the time dependent part.