A note on fractional moments for the one-dimensional continuum Anderson model

TL;DR

This paper proves dynamical localization in the one-dimensional continuum Anderson model by demonstrating exponential decay of fractional moments of the Green function across all energies and single-site distributions.

Contribution

It provides a rigorous proof of dynamical localization using the fractional moments method for the continuum Anderson model.

Findings

Exponential decay of fractional moments of the Green function at all energies.

Dynamical localization characterized by exponential decay of spatial correlations.

Applicable to any single-site distribution with bounded, compact support.

Abstract

We give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the one-dimensional continuum Anderson model via the fractional moments method. This follows via exponential decay of fractional moments of the Green function, which is shown to hold at arbitrary energy and for any single-site distribution with bounded, compactly supported density.