Anderson localization for weakly interacting multi-particle models in the continuum

TL;DR

This paper proves spectral and dynamical localization for weakly interacting multi-particle Anderson models in the continuum, demonstrating stability of single-particle bounds under small interactions using multi-scale analysis.

Contribution

It extends localization results to multi-particle continuum models, showing stability of single-particle bounds with small inter-particle interactions.

Findings

Spectral exponential localization established.

Strong dynamical localization proven.

Results hold under minimal regularity assumptions on the potential.

Abstract

For the weakly interacting one-dimensional multi-particle Anderson model in the continuum space of configurations, we prove the spectral exponential and the strong dynamical localization. The results require the interaction amplitude to be sufficiently small. The general strategy uses the multi-scale analysis bounds. Actually, we show that the multi-scale analysis bounds of the single particle model remain stable when passing to multi-particle systems, provided that the inter-particle interaction is sufficiently small. The common probability distribution of the i.i.d. random external potential in the Anderson model, is only needed to be log-H\"older continuous.