Multi-particle localization at low energy for the multi-dimensional continuous Anderson model

TL;DR

This paper proves that multi-particle Anderson models in the continuum exhibit spectral exponential and strong dynamical localization at low energies, under certain assumptions on the potential and interactions, using multi-scale analysis.

Contribution

It establishes almost sure constancy of the spectrum's lower edges and proves localization for multi-particle continuum models, extending previous results to more general settings.

Findings

Spectral exponential localization at low energy

Strong dynamical localization at low energy

Almost sure constancy of spectrum's lower edges

Abstract

We study the multi-particle Anderson model in the continuum and show that under some mild assumptions on the random external potential and the inter-particle interaction, for any finite number of particles, the multi-particle lower edges of the spectrum are almost surely constant in absence of ergodicity. We stress that this result is not quite obvious and has to be handled carefully. In addition, we prove the spectral exponential and the strong dynamical localization of the continuous multi-particle Anderson model at low energy. The proof based on the multi-particle multi-scale analysis bounds, needs the values of the external random potential to be independent and identically distributed (i.i.d.) whose common probability distribution is at least Log-H\"older continuous.