Optimal Wegner estimate and the density of states for N-body, interacting Schrodinger operators with random potentials

TL;DR

This paper establishes optimal Wegner estimates and analyzes the density of states for N-body interacting Schrödinger operators with random potentials, extending previous results and applying advanced unique continuation principles.

Contribution

It introduces an optimal one-volume Wegner estimate for interacting N-particle systems with random potentials, extending previous work and applying new unique continuation techniques.

Findings

Proved an optimal one-volume Wegner estimate for N-body systems.

Established Lipschitz continuity of the integrated density of states.

Extended results to Delone-Anderson type random potentials.

Abstract

We prove an optimal one-volume Wegner estimate for interacting systems of $N$ quantum particles moving in the presence of random potentials. The proof is based on the scale-free unique continuation principle recently developed for the 1-body problem by Rojas-Molina and Veseli\`c \cite{RM-V1} and extended to spectral projectors by Klein \cite{klein1}. These results extend of our previous results in \cite{CHK:2003,CHK:2007}. We also prove a two-volume Wegner estimate as introduced in \cite{chulaevsky-suhov1}. The random potentials are generalized Anderson-type potentials in each variable with minimal conditions on the single-site potential aside from positivity. Under additional conditions, we prove the Lipschitz continuity of the integrated density of states (IDS) This implies the existence and local finiteness of the density of states. We also apply these techniques to interacting $N$-particle Schr\"odinger operators with Delone-Anderson type random external potentials.