Optimal Wegner Estimate and the Density of States for N-body, Interacting Schr\"odinger Operators with Random Potentials

#### P. D. Hislop, F. Klopp

2015, v.21, №3, 519-536

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\tire Molina and Veseli\c \cite{RM-V1} and extended to spectral projectors by Klein \cite{klein1}. These results extend 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\tire Anderson type random external potentials.
for $N$ interacting quantum particles moving in the presence of random potentials. The random potentials are generalized Anderson-type potentials in each variable with minimal conditions on the single-site potential aside from %positivity. The Lipschitz continuity of the IDS implies the existence and local finiteness of the density of states.
These results follow from an optimal one-volume Wegner estimate 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}. We also apply these techniques to interacting $N$-particle Schr\"odinger operators with Delone-Anderson type random external potentials.

Keywords: many-body quantum theory, random operators, localization