^{1,2}

^{3}

^{1}

^{2}

^{3}

For a system of

Recently,
models describing interacting quantum particles in a random potential have been
studied (see, e.g., [

First, we prove that if the Hamiltonian of the single
particle in the “homogeneous” media admits an integrated density of states
(IDS), then, so does the interacting

Note that, in general, knowledge of the integrated density of states is not yielding estimates for the normalized counting functions of the finite volume restrictions of the random operator; such information is also very valuable as it is a major tool in the study of the spectrum. Therefore, the second aim of this note is to provide estimates on the finite volume normalized counting function which lead to a Wegner estimate. The proof uses the ideology and tools developed for the one-particle Hamiltonian.

The
noninteracting

the operator

Classical models for which the IDS is known to exist
include periodic, quasiperiodic, and ergodic random Schrödinger operators (see,
e.g., [

In the definition of the density of states, we could also have considered the case of Neumann or other boundary conditions.

The interacting

Finally, we make one more assumption on both

the operator

We now
compute the IDS for the

Recall that, by assumption (H.1.b), the
single particle model

Let

The IDS for the noninteracting

The operator

Let us now say a word on the boundary conditions chosen to define
the IDS. Here, we chose to define it as an infinite-volume limit of the
normalized counting for Dirichlet eigenvalues. Clearly, if we know that the single
particle Hamiltonian has an IDS defined as the
infinite-volume limit of the normalized counting for Neumann eigenvalues, so
does the noninteracting

Let

Assume (H.1), (H.2), and (H.3) are satisfied. For any

Assume (H.1), (H.2), and (H.3) are satisfied. The IDS for the interacting

In Corollary

for the Fermi statistics, the Fermi integrated
density of states

for the Bose statistics, the Bose integrated
density of states

Assume (H.1), (H.2), and (H.3)
are satisfied. One has

We take some

By (

for any

By [

In the
interacting multiparticle Anderson model, we consider a random external
potential, that is,

One of the
interesting properties of the integrated density of states is its regularity;
it is well known to play an important role in the theory of localization for
random one-particle models (see, e.g., [

On the other
hand, Corollary

We now prove a Wegner estimate; for convenience, we assume the following.

The single-site
potential

For the proof of a Wegner estimate in the
interacting

Let us assume (H.A.2) and (H.A.3), and
let

Let

Let one assumes (H.A.2) and (H.A.3),
and let

For every

By (H.A.2) and (H.A.3), we get a

Under the assumptions (H.A.2) and (H.A.3), we
have