# The Integrated Density of States for  an Interacting

Authors: Frédéric Klopp; Heribert Zenk
Journal: Advances in Mathematical Physics
 (2009)

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## Abstract

For a system ofn interacting particles moving in the
background of a “homogeneous” potential, we show that if the
single particle Hamiltonian admits a density of states, so does the interacting n-particle Hamiltonian. Moreover, this integrated density of states
coincides with that of the free particle Hamiltonian. For the
interacting n-particle Anderson model, we prove regularity
properties of the integrated density of states by establishing a
Wegner estimate.


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## Body

## 1. Introduction

Recently,
models describing interacting quantum particles in a random potential have been
studied (see, e.g., [1–3]). We consider n interacting particles moving in a
“homogeneous” potential in the d-dimensional configuration space ℝd.
A typical example of what we mean by a “homogeneous” potential is an Anderson
or alloy-type random potential. The goal of the present
paper is twofold.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 interactingn-particle
Hamiltonian. The proof consists of two steps.
First, we prove the claim for the noninteracting n-particle system and in a second step, we show
that the IDS for noninteracting and interacting
system is the same. These two steps allow an
application to the interacting n-particle Anderson model in ℝd.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.


### 1.1. The Interacting Multiparticle Model

The
noninteractingn-particle Hamiltonian satisfies H0n=−Δ+Vextn where the Laplacian −Δ on ℝnd describes the free kinetic energy of the n particles. As all the particles are in the
same background, the potential Vextn is of the form(1.1)Vextn(x1,…,xn)=∑k=1nV1(xk).Hence, the noninteracting n-particle Hamiltonian is a sum of one-particle
Hamiltonians H1=−Δ+V1.
On the one particle potential V1,
we assume that(H.1.a)
(V1)+:=max{V1,0} is locally square integrable and (V1)−:=max{−V1,0} is an infinitesimally −Δ-bounded potential, that is, 𝒟((V1)−)⊇𝒟(−Δ) and for all α>0,
there exists γ(α)<∞,
such that for all ϕ∈𝒟(−Δ)(1.2)∥(V1)−ϕ∥≤α∥Δϕ∥+γ(α)∥ϕ∥,(H.1.b)
the operatorH1 admits an integrated density of states, say N1,
that is, if H0.L1 denotes the Dirichlet restriction of H1 to a cube Λ(0,L) centered at 0 of side-length, L,
then the following limit exists(1.3)N1(E):=limL→+∞L−dTrace(1]−∞,E](H0.L1)).
Assumption (H.1.a) implies essential self-adjointness of −Δ+V1 on C0∞(ℝd) by [4, Theorem X.29]. Indeed,(1.4)Vextn=(Vextn)+−(Vextn)−,(Vextn)±(x1,…,xn):=∑j=1n(V1)±(xj),where(i)
(Vextn)− is infinitesimally −Δ-bounded, that is, (1.2) holds for the same
constants and the Laplacian over ℝnd;(ii)
(Vextn)+ is nonnegative locally square integrable.
The self-adjoint extensions of −Δ+V1 and −Δ+Vextn are again denoted by H1 and H0n;
they are bounded from what follows.Classical models for which the IDS is known to exist
include periodic, quasiperiodic, and ergodic random Schrödinger operators (see,
e.g., [5]).In the definition of the density of states, we could
also have considered the case of Neumann or other boundary conditions.The interactingn-particle Hamiltonian is of the
form(1.5)Hn:=−Δ+Vin+Vextn,where(1.6)Vin(x1,…,xn):=∑1≤k<l≤nV(xk−xl)is a localized repulsive
interaction potential generated by the particles; so we assume
that(H.2)
V:ℝd→ℝ is measurable nonnegative locally square
integrable and V tends to 0 at infinity.
The standard
repulsive interaction in three-dimensional space is of course the Coulomb
interaction V(x)=1/|x|.
In some cases, due to screening, it must be replaced by the Yukawa's
interaction V(x)=e−|x|/|x|.Finally, we make one more assumption on bothV1 and V; we assume that(H.3)
the operatorVin(H0n−i)−1 is bounded.
Assumption (H.3) is satisfied in the case of the Coulomb and Yukawa potential for those V1 satisfying (H.1.a); H0n is self-adjoint on 𝒟(H0n)⊆𝒟(−Δ),
hence ∥Vin(H0n−i)−1∥≤∥Vin(−Δ−i)−1∥⋅∥(−Δ−i)(H0n−i)−1∥,
where ∥(−Δ−i)(H0n−i)−1∥<∞ due to closed graph theorem and ∥Vin(−Δ−i)−1∥<∞ for Coulomb and Yukawa's interaction
potentials Vin;
see [4, Theorem X.16].


## 1.1. The Interacting Multiparticle Model

The
noninteractingn-particle Hamiltonian satisfies H0n=−Δ+Vextn where the Laplacian −Δ on ℝnd describes the free kinetic energy of the n particles. As all the particles are in the
same background, the potential Vextn is of the form(1.1)Vextn(x1,…,xn)=∑k=1nV1(xk).Hence, the noninteracting n-particle Hamiltonian is a sum of one-particle
Hamiltonians H1=−Δ+V1.
On the one particle potential V1,
we assume that(H.1.a)
(V1)+:=max{V1,0} is locally square integrable and (V1)−:=max{−V1,0} is an infinitesimally −Δ-bounded potential, that is, 𝒟((V1)−)⊇𝒟(−Δ) and for all α>0,
there exists γ(α)<∞,
such that for all ϕ∈𝒟(−Δ)(1.2)∥(V1)−ϕ∥≤α∥Δϕ∥+γ(α)∥ϕ∥,(H.1.b)
the operatorH1 admits an integrated density of states, say N1,
that is, if H0.L1 denotes the Dirichlet restriction of H1 to a cube Λ(0,L) centered at 0 of side-length, L,
then the following limit exists(1.3)N1(E):=limL→+∞L−dTrace(1]−∞,E](H0.L1)).
Assumption (H.1.a) implies essential self-adjointness of −Δ+V1 on C0∞(ℝd) by [4, Theorem X.29]. Indeed,(1.4)Vextn=(Vextn)+−(Vextn)−,(Vextn)±(x1,…,xn):=∑j=1n(V1)±(xj),where(i)
(Vextn)− is infinitesimally −Δ-bounded, that is, (1.2) holds for the same
constants and the Laplacian over ℝnd;(ii)
(Vextn)+ is nonnegative locally square integrable.
The self-adjoint extensions of −Δ+V1 and −Δ+Vextn are again denoted by H1 and H0n;
they are bounded from what follows.Classical models for which the IDS is known to exist
include periodic, quasiperiodic, and ergodic random Schrödinger operators (see,
e.g., [5]).In the definition of the density of states, we could
also have considered the case of Neumann or other boundary conditions.The interactingn-particle Hamiltonian is of the
form(1.5)Hn:=−Δ+Vin+Vextn,where(1.6)Vin(x1,…,xn):=∑1≤k<l≤nV(xk−xl)is a localized repulsive
interaction potential generated by the particles; so we assume
that(H.2)
V:ℝd→ℝ is measurable nonnegative locally square
integrable and V tends to 0 at infinity.
The standard
repulsive interaction in three-dimensional space is of course the Coulomb
interaction V(x)=1/|x|.
In some cases, due to screening, it must be replaced by the Yukawa's
interaction V(x)=e−|x|/|x|.Finally, we make one more assumption on bothV1 and V; we assume that(H.3)
the operatorVin(H0n−i)−1 is bounded.
Assumption (H.3) is satisfied in the case of the Coulomb and Yukawa potential for those V1 satisfying (H.1.a); H0n is self-adjoint on 𝒟(H0n)⊆𝒟(−Δ),
hence ∥Vin(H0n−i)−1∥≤∥Vin(−Δ−i)−1∥⋅∥(−Δ−i)(H0n−i)−1∥,
where ∥(−Δ−i)(H0n−i)−1∥<∞ due to closed graph theorem and ∥Vin(−Δ−i)−1∥<∞ for Coulomb and Yukawa's interaction
potentials Vin;
see [4, Theorem X.16].


## 2. The Integrated Density of States

We now
compute the IDS for then-particle model. Let ΛL=Λ(0,L) be the cube in ℝd centered at 0 with side-length L and write ΛLn=ΛL×⋯×ΛL for the product of n copies of ΛL.
We denote the restriction of the interacting n-particle Hamiltonian Hn to ΛLn with Dirichlet boundary conditions by HLn.
Clearly assumptions (H.2) and (H.1.a) guarantee that HLn is bounded from what follows with compact resolvent.
Hence, for any E∈ℝ,
one defines the normalized counting functions(2.1)NL(E):=L−ndTrace(1]−∞,E](HLn)).As usual, N,
the IDS of Hn is defined as the limit of NL(E) when L→+∞.
Equivalently, one can define the density of states measure applied to a test
function φ as the limit of L−ndTrace[φ(HLn)].
If the limit exists, it defines a nonnegative measure. It is a classical result
that the existence of that limit (for all test functions) or that of NL(E)  is equivalent
[5].


### 2.1. The IDS for the Noninteractingn-Particle System

Recall that, by assumption (H.1.b), the
single particle modelH1 admits an IDS (see [5]) and a density of states
measure denoted, respectively, by N1 and ν1.LetH0,Ln be the restriction of H0n to ΛLn with Dirichlet boundary conditions. One has
the following lemma.Lemma 2.1.
The IDS for the noninteractingn-particle Boltzmann model given
by(2.2)Nni(E):=limL→∞1LndTrace(1]−∞,E](H0,Ln))exists and
satisfies(2.3)Nni=N1∗ν1∗⋯∗ν1. Let us comment on
this result. First, the convolution product in (2.3) makes sense as all the
measures and functions are supported on half-axes of the form [a,+∞);
this results from assumption (H.1.a). When the field V1 is not bounded from what follows, one will need some
estimate on the decay of N1 and ν1 near −∞ to make sense of (2.3) (and to prove it); such
estimates are known for some models (see, e.g., [5, 6]).Proof.
The operatorH0n is the sum of n commuting Hamiltonians, each of which is
unitarily equivalent to H1;
so is H0,Ln,
its restriction to the cube ΛLn.
As the sum decomposition of H0n commutes with the restriction to ΛLn,
the eigenvalues of H0,Ln are exactly the sum of n eigenvalues of H1 restricted to ΛL.
This immediately yields that(2.4)Trace(1]−∞,E](H0,Ln))=(N^1L*ν^1L*⋯*ν^1L)(E),where N^1L(E) is the eigenvalue counting function for H1 restricted to ΛL,
and ν^1L  is its counting
measure (i.e., dN^1L). The normalized counting function and
measure, N1L and ν1L,
are defined as(2.5)N1L=1LdN^1L,ν1L=1Ldν^1L.The existence of the density of
states of H1 then exactly says that N1L and ν1L converge, respectively, to N1 and ν1.
The convergence of N1L*ν1L*⋯*ν1L to N1*ν1*⋯*ν1 is then guaranteed as the convolution is
bilinear bicontinuous operation on distributions. This completes the proof of Lemma 2.1.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 noninteractingn-body Hamiltonian. Moreover, in the case when
the two limits coincide for the one-body Hamiltonian, they also coincide for
the noninteracting n-body Hamiltonian. Using Dirichlet-Neumann
bracketing, one then sees that the integrated densities of states for both the
one-body and noninteracting n-body Hamiltonian for positive mixed boundary
conditions also exist and coincide with that defined with either Dirichlet or
Neumann boundary conditions.


### 2.2. Existence of the IDS for the Interactingn-Particle System

LetHLn denote the restriction of Hn to the box ΛLn with Dirichlet boundary conditions. Our main
result is.Theorem 2.2.
Assume (H.1), (H.2), and (H.3) are satisfied. For anyφ∈C0∞(ℝ),
one has(2.6)1LndTrace[φ(HLn)−φ(H0,Ln)]→L→∞0.As the density of
states measure of Hn is defined by(2.7)〈φ,dN〉=limL→+∞1LndTrace[φ(HLn)],we immediately get
the following corollary.Corollary 2.3.
Assume (H.1), (H.2), and (H.3) are satisfied. The IDS for the interactingn-particle Boltzmann model Hn exists and coincides with that of the
noninteracting model H0n;
hence, it satisfies(2.8)N=Nni=N1∗ν1∗⋯∗ν1.Note that, in view of
the remark concluding Section 2.1, we see that the integrated density of states
of the interacting n-body Hamiltonian is independent of the
boundary conditions if that of the one-body Hamiltonian is.In Corollary2.3, we dealt with the Boltzmann
statistic, that is, without statistic. Theorem 2.2 stays clearly true for both
the Fermi and the Bose statistics, that is, if one restricts to the subspaces
of symmetric and antisymmetric functions. One defines the
following:(i)
for the Fermi statistics, the Fermi integrated
density of states(2.9)〈φ,dNF〉=limL→+∞n!LndTrace⋀nL2(ΛL1)[φ(HLn)],where ⋀nL2(ΛL1) denotes n-fold antisymmetric tensor product of L2(ΛL1);(ii)
for the Bose statistics, the Bose integrated
density of states(2.10)〈φ,dNB〉=limL→+∞n!LndTrace⊕snL2(ΛL1)[φ(HLn)],where ⊕snL2(ΛL1) denotes n-fold symmetric tensor product of L2(ΛL1).
Let us now discuss shortly the Bose and Fermi counting functions (i.e., the eigenvalue
counting functions of the Hamiltonian restricted to a finite cube) in the free
case (i.e., when the interaction vanishes). Consider the cube ΛL1 and let E1(L)≤E2(L)≤⋯ be the eigenvalue of the single particle
Hamiltonian repeated according to multiplicity. The three counting functions
are then given by(2.11)#L(E):=#{eigenvaluesofH0,LnonL2(ΛLn)lessthanE}=#{(j1,j2,…,jn):Ej1(L)+Ej2(L)+⋯+Ejn(L)≤E},#LF(E):=#{eigenvaluesofH0,Lnon⋀nL2(ΛL1)lessthanE}=#{(j1,j2,…,jn):j1<j2<⋯<jn,Ej1(L)+⋯+Ejn(L)≤E},#LB(E):=#{eigenvaluesofH0,Lnon⊕snL2(ΛL1)lessthanE}=#{(j1,j2,…,jn):j1≤j2≤⋯≤jn,Ej1(L)+⋯+Ejn(L)≤E}.Hence,(2.12)n!#LF(E)≤#L(E)≤n!#LB(E).Uniformly in L,
the eigenvalues (Ej(L))j≥1 are lower bounded by, say, −C.
Hence, if Ej1(L)+Ej2(L)+⋯+Ejn(L)≤E,
then, for k=1,…,n,
one has Ejk(L)≤E+Cn so that jk≤N^1L(E+Cn)=N1L(E+Cn)Ld.
This implies that(2.13)0≤#LB(E)−#LF(E)=#{(j1,j2,…,jn);j1≤j2≤⋯≤jn,∃k<ls.t.jk=jlEj1(L)+Ej2(L)+⋯+Ejn(L)≤E}≤C˜Ld(n−1).Thus, dividing (2.12) and (2.13)
by Lnd and taking the limit L→+∞,
we obtain that the free Fermi and Bose density of states are equal to the
Boltzmann one. Theorem 2.2 then gives the following
corollary.Corollary 2.4.
Assume (H.1), (H.2), and (H.3)
are satisfied. One hasN=NB=NF.Proof.
We take someq>nd/2 and specify the appropriate choice later on.
By assumptions (H.1.a) and (H.2), there exists ζ>0 such that(2.14)−∞<−ζ≤min(infL≥1{inf[σ(H0,Ln)∪σ(HLn)]},inf[σ(H0n)∪σ(Hn)]).Let γ=γ(1/2) be given by (1.2) for α=1/2.
Fix λ0>ζ+2γ+1.
By (2.14), we only need to prove (2.6) for φ∈C0∞(ℝ) supported in (−ζ−1,+∞).
For such a function, let φ˜ be an almost analytic extension of the
function x↦(x+λ0)qφ(x)∈C0∞(ℝ),
that is, φ˜ satisfies
(i)
φ˜∈𝒮({z∈ℂ:|ℑz|<1},(ii)
for anyk∈ℕ,
the family of functions (x↦(∂φ˜/∂z¯)(x+iy)|y|−k)0<|y|<1 is bounded in 𝒮(ℝ).
The functional
calculus based on the Helffer-Sjöstrand formula implies(2.15)φ(HLn)−φ(H0,Ln)=i2π∫ℂ∂φ˜∂z¯(z)[(HLn+λ0)−q(HLn−z)−1−(H0,Ln+λ0)−q(H0,Ln−z)−1]dz⋀dz¯.In the following, we apply an
idea, which has already been used in [6, 7] and which simplifies in this situation. Using
resolvent equality, the integrand in (2.15) is written as(2.16)(HLn+λ0)−q(HLn−z)−1−(H0,Ln+λ0)−q(H0,Ln−z)−1=(H0,Ln+λ0)−q[(HLn−z)−1−(H0,Ln−z)−1]+[(HLn+λ0)−q−(H0,Ln+λ0)−q](HLn−z)−1=−(H0,Ln+λ0)−q(H0,Ln−z)−1(Vin)(HLn−z)−1−∑l=1q(H0,Ln+λ0)l−q−1(Vin)(HLn+λ0)−l(HLn−z)−1.Estimating the trace of (2.16),
we choose ε>0 and write(2.17)Vin=Vin⋅1{|Vin|≤ε}+Vin⋅1{|Vin|>ε}and note that Vin⋅1{|Vin|≤ε} is bounded by ∥Vin⋅1{|Vin|≤ε}∥≤ε.
As V is nonnegative, one has(2.18)supp(Vin⋅1{|Vin|>ε})⊆⋃j=1n⋃i=1i≠jn{(x1,…,xn)∈ℝnd:V(xi−xj)≥εn(n−1)}.As, by assumption (H.2), V tends to 0 at infinity, (2.18) implies that there exists 0<C(n;ε) (independent of L) such that(2.19)μ({|Vin|>ε}∩ΛLn)≤C(n,ε)L(n−1)d,where μ(⋅) denotes the Lebesgue measure. Using
decomposition (2.17) of Vin,
we obtain(2.20)|Trace(H0,Ln+λ0)−q(H0,Ln−z)−1(Vin)(HLn−z)−1|≤ε|ℑz|2Trace|(H0,Ln+λ0)−q|+1|ℑz|∥(Vin)(HLn−z)−1∥⋅Trace|(H0,Ln+λ0)−q1{|Vin|>ε}∩ΛLn|≤ε|ℑz|2∥(H0,Ln+λ0)−1∥𝒯qq+1|ℑz|2∥(H0,Ln+λ0)−1∥𝒯qq−1⋅∥(H0,Ln+λ0)−11{|Vin|>ε}∩ΛLn∥𝒯q⋅∥(Vin)(H0,Ln+λ0)−1∥,where ∥⋅∥𝒯q denotes the qth Schatten class norm (see [8]) and we used Hölder's
inequality. In the same way, the cyclicity of the trace yields(2.21)|Trace(H0,Ln+λ0)l−q−1(Vin)(HLn+λ0)−l(HLn−z)−1|≤Trace|(HLn+λ0)−l(H0,Ln+λ0)l−q−1(Vin)(HLn−z)−1|≤∥(HLn+λ0)−l(H0,Ln+λ0)l∥⋅Trace|(H0,Ln+λ0)−q−1(Vin)(HLn−z)−1|≤C|ℑz|∥(H0,Ln+λ0)−1∥𝒯qq−1⋅∥(H0,Ln+λ0)−11{|Vin|>ε}∩ΛLn∥𝒯q⋅∥(Vin)(H0,Ln+λ0)−1∥+Cε|ℑz|∥(H0,Ln+λ0)−1∥𝒯qq.We are now left with estimating ∥(H0,Ln+λ0)−1∥𝒯q and ∥(H0,Ln+λ0)−11{|Vin|>ε}∩ΛLn∥𝒯q for q sufficiently large, depending on nd.
Therefore, we compute(2.22)∥(H0,Ln+λ0)−11{|Vin|>ε}∩ΛLn∥𝒯q≤∥(H0,Ln+λ0)−1(−ΔΛLn+λ0)1/2∥𝒯2q⋅∥(−ΔΛLn+λ0)−(1/2)1{|Vin|>ε}∩ΛLn∥𝒯2q,where −ΔΛLn is the Dirichlet Laplacian on ΛLn.
We use the decomposition (1.4). As the Laplacians are positive, the
infinitesimal −Δ-boundedness on (Vextn)−, [4, Theorem X.18]
and the definition of γ imply the following form
bound:(2.23)|〈ϕ,(Vextn)−ϕ〉|≤12〈ϕ,−ΔΛLnϕ〉+γ∥ϕ∥2.As λ0>2γ+1,
one has(2.24)H0,Ln+λ0≥−ΔΛLn+(Vextn)−+λ0≥12(−ΔΛLn−2γ+2λ0)≥12(−ΔΛLn+λ0).Thus, the operator H0,Ln+λ0 is invertible and(2.25)(H0,Ln+λ0)−1≤2(−ΔΛLn+λ0)−1.Let (μj̲)j̲ and (ϕj̲)j̲,
respectively, denote the eigenvalues and eigenfunctions of the Dirichlet
Laplacian −ΔΛLn (the index j̲ runs over (ℕnd)*). For q∈ℕ such that 2q>nd,
we compute(2.26)∥(H0,Ln+λ0)−1(−ΔΛLn+λ0)1/2∥τ2q2q=∑j̲∈ℕnd(μj̲(−ΔΛLn)+λ0)q〈ϕj̲,(H0,Ln+λ0)−1ϕj̲〉2q≤22q∑j̲∈ℕnd(μj̲(−ΔΛLn)+λ0)q〈ϕj̲,(−ΔΛLn+λ0)−1ϕj̲〉)2q=22q∑j̲∈ℕnd(μj̲(−ΔΛLn)+λ0)−q≤CLnd.The last estimate is a direct
computation using the explicit form of the Dirichlet eigenvalues.
By [6, Lemma 2.2], we know that, for q∈ℕ such that 2q>nd,
there exists Cq>0 such that, for any measurable subset Λ′⊆ΛLn,
one has(2.27)∥(−ΔΛLn+λ0)−1/21Λ′∥𝒯2q2q≤Cqμ(Λ′).Choosing Λ′={|Vin|>ε}∩ΛLn and taking (2.19) into account, then by
combining estimates (2.20)–(2.27), we get that there exists c,
depending only on q (and the bound in assumption (H.3)), such
that(2.28)Trace|(HLn+λ0)−q(HLn−z)−1−(H0,Ln+λ0)−q(H0,Ln−z)−1|≤c(ε|ℑz|2Lnd+1|ℑz|2Lnd−(d/2q)+ε|ℑz|Lnd+1|ℑz|Lnd−(d/2q)).By using this inequality in
(2.15), we get (2.6) as φ˜ being almost analytic, ∂¯φ˜(z) vanishes to any order in ℑz  as z approaches the real line. Thus, we completed
the proof of Theorem 2.2.


## 2.1. The IDS for the Noninteractingn-Particle System

Recall that, by assumption (H.1.b), the
single particle modelH1 admits an IDS (see [5]) and a density of states
measure denoted, respectively, by N1 and ν1.LetH0,Ln be the restriction of H0n to ΛLn with Dirichlet boundary conditions. One has
the following lemma.Lemma 2.1.
The IDS for the noninteractingn-particle Boltzmann model given
by(2.2)Nni(E):=limL→∞1LndTrace(1]−∞,E](H0,Ln))exists and
satisfies(2.3)Nni=N1∗ν1∗⋯∗ν1. Let us comment on
this result. First, the convolution product in (2.3) makes sense as all the
measures and functions are supported on half-axes of the form [a,+∞);
this results from assumption (H.1.a). When the field V1 is not bounded from what follows, one will need some
estimate on the decay of N1 and ν1 near −∞ to make sense of (2.3) (and to prove it); such
estimates are known for some models (see, e.g., [5, 6]).Proof.
The operatorH0n is the sum of n commuting Hamiltonians, each of which is
unitarily equivalent to H1;
so is H0,Ln,
its restriction to the cube ΛLn.
As the sum decomposition of H0n commutes with the restriction to ΛLn,
the eigenvalues of H0,Ln are exactly the sum of n eigenvalues of H1 restricted to ΛL.
This immediately yields that(2.4)Trace(1]−∞,E](H0,Ln))=(N^1L*ν^1L*⋯*ν^1L)(E),where N^1L(E) is the eigenvalue counting function for H1 restricted to ΛL,
and ν^1L  is its counting
measure (i.e., dN^1L). The normalized counting function and
measure, N1L and ν1L,
are defined as(2.5)N1L=1LdN^1L,ν1L=1Ldν^1L.The existence of the density of
states of H1 then exactly says that N1L and ν1L converge, respectively, to N1 and ν1.
The convergence of N1L*ν1L*⋯*ν1L to N1*ν1*⋯*ν1 is then guaranteed as the convolution is
bilinear bicontinuous operation on distributions. This completes the proof of Lemma 2.1.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 noninteractingn-body Hamiltonian. Moreover, in the case when
the two limits coincide for the one-body Hamiltonian, they also coincide for
the noninteracting n-body Hamiltonian. Using Dirichlet-Neumann
bracketing, one then sees that the integrated densities of states for both the
one-body and noninteracting n-body Hamiltonian for positive mixed boundary
conditions also exist and coincide with that defined with either Dirichlet or
Neumann boundary conditions.


## 2.2. Existence of the IDS for the Interactingn-Particle System

LetHLn denote the restriction of Hn to the box ΛLn with Dirichlet boundary conditions. Our main
result is.Theorem 2.2.
Assume (H.1), (H.2), and (H.3) are satisfied. For anyφ∈C0∞(ℝ),
one has(2.6)1LndTrace[φ(HLn)−φ(H0,Ln)]→L→∞0.As the density of
states measure of Hn is defined by(2.7)〈φ,dN〉=limL→+∞1LndTrace[φ(HLn)],we immediately get
the following corollary.Corollary 2.3.
Assume (H.1), (H.2), and (H.3) are satisfied. The IDS for the interactingn-particle Boltzmann model Hn exists and coincides with that of the
noninteracting model H0n;
hence, it satisfies(2.8)N=Nni=N1∗ν1∗⋯∗ν1.Note that, in view of
the remark concluding Section 2.1, we see that the integrated density of states
of the interacting n-body Hamiltonian is independent of the
boundary conditions if that of the one-body Hamiltonian is.In Corollary2.3, we dealt with the Boltzmann
statistic, that is, without statistic. Theorem 2.2 stays clearly true for both
the Fermi and the Bose statistics, that is, if one restricts to the subspaces
of symmetric and antisymmetric functions. One defines the
following:(i)
for the Fermi statistics, the Fermi integrated
density of states(2.9)〈φ,dNF〉=limL→+∞n!LndTrace⋀nL2(ΛL1)[φ(HLn)],where ⋀nL2(ΛL1) denotes n-fold antisymmetric tensor product of L2(ΛL1);(ii)
for the Bose statistics, the Bose integrated
density of states(2.10)〈φ,dNB〉=limL→+∞n!LndTrace⊕snL2(ΛL1)[φ(HLn)],where ⊕snL2(ΛL1) denotes n-fold symmetric tensor product of L2(ΛL1).
Let us now discuss shortly the Bose and Fermi counting functions (i.e., the eigenvalue
counting functions of the Hamiltonian restricted to a finite cube) in the free
case (i.e., when the interaction vanishes). Consider the cube ΛL1 and let E1(L)≤E2(L)≤⋯ be the eigenvalue of the single particle
Hamiltonian repeated according to multiplicity. The three counting functions
are then given by(2.11)#L(E):=#{eigenvaluesofH0,LnonL2(ΛLn)lessthanE}=#{(j1,j2,…,jn):Ej1(L)+Ej2(L)+⋯+Ejn(L)≤E},#LF(E):=#{eigenvaluesofH0,Lnon⋀nL2(ΛL1)lessthanE}=#{(j1,j2,…,jn):j1<j2<⋯<jn,Ej1(L)+⋯+Ejn(L)≤E},#LB(E):=#{eigenvaluesofH0,Lnon⊕snL2(ΛL1)lessthanE}=#{(j1,j2,…,jn):j1≤j2≤⋯≤jn,Ej1(L)+⋯+Ejn(L)≤E}.Hence,(2.12)n!#LF(E)≤#L(E)≤n!#LB(E).Uniformly in L,
the eigenvalues (Ej(L))j≥1 are lower bounded by, say, −C.
Hence, if Ej1(L)+Ej2(L)+⋯+Ejn(L)≤E,
then, for k=1,…,n,
one has Ejk(L)≤E+Cn so that jk≤N^1L(E+Cn)=N1L(E+Cn)Ld.
This implies that(2.13)0≤#LB(E)−#LF(E)=#{(j1,j2,…,jn);j1≤j2≤⋯≤jn,∃k<ls.t.jk=jlEj1(L)+Ej2(L)+⋯+Ejn(L)≤E}≤C˜Ld(n−1).Thus, dividing (2.12) and (2.13)
by Lnd and taking the limit L→+∞,
we obtain that the free Fermi and Bose density of states are equal to the
Boltzmann one. Theorem 2.2 then gives the following
corollary.Corollary 2.4.
Assume (H.1), (H.2), and (H.3)
are satisfied. One hasN=NB=NF.Proof.
We take someq>nd/2 and specify the appropriate choice later on.
By assumptions (H.1.a) and (H.2), there exists ζ>0 such that(2.14)−∞<−ζ≤min(infL≥1{inf[σ(H0,Ln)∪σ(HLn)]},inf[σ(H0n)∪σ(Hn)]).Let γ=γ(1/2) be given by (1.2) for α=1/2.
Fix λ0>ζ+2γ+1.
By (2.14), we only need to prove (2.6) for φ∈C0∞(ℝ) supported in (−ζ−1,+∞).
For such a function, let φ˜ be an almost analytic extension of the
function x↦(x+λ0)qφ(x)∈C0∞(ℝ),
that is, φ˜ satisfies
(i)
φ˜∈𝒮({z∈ℂ:|ℑz|<1},(ii)
for anyk∈ℕ,
the family of functions (x↦(∂φ˜/∂z¯)(x+iy)|y|−k)0<|y|<1 is bounded in 𝒮(ℝ).
The functional
calculus based on the Helffer-Sjöstrand formula implies(2.15)φ(HLn)−φ(H0,Ln)=i2π∫ℂ∂φ˜∂z¯(z)[(HLn+λ0)−q(HLn−z)−1−(H0,Ln+λ0)−q(H0,Ln−z)−1]dz⋀dz¯.In the following, we apply an
idea, which has already been used in [6, 7] and which simplifies in this situation. Using
resolvent equality, the integrand in (2.15) is written as(2.16)(HLn+λ0)−q(HLn−z)−1−(H0,Ln+λ0)−q(H0,Ln−z)−1=(H0,Ln+λ0)−q[(HLn−z)−1−(H0,Ln−z)−1]+[(HLn+λ0)−q−(H0,Ln+λ0)−q](HLn−z)−1=−(H0,Ln+λ0)−q(H0,Ln−z)−1(Vin)(HLn−z)−1−∑l=1q(H0,Ln+λ0)l−q−1(Vin)(HLn+λ0)−l(HLn−z)−1.Estimating the trace of (2.16),
we choose ε>0 and write(2.17)Vin=Vin⋅1{|Vin|≤ε}+Vin⋅1{|Vin|>ε}and note that Vin⋅1{|Vin|≤ε} is bounded by ∥Vin⋅1{|Vin|≤ε}∥≤ε.
As V is nonnegative, one has(2.18)supp(Vin⋅1{|Vin|>ε})⊆⋃j=1n⋃i=1i≠jn{(x1,…,xn)∈ℝnd:V(xi−xj)≥εn(n−1)}.As, by assumption (H.2), V tends to 0 at infinity, (2.18) implies that there exists 0<C(n;ε) (independent of L) such that(2.19)μ({|Vin|>ε}∩ΛLn)≤C(n,ε)L(n−1)d,where μ(⋅) denotes the Lebesgue measure. Using
decomposition (2.17) of Vin,
we obtain(2.20)|Trace(H0,Ln+λ0)−q(H0,Ln−z)−1(Vin)(HLn−z)−1|≤ε|ℑz|2Trace|(H0,Ln+λ0)−q|+1|ℑz|∥(Vin)(HLn−z)−1∥⋅Trace|(H0,Ln+λ0)−q1{|Vin|>ε}∩ΛLn|≤ε|ℑz|2∥(H0,Ln+λ0)−1∥𝒯qq+1|ℑz|2∥(H0,Ln+λ0)−1∥𝒯qq−1⋅∥(H0,Ln+λ0)−11{|Vin|>ε}∩ΛLn∥𝒯q⋅∥(Vin)(H0,Ln+λ0)−1∥,where ∥⋅∥𝒯q denotes the qth Schatten class norm (see [8]) and we used Hölder's
inequality. In the same way, the cyclicity of the trace yields(2.21)|Trace(H0,Ln+λ0)l−q−1(Vin)(HLn+λ0)−l(HLn−z)−1|≤Trace|(HLn+λ0)−l(H0,Ln+λ0)l−q−1(Vin)(HLn−z)−1|≤∥(HLn+λ0)−l(H0,Ln+λ0)l∥⋅Trace|(H0,Ln+λ0)−q−1(Vin)(HLn−z)−1|≤C|ℑz|∥(H0,Ln+λ0)−1∥𝒯qq−1⋅∥(H0,Ln+λ0)−11{|Vin|>ε}∩ΛLn∥𝒯q⋅∥(Vin)(H0,Ln+λ0)−1∥+Cε|ℑz|∥(H0,Ln+λ0)−1∥𝒯qq.We are now left with estimating ∥(H0,Ln+λ0)−1∥𝒯q and ∥(H0,Ln+λ0)−11{|Vin|>ε}∩ΛLn∥𝒯q for q sufficiently large, depending on nd.
Therefore, we compute(2.22)∥(H0,Ln+λ0)−11{|Vin|>ε}∩ΛLn∥𝒯q≤∥(H0,Ln+λ0)−1(−ΔΛLn+λ0)1/2∥𝒯2q⋅∥(−ΔΛLn+λ0)−(1/2)1{|Vin|>ε}∩ΛLn∥𝒯2q,where −ΔΛLn is the Dirichlet Laplacian on ΛLn.
We use the decomposition (1.4). As the Laplacians are positive, the
infinitesimal −Δ-boundedness on (Vextn)−, [4, Theorem X.18]
and the definition of γ imply the following form
bound:(2.23)|〈ϕ,(Vextn)−ϕ〉|≤12〈ϕ,−ΔΛLnϕ〉+γ∥ϕ∥2.As λ0>2γ+1,
one has(2.24)H0,Ln+λ0≥−ΔΛLn+(Vextn)−+λ0≥12(−ΔΛLn−2γ+2λ0)≥12(−ΔΛLn+λ0).Thus, the operator H0,Ln+λ0 is invertible and(2.25)(H0,Ln+λ0)−1≤2(−ΔΛLn+λ0)−1.Let (μj̲)j̲ and (ϕj̲)j̲,
respectively, denote the eigenvalues and eigenfunctions of the Dirichlet
Laplacian −ΔΛLn (the index j̲ runs over (ℕnd)*). For q∈ℕ such that 2q>nd,
we compute(2.26)∥(H0,Ln+λ0)−1(−ΔΛLn+λ0)1/2∥τ2q2q=∑j̲∈ℕnd(μj̲(−ΔΛLn)+λ0)q〈ϕj̲,(H0,Ln+λ0)−1ϕj̲〉2q≤22q∑j̲∈ℕnd(μj̲(−ΔΛLn)+λ0)q〈ϕj̲,(−ΔΛLn+λ0)−1ϕj̲〉)2q=22q∑j̲∈ℕnd(μj̲(−ΔΛLn)+λ0)−q≤CLnd.The last estimate is a direct
computation using the explicit form of the Dirichlet eigenvalues.
By [6, Lemma 2.2], we know that, for q∈ℕ such that 2q>nd,
there exists Cq>0 such that, for any measurable subset Λ′⊆ΛLn,
one has(2.27)∥(−ΔΛLn+λ0)−1/21Λ′∥𝒯2q2q≤Cqμ(Λ′).Choosing Λ′={|Vin|>ε}∩ΛLn and taking (2.19) into account, then by
combining estimates (2.20)–(2.27), we get that there exists c,
depending only on q (and the bound in assumption (H.3)), such
that(2.28)Trace|(HLn+λ0)−q(HLn−z)−1−(H0,Ln+λ0)−q(H0,Ln−z)−1|≤c(ε|ℑz|2Lnd+1|ℑz|2Lnd−(d/2q)+ε|ℑz|Lnd+1|ℑz|Lnd−(d/2q)).By using this inequality in
(2.15), we get (2.6) as φ˜ being almost analytic, ∂¯φ˜(z) vanishes to any order in ℑz  as z approaches the real line. Thus, we completed
the proof of Theorem 2.2.


## 3. Application to the Interacting Multiparticle Anderson Model

In the
interacting multiparticle Anderson model, we consider a random external
potential, that is,V1=V1(ω).
The one particle Anderson potential is of the form(3.1)V1(ω,x)=∑j∈ℤdωju(x−j),with a family ωj:Ω→ℝ of random variables on (Ω,ℙ).
This one-particle models leads us to the n-particle random “background”
potential(3.2)Vn(ω,x1,…,xn)=∑k=1nV1(ω,xk)and the interacting n-particle Hamiltonian reads as(3.3)Hn(ω)=−Δ+Vin+Vn(ω).For the Anderson model, it is
known under rather general assumptions that, for a given energy, the normalized
counting function defined in assumption (H.1.b) converges almost surely (see,
e.g., [5, 9]). The limit is a
nondecreasing function of E.
Its discontinuity set is countable. By [9, pp. 311f], for almost every ω,
except at this set, the normalized counting function defined in assumption
(H.1.b) then converges. On this set of full measure, we can now apply the
results of the last section and get a ℙ-almost sure integrated density of states Nni=N for both, the noninteracting and interacting n-particle system. Note that only translations along a “diagonal”
vector (j,j,…,j)∈ℤnd leave Hn(ω) invariant. Hence, for an application of
ergodic theorems (as in the one particle case) for the proof of existence and ℙ-almost sure constancy of N,
there are typically too few ergodic transformations.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., [10]). Usually, it comes into play through a Wegner
estimate, that is, an estimate of the type(3.4)𝔼(Trace1]E0,E0+η](HΛn))≤CWη|Λ|.On the other
hand, Corollary2.3 directly relates the regularity of the IDS of the
interacting system to that of the IDS of the single particle Hamiltonian. The
regularity of the IDS of the single particle has been the subject of a lot of
interest recently (see, e.g., [11, 12]).We now prove a
Wegner estimate; for convenience, we assume the following.(H.A.2)
The single-site
potentialu is nonnegative, compactly supported, ∥u∥L∞≤1 and that there is some c>0,
such that u(x)≥c for x∈[−(1/2),1/2]d.For the proof of a Wegner estimate in the
interactingn-particle Anderson model, we can choose rather
general probabilistic hypothesis like in [13]:(H.A.3)
(ωj:Ω→ℝ)j∈ℤd is a family of bounded random variables on the
probability space (Ω,ℙ).
When μj denotes the conditional probability measure
for ωj at site j∈ℤd conditioned on all the other random variables (ωi)i≠j,
that is, for all A∈ℬ(ℝ),(3.5)μj(A)=ℙ({ωj∈A∣(ωi)i≠j}),then, a Wegner estimate à
la [13] uses the quantity(3.6)s(η):=supj∈ℤd𝔼{supE∈ℝμj([E,E+η])}and is stated as
follows.Theorem 3.1.
Let us assume (H.A.2) and (H.A.3), and
letΛ⊆ℝnd be a bounded open cube of side length ≥1, let HΛn(ω)  be  the restriction
of Hn(ω) to Λ with Dirichlet boundary conditions. Then,
there exists an increasing function(3.7)CW:ℝ→[0,∞[E0↦CW(E0),such that for all η>0(3.8)𝔼(Trace1]E0,E0+η](HΛn))≤CW(E0)s(η)|Λ|.In
order to prove Theorem 3.1, we prove two preparatory lemmas.Lemma 3.2.
LetΛ⊆ℝnd be an open bounded cube, then the restrictions Hi,Λn and Hi,Λ,Nn of Hin=−Δ+Vin to Λ with Dirichlet or
Neumann boundary conditions define self-adjoint operators with compact
resolvent.Proof.
Vin is infinitesimally −Δ form bounded according to [4, Theorem X.18], so the
infinitesimal form bound(3.9)|〈Ψ,VinΨ〉|≤ε∥∇Ψ∥2+bε∥Ψ∥2is true for Ψ∈H1(ℝnd),
in particular (3.9) is true for Ψ∈𝒟(−ΔΛ)=H01(Λ)⊆H1(ℝnd).
Hence, the form sum defines via representation theorem a self-adjoint operator Hi,Λn=−ΔΛ+Vin|Λ.
The eigenvalues μk(−ΔΛ) tend to infinity, so by the minimax principle
and (3.9), we see that Hi,Λn has compact resolvent. The proof
of(3.10)〈Ψ,VinΨ〉≤ε∥∇Ψ∥2+cε∥Ψ∥2,Ψ∈H1(Λ)uses the extension operator EΛ′:H1(Λ)→H01(Λ′) to Λ′:={x∈ℝnd:dist(x,Λ)<1},
which has the properties EΛ′Ψ|Λ=Ψ,∥EΛ′Ψ∥H1≤c1∥Ψ∥H1 and ∥EΛ′Ψ∥L2≤c2∥Ψ∥L2;
see [14, Satz 5.6 and
Folgerung 5.2]. For EΛ′Ψ∈H01(Λ′)⊆H1(ℝnd),
we use (3.9), hence by Vin≥0 and the above properties of EΛ′ we get for Ψ∈H1(Λ),(3.11)0≤〈Ψ,VinΨ〉≤〈EΛ′Ψ,VinEΛ′Ψ〉≤ε∥∇(EΛ′Ψ)∥L22+bε∥EΛ′Ψ∥L22=ε∥(EΛ′Ψ)∥H12+(bε−ε)∥EΛ′Ψ∥L22≤εc12∥∇Ψ∥2+(c22(bε−ε)+εc12)∥Ψ∥2,which is (3.10). With (3.10) at
hand, the proof for Neumann boundary conditions is similar to the Dirichlet
case.Lemma 3.3.
Let one assumes (H.A.2) and (H.A.3),
and letΛ⊆ℝnd be a bounded open cube, j=(j1,…,jn)∈ℤnd with Λj:=Λ∩Λ(j,1)≠∅ (here, Λ(j,1)={|x−jk|≤1/2,1≤k≤n}), then for every f∈L2(Λj),(3.12)𝔼{〈f,1]E0,E0+η](HΛn)f〉}≤8c2s(η)∥f∥2.Proof.
For everyj∈ℤd,
we define uj:ℝnd→ℝ by(3.13)uj(x1,…,xn):=∑k=1nu(xk−j)and set ω˜j=(ωl)l≠j.
Fix a component of j,
say j1,
then we get a decomposition(3.14)Vn(ω,x1,…,xn)=ωj1uj1(x1,…,xn)+∑l∈ℤdl≠j1ωlul(x1,…,xn)of the random potential Vn(ω),
and the same is true for HΛn(ω):(3.15)HΛn(ω)=−ΔΛ+∑l∈ℤdl≠j1ωlul1Λ+ωj1uj11Λ=:H˜Λn(ω˜j1)+ωj1uj11Λ.By the covering condition u1[−1/2,1/2]d≥c on the single site-potential u,
we get uj1≥c1Λj,
hence we can write f=guj1,
where g(x)=f(x)/uj1(x) almost everywhere, so ∥g∥≤c−1∥f∥.
By spectral calculus,(3.16)∫E0E0+ηdE〈φ,ℑ(H−E−iη)−1φ〉≥π4〈φ,1]E0,E0+η](H)φ〉,for every self-adjoint H,
see [13], (3.9). The
equalities and estimates in (3.15) and (3.16) allow us to put the problem into a
form, where the results of spectral averaging, [11, Section 3], apply(3.17)𝔼〈f,1]E0,E0+η](HΛn)f〉=𝔼∫ℝdμj1(ωj1)〈g,uj11]E0,E0+η](H˜Λn(ω˜j1)+ωj1uj1)uj1g〉≤4π𝔼∫ℝdμj1(ωj1)∫E0E0+ηdEℑ〈g,uj1(H˜Λn(ω˜)+ωj1uj1−E−iη)−1uj1g〉≤8c2∥f∥2s(η).Proof.
By (H.A.2) and (H.A.3), we get aℙ-almost sure bound ∥Vn(ω)∥≤|Vn|,
then Lemma 3.2 implies that the restrictions HΛn(ω) and HΛ,Nn(ω) of Hn(ω) to a bounded open cube with Dirichlet or
Neumann boundary conditions define self-adjoint operators with compact
resolvent ℙ-almost sure. Let J:={j∈ℤnd:Λ(j,1)∩Λ≠∅} and for j∈J set Λj:=Λ(j,1)∩Λ.
Then Λ′:=Λ∖∪j∈JΛj has Lebesgue measure 0,
so by [15, XIII.15, Propositions 3 and 4], we have(3.18)−ΔΛ≥−ΔΛ,N≥−ΔΛ∖Λ′,N=⨁j∈J(−ΔΛj,N).So with Hi,Λj,Nn defined in Lemma 3.2, we get ℙ-almost sure:(3.19)HΛn(ω)≥HΛ,Nn:=⨁j∈JHi,Λj,Nn−|Vn|.By spectral calculus,(3.20)Trace(1]E0,E0+η](HΛn(ω)))≤eE0+ηTrace(1]E0,E0+η](HΛn(ω))e−HΛn(ω)).Let (φk(ω))k∈ℕ be the orthogonal basis of L2(Λ) consisting out of eigenvectors of HΛn(ω) to eigenvalues μk(ω) and let M(ω):={k∈ℕ:μk(ω)∈]E0,E0+η]},
then(3.21)Trace(1]E0,E0+η](HΛn(ω))e−HΛn(ω))=∑k∈M(ω)e−〈φk(ω),HΛn(ω)φk(ω)〉≤∑k∈M(ω)e−〈φk(ω),HΛ,Nnφk(ω)〉≤∑k∈M(ω)〈φk(ω),e−HΛ,Nnφk(ω)〉=Trace(1]E0,E0+η](HΛn(ω))e−HΛ,Nn),where the last estimate follows
from Jensen's inequality. Let (ϕk,j)k∈ℕ be an orthonormal basis of L2(Λj) consisting of eigenvectors of Hi,Λj,Nn to the eigenvalues Ek,j,
then(3.22)Trace(1]E0,E0+η](HΛn(ω))e−HΛ,Nn)=∑k∈ℕ∑j∈J〈ϕk,j,1]E0,E0+η](HΛn(ω))ϕk,j〉e−Ek,j+|Vn|.As ϕk,j∈L2(Λj) and ∥ϕk,j∥≤1,
Lemma 3.3 implies(3.23)𝔼〈ϕk,j,1]E0,E0+η](HΛn(ω))ϕk,j〉≤8c2s(η).As Vin is nonnegative, the eigenvalues Ek,j of Hi,Λj,Nn=−ΔΛj,N+Vin|Λj are estimated from what follows by the eigenvalues of −ΔΛj,N.
These are known explicitly, see [15, page 266], which can be used to estimate(3.24)∑k∈ℕ∑j∈Je−Ek,j≤Card(J)(eπ2eπ2−1)nd.If the side-length of Λ is bigger than 1,
then Card(J)≤3nd|Λ|,
so when applying expectation value to the chain of inequalities (3.20) to
(3.24), it implies(3.25)𝔼(Trace1]E0,E0+η](HΛn))≤eE0+η+|Vn|(3eπ2eπ2−1)nd8c2s(η)|Λ|.Under the assumptions (H.A.2) and (H.A.3), we
have(3.26)N(E)=𝔼(N(E,⋅)1Ω′)=𝔼(N(E,⋅)),hence by the Wegner estimate we
can deduce regularity properties of N from those of the conditioned measures (μj)j∈ℤd via(3.27)0≤N(E+η)−N(E)≤CW(E+η)s(η).



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