The Integrated Density of States for

Volume 2009, Article ID 679827,15pages doi:10.1155/2009/679827

Research Article

The Integrated Density of States for

an Interacting Multiparticle Homogeneous Model and Applications to the Anderson Model

Fr ´ed ´eric Klopp

and Heribert Zenk

1LAGA, Institut Galil´ee, Universit´e Paris-Nord, 93430 Villetaneuse, France

2Institut Universitaire de France, 75005 Paris, France

3Mathematisches Institut, Ludwig-Maximilians-Universit¨at, Theresienstraße 39, 80333 M ¨unchen, Germany

Correspondence should be addressed to Heribert Zenk,[email protected] Received 8 July 2008; Revised 9 October 2008; Accepted 13 January 2009

Recommended by Valentin Zagrebnov

For a system ofninteracting 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 interactingn-particle Anderson model, we prove regularity properties of the integrated density of states by establishing a Wegner estimate.

Copyrightq2009 F. Klopp and H. Zenk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Recently, models describing interacting quantum particles in a random potential have been studiedsee, e.g.,1–3. We considerninteracting particles moving in a “homogeneous”

potential in thed-dimensional configuration spaceR^d.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 statesIDS, 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 inR^d.

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 satisfiesH₀ⁿ −Δ V_extⁿ where the Laplacian−Δ onR^nddescribes the free kinetic energy of thenparticles. As all the particles are in the same background, the potentialV_extⁿ is of the form

V_extⁿ

x1, . . . , xn

ⁿ

k1

V¹ xk

. 1.1

Hence, the noninteractingn-particle Hamiltonian is a sum of one-particle HamiltoniansH¹

−Δ V¹.On the one particle potentialV¹,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,DV¹₋⊇ D−Δand for allα >0, there existsγα<∞,such that for allφ∈ D−Δ

V¹₋φ≤αΔφγαφ, 1.2 H.1.bthe operator H¹ admits an integrated density of states, say N₁, that is, if H_0.L¹ denotes the Dirichlet restriction ofH¹to a cubeΛ0, Lcentered at 0 of side-length, L,then the following limit exists

N1E: lim

L→∞L^−dTrace 1−∞,E

H_0.L¹

. 1.3

AssumptionH.1.aimplies essential self-adjointness of−Δ V¹onC^∞₀ R^dby4, Theorem X.29. Indeed,

V_extⁿ V_extⁿ

− V_extⁿ

−,

V_extⁿ

±

x₁, . . . , x_n :ⁿ

j1

V¹

±

x_j

, 1.4

where

i V_extⁿ ₋is infinitesimally−Δ-bounded, that is,1.2holds for the same constants and the Laplacian overR^nd;

ii V_extⁿ is nonnegative locally square integrable.

The self-adjoint extensions of−Δ V¹ and−Δ V_extⁿ are again denoted byH¹ andH₀ⁿ; they are bounded from what follows.

Classical models for which the IDS is known to exist include periodic, quasiperiodic, and ergodic random Schr ¨odinger operatorssee, 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

Hⁿ:−Δ V_iⁿV_extⁿ , 1.5

where

V_iⁿ

x1, . . . , xn

:

1≤k<l≤n

V xk−xl

1.6

is a localized repulsive interaction potential generated by the particles; so we assume that H.2V : R^d → Ris measurable nonnegative locally square integrable andV tends to 0

at infinity.

The standard repulsive interaction in three-dimensional space is of course the Coulomb interactionVx 1/|x|.In some cases, due to screening, it must be replaced by the Yukawa’s interactionVx e^−|x|/|x|.

Finally, we make one more assumption on bothV¹andV; we assume that H.3the operatorV_iⁿH₀ⁿ−i⁻¹is bounded.

AssumptionH.3is satisfied in the case of the Coulomb and Yukawa potential for those V¹ satisfying H.1.a; H₀ⁿ is self-adjoint on DH₀ⁿ ⊆ D−Δ, hence V_iⁿH₀ⁿ−i⁻¹ ≤ V_iⁿ−Δ−i⁻¹ · −Δ−iH₀ⁿ −i⁻¹,where−Δ−iH₀ⁿ−i⁻¹ < ∞due to closed graph theorem andV_iⁿ−Δ−i⁻¹ < ∞for Coulomb and Yukawa’s interaction potentialsV_iⁿ; see 4, Theorem X.16.

2. The Integrated Density of States

We now compute the IDS for then-particle model. LetΛL Λ0, Lbe the cube inR^dcentered at 0 with side-lengthLand writeΛⁿ_L ΛL×· · ·×ΛLfor the product ofncopies ofΛL.We denote the restriction of the interactingn-particle HamiltonianHⁿ toΛⁿ_L with Dirichlet boundary conditions byH_Lⁿ.Clearly assumptionsH.2andH.1.aguarantee thatH_Lⁿis bounded from what follows with compact resolvent. Hence, for any E ∈ R,one defines the normalized counting functions

NLE:L^−ndTrace 1−∞,E

H_Lⁿ

. 2.1

As usual,N,the IDS ofHⁿ is defined as the limit ofN_LEwhenL → ∞.Equivalently, one can define the density of states measure applied to a test function ϕ as the limit of L^−ndTraceϕH_Lⁿ.If the limit exists, it defines a nonnegative measure. It is a classical result that the existence of that limitfor all test functionsor that ofN_LEis equivalent5.

2.1. The IDS for the Noninteractingn-Particle System

Recall that, by assumptionH.1.b, the single particle modelH¹admits an IDSsee5and a density of states measure denoted, respectively, byN₁andν₁.

LetH_0,Lⁿ be the restriction ofH₀ⁿ toΛⁿ_L with Dirichlet boundary conditions. One has the following lemma.

Lemma 2.1. The IDS for the noninteractingn-particle Boltzmann model given by NniE: lim

L→ ∞

1

L^ndTrace 1_−∞,E

H_0,Lⁿ

2.2

exists and satisfies

NniN1∗ν1∗ · · · ∗ν1. 2.3 Let us comment on this result. First, the convolution product in2.3makes sense as all the measures and functions are supported on half-axes of the forma,∞; this results from assumptionH.1.a. When the fieldV¹is not bounded from what follows, one will need some estimate on the decay ofN₁ andν₁ near−∞to make sense of2.3 and to prove it;

such estimates are known for some modelssee, e.g.,5,6.

Proof ofLemma 2.1. The operator H₀ⁿ is the sum of n commuting Hamiltonians, each of which is unitarily equivalent to H¹; so is H_0,Lⁿ ,its restriction to the cube Λⁿ_L. As the sum decomposition ofH₀ⁿcommutes with the restriction toΛⁿ_L,the eigenvalues ofH_0,Lⁿ are exactly the sum ofneigenvalues ofH¹restricted toΛL.This immediately yields that

Trace 1_−∞,E

H_0,Lⁿ

N₁^L∗ν^L₁∗ · · · ∗ν^L₁

E, 2.4 where N₁^LE is the eigenvalue counting function for H¹ restricted to ΛL, and ν^L₁ is its counting measurei.e.,dN₁^L. The normalized counting function and measure,N₁^Landν₁^L, are defined as

N₁^L 1

L^dN₁^L, ν₁^L 1

L^dν₁^L. 2.5

The existence of the density of states of H¹ then exactly says that N₁^L and ν^L₁ converge, respectively, to N1 and ν1. The convergence of N₁^L∗ν^L₁∗ · · · ∗ν^L₁ to N1∗ν1∗ · · · ∗ν1 is then guaranteed as the convolution is bilinear bicontinuous operation on distributions. This completes the proof ofLemma 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 noninteractingn-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

LetH_Lⁿ denote the restriction ofHⁿ to the boxΛⁿ_Lwith Dirichlet boundary conditions. Our main result is.

Theorem 2.2. Assume (H.1), (H.2), and (H.3) are satisfied. For anyϕ∈C^∞₀ R,one has

1

L^ndTrace ϕ

H_Lⁿ

−ϕ

H_0,L^{n L→ ∞}−→ 0. 2.6

As the density of states measure ofHⁿis defined by ϕ, dN lim

L→∞

1

L^ndTrace ϕ

H_Lⁿ , 2.7

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 modelHⁿexists and coincides with that of the noninteracting modelH₀ⁿ; hence, it satisfies NNniN1∗ν1∗ · · · ∗ν1. 2.8 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 Corollary 2.3, we dealt with the Boltzmann statistic, that is, without statistic.

Theorem 2.2stays 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:

ifor the Fermi statistics, the Fermi integrated density of states ϕ, dN^F

lim

L→∞

n!

L^nd Trace_∧nL2Λ¹_L ϕ

H_Lⁿ , 2.9

where∧nL²Λ¹_Ldenotesn-fold antisymmetric tensor product ofL²Λ¹_L; iifor the Bose statistics, the Bose integrated density of states

ϕ, dN^B lim

L→∞

n!

L^nd Trace_⊕^s

nL²Λ¹_L

ϕ

H_Lⁿ , 2.10

where⊕^snL²Λ¹_Ldenotesn-fold symmetric tensor product ofL²Λ¹_L.

Let us now discuss shortly the Bose and Fermi counting functions i.e., the eigenvalue counting functions of the Hamiltonian restricted to a finite cubein the free casei.e., when the interaction vanishes. Consider the cubeΛ¹_Land letE₁L≤E₂L≤ · · · be the eigenvalue of the single particle Hamiltonian repeated according to multiplicity. The three counting functions are then given by

#_LE:#

eigenvalues ofH_0,Lⁿ onL² Λⁿ_L

less thanE #

j₁, j₂, . . . , j_n

:E_j1L E_j2L · · ·E_jnL≤E ,

#^F_LE:#

eigenvalues ofH_0,Lⁿ on∧nL² Λ¹_L

less thanE #

j₁, j₂, . . . , j_n

:j₁< j₂<· · ·< j_n, E_j1L · · ·E_jnL≤E ,

#^B_LE:#

eigenvalues ofH_0,Lⁿ on⊕^snL² Λ¹_L

less thanE #

j1, j2, . . . , jn

:j1≤j2≤ · · · ≤jn, Ej1L · · ·EjnL≤E .

2.11

Hence,

n!#^F_LE≤#LE≤n!#^B_LE. 2.12 Uniformly inL,the eigenvaluesEjL_j≥1are lower bounded by, say,−C.Hence, ifE_j1L E_j2L· · ·EjnL≤E,then, fork1, . . . , n,one hasE_jkL≤ECnso thatj_k≤N₁^LECn N₁^LECnL^d.This implies that

0≤#^B_LE−#^F_LE #

⎧⎨

⎩

j1, j2, . . . , jn

;

j1 ≤j2≤ · · · ≤jn, ∃k < l s.t. jkjl

E_j1L E_j2L · · ·E_jnL≤E

⎫⎬

⎭

≤CL ^dn−1.

2.13

Thus, dividing2.12and2.13byL^ndand taking the limitL → ∞,we obtain that the free Fermi and Bose density of states are equal to the Boltzmann one.Theorem 2.2then gives the following corollary.

Corollary 2.4. Assume (H.1), (H.2), and (H.3) are satisfied. One hasNN^BN^F.

Proof ofTheorem 2.2. We take someq > nd/2 and specify the appropriate choice later on. By assumptionsH.1.aandH.2, there existsζ >0 such that

−∞<−ζ≤min

infL≥1

inf σ

H_0,Lⁿ

∪σ

H_Lⁿ ,inf σ

H₀ⁿ

∪σ

Hⁿ . 2.14

Letγγ1/2be given by1.2forα1/2.Fixλ₀> ζ2γ1.

By2.14, we only need to prove2.6forϕ∈C₀^∞Rsupported in−ζ−1,∞.For such a function, letϕbe an almost analytic extension of the functionx→xλ₀^qϕx∈C₀^∞R, that is,ϕsatisfies

iϕ∈ S{z∈C:|Iz|<1},

iifor anyk∈N,the family of functionsx→∂ϕ/∂zx iy|y|^−k0<|y|<1is bounded inSR.

The functional calculus based on the Helffer-Sj ¨ostrand formula implies

ϕ H_Lⁿ

−ϕ H_0,Lⁿ

i 2π

C

∂ϕ

∂zz H_Lⁿλ0

_−q

H_Lⁿ−z₋₁

−

H_0,Lⁿ λ0

_−q

H_0,Lⁿ −z₋₁

dz∧dz.

2.15 In the following, we apply an idea, which has already been used in6,7and which simplifies in this situation. Using resolvent equality, the integrand in2.15is written as

H_Lⁿλ0

_−q

H_Lⁿ−z₋₁

−

H_0,Lⁿ λ0

_−q

H_0,Lⁿ −z₋₁

H_0,Lⁿ λ_0−q

H_Lⁿ−z₋₁

−

H_0,Lⁿ −z₋₁

H_Lⁿλ_0−q

−

H_0,Lⁿ λ_0−q

H_Lⁿ−z₋₁ −

H_0,Lⁿ λ_0−q

H_0,Lⁿ −z₋₁ V_iⁿ

H_Lⁿ−z₋₁

− q

l1

H_0,Lⁿ λ0

_l−q−1 V_iⁿ

H_Lⁿλ0

_−l

H_Lⁿ−z₋₁ .

2.16

Estimating the trace of2.16, we chooseε >0 and write

V_iⁿV_iⁿ·1{|V_iⁿ|≤ε}V_iⁿ·1{|V_iⁿ|>ε} 2.17

and note thatV_iⁿ·1{|V_iⁿ|≤ε}is bounded byV_iⁿ·1{|V_iⁿ|≤ε} ≤ε.AsVis nonnegative, one has

supp

V_iⁿ·1{|V_iⁿ|>ε}

⊆ ⁿ

j1

n i /i1j

x1, . . . , xn

∈R^nd :V xi−xj

≥ ε nn−1

. 2.18

As, by assumptionH.2,V tends to 0 at infinity,2.18implies that there exists 0 < Cn;ε independent ofLsuch that

μV_iⁿ> ε

∩Λⁿ_L

≤Cn, εL^n−1d, 2.19

whereμ·denotes the Lebesgue measure. Using decomposition2.17ofV_iⁿ,we obtain Trace

H_0,Lⁿ λ_0−q

H_0,Lⁿ −z₋₁ V_iⁿ

H_Lⁿ−z₋₁

≤ ε

|Iz|²TraceH_0,Lⁿ λ_0−q 1

|Iz|V_iⁿ

H_Lⁿ−z₋₁

·TraceH_0,Lⁿ λ0

_−q

1_{|Viⁿ_|>ε}∩Λⁿ_L

≤ ε

|Iz|²H_0,Lⁿ λ0

₋₁^q

Tq 1

|Iz|²H_0,Lⁿ λ0

₋₁^q−1

Tq

·H_0,Lⁿ λ0

₋₁

1{|V_iⁿ|>ε}∩Λⁿ_L

Tq·V_iⁿ

H_0,Lⁿ λ0

₋₁,

2.20

where · Tq denotes theqth Schatten class normsee8and we used H ¨older’s inequality.

In the same way, the cyclicity of the trace yields Trace

H_0,Lⁿ λ_0l−q−1 V_iⁿ

H_Lⁿλ_0−l

H_Lⁿ−z₋₁

≤TraceH_Lⁿλ0

_−l

H_0,Lⁿ λ0

_l−q−1 V_iⁿ

H_Lⁿ−z₋₁

≤H_Lⁿλ_0−l

H_0,Lⁿ λ₀l·TraceH_0,Lⁿ λ_0−q−1 V_iⁿ

H_Lⁿ−z₋₁

≤ C

|Iz|H_0,Lⁿ λ0

₋₁^q−1

Tq ·

H_0,Lⁿ λ0

₋₁

1_{|Viⁿ_|>ε}∩Λⁿ_LTq·V_iⁿ

H_0,Lⁿ λ0

₋₁

C ε

|Iz|H_0,Lⁿ λ0

₋₁^q

Tq.

2.21

We are now left with estimating H_0,Lⁿ λ₀⁻¹_Tq and H_0,Lⁿ λ₀⁻¹1_{|Viⁿ_|>ε}∩Λⁿ_LTq for q sufficiently large, depending onnd.Therefore, we compute

H_0,Lⁿ λ0

₋₁

1_{|Viⁿ_|>ε}∩Λⁿ_L

Tq≤H_0,Lⁿ λ0

₋₁

−Δ_Λⁿ_Lλ0

_1/2

T2q

·−ΔΛⁿ_Lλ0

_−1/2

1{|V_iⁿ|>ε}∩Λⁿ_L

T2q,

2.22

where −ΔΛⁿ_L is the Dirichlet Laplacian on Λⁿ_L. We use the decomposition 1.4. As the Laplacians are positive, the infinitesimal−Δ-boundedness onV_extⁿ ₋,4, Theorem X.18and the definition ofγimply the following form bound:

φ,V_extⁿ ₋φ≤ 1 2

φ,−Δ_Λⁿ_Lφ

γφ². 2.23

Asλ0>2γ1,one has

H_0,Lⁿ λ₀≥ −Δ_Λⁿ_L V_extⁿ

−λ₀ ≥ 1 2

−Δ_Λⁿ_L−2γ2λ₀

≥ 1 2

−Δ_Λⁿ_Lλ₀

. 2.24

Thus, the operatorH_0,Lⁿ λ0is invertible and H_0,Lⁿ λ₀₋₁

≤2

−Δ_Λⁿ_Lλ₀₋₁

. 2.25

Letμj_j andφj_j,respectively, denote the eigenvalues and eigenfunctions of the Dirichlet Laplacian−Δ_Λⁿ_Lthe indexjruns overN^nd∗. Forq∈Nsuch that 2q > nd,we compute

H_0,Lⁿ λ₀₋₁

−Δ_Λⁿ_Lλ₀1/2^2q

τ2q

j∈N^nd

μ_j

−Δ_Λⁿ_L λ₀q

φ_j,

H_0,Lⁿ λ₀₋₁ φ_j2^q

≤2^2q

j∈N^nd

μj

−ΔΛⁿ_L λ0

_q φj,

−ΔΛⁿ_Lλ0

₋₁ φj

_2q

2^2q

j∈N^nd

μj

−ΔΛⁿ_L λ0

_−q

≤CL^nd.

2.26 The last estimate is a direct computation using the explicit form of the Dirichlet eigenvalues.

By6, Lemma 2.2, we know that, forq∈Nsuch that 2q > nd,there existsC_q >0 such that, for any measurable subsetΛ⊆Λⁿ_L,one has

−ΔΛⁿ_Lλ_0−1/2 1Λ^2q_T

2q≤C_qμΛ. 2.27

ChoosingΛ {|V_iⁿ|> ε} ∩Λⁿ_Land taking2.19into account, then by combining estimates 2.20–2.27, we get that there existsc,depending only onqand the bound in assumption H.3, such that

TraceH_Lⁿλ0

_−q

H_Lⁿ−z₋₁

−

H_0,Lⁿ λ0

_−q

H_0,Lⁿ −z₋₁

≤c ε

|Iz|²L^nd 1

|Iz|²L^nd−d/2q ε

|Iz|L^nd 1

|Iz|L^nd−d/2q

.

2.28

By using this inequality in2.15, we get2.6asϕbeing almost analytic,∂ϕz vanishes to any order inIzaszapproaches the real line. Thus, we completed the proof ofTheorem 2.2.

3. Application to the Interacting Multiparticle Anderson Model

In the interacting multiparticle Anderson model, we consider a random external potential, that is,V¹V¹ω.The one particle Anderson potential is of the form

V¹ω, x

j∈Z^d

ωjux−j, 3.1

with a familyωj :Ω → Rof random variables onΩ,P.This one-particle models leads us to then-particle random “background” potential

Vⁿ

ω, x₁, . . . , x_n ⁿ

k1

V¹ ω, x_k

3.2

and the interactingn-particle Hamiltonian reads as

Hⁿω −Δ V_iⁿVⁿω. 3.3

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 surelysee, e.g., 5,9. The limit is a nondecreasing function of E.Its discontinuity set is countable. By9, pp. 311f, for almost everyω,except at this set, the normalized counting function defined in assumptionH.1.bthen converges. On this set of full measure, we can now apply the results of the last section and get aP-almost sure integrated density of states Nni N for both, the noninteracting and interacting n-particle system. Note that only translations along a “diagonal” vectorj, j, . . . , j ∈ Z^nd leaveHⁿωinvariant. Hence, for an application of ergodic theoremsas in the one particle casefor the proof of existence and P-almost sure constancy ofN,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

E

Trace1E0,E0η H_Λⁿ

≤CWη|Λ|. 3.4

On the other hand, Corollary 2.3 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 recentlysee, e.g.,11,12.

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

H.A.2The single-site potentialuis nonnegative, compactly supported,uL^∞ ≤1 and that there is somec >0,such thatux≥cforx∈−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 in13:

H.A.3 ωj:Ω → R_j∈Zd is a family of bounded random variables on the probability space Ω,P.

Whenμ_jdenotes the conditional probability measure forω_jat sitej ∈Z^dconditioned on all the other random variablesωi_{i /j},that is, for allA∈ BR,

μjA P{ωj∈A|ωi_{i /j}}, 3.5

then, a Wegner estimate `a la13uses the quantity sη:sup

j∈Z^dE

sup

E∈Rμj

E, Eη 3.6

and is stated as follows.

Theorem 3.1. Let us assume (H.A.2) and (H.A.3), and letΛ⊆ R^ndbe a bounded open cube of side length≥1,letH_Λⁿωbe the restriction ofHⁿωtoΛwith Dirichlet boundary conditions. Then, there exists an increasing function

C_W :R−→0,∞ E0→CW

E0

, 3.7

such that for allη >0

E

Trace1E0,E0η H_Λⁿ

≤CW

E0

sη|Λ|. 3.8

In order to proveTheorem 3.1, we prove two preparatory lemmas.

Lemma 3.2. LetΛ ⊆R^ndbe an open bounded cube, then the restrictionsH_i,Λⁿ andH_i,Λ,Nⁿ ofH_iⁿ

−Δ V_iⁿ toΛ with Dirichlet or Neumann boundary conditions define self-adjoint operators with compact resolvent.

Proof. V_iⁿ is infinitesimally −Δ form bounded according to 4, Theorem X.18, so the infinitesimal form bound

Ψ, V_iⁿΨ≤ε∇Ψ²bεΨ² 3.9 is true forΨ∈H¹R^nd,in particular3.9is true forΨ∈ D−Δ_Λ H₀¹Λ⊆H¹R^nd.Hence, the form sum defines via representation theorem a self-adjoint operatorH_i,Λⁿ −Δ_ΛV_iⁿ|_Λ. The eigenvaluesμk−ΔΛtend to infinity, so by the minimax principle and3.9, we see that H_i,Λⁿ has compact resolvent. The proof of

Ψ, V_iⁿΨ

≤ε∇Ψ²cεΨ², Ψ∈H¹Λ 3.10 uses the extension operatorE_Λ:H¹Λ → H₀¹ΛtoΛ:{x∈R^nd: distx,Λ<1},which has the propertiesE_ΛΨ|Λ Ψ,EΛΨH¹≤c₁ΨH¹andEΛΨL² ≤c₂ΨL²; see14, Satz 5.6 and Folgerung 5.2. ForE_ΛΨ ∈ H₀¹Λ ⊆ H¹R^nd,we use3.9, hence byV_iⁿ ≥ 0 and the above properties ofE_Λwe get forΨ∈H¹Λ,

0≤

Ψ, V_iⁿΨ

≤

E_ΛΨ, V_iⁿE_ΛΨ

≤ε∇

E_ΛΨ²

L²b_εE_ΛΨ²

L²

εE_ΛΨ²

H¹

bε−εE_ΛΨ²

L² ≤εc²₁∇Ψ² c²₂

bε−ε εc₁²

Ψ²,

3.11

which is3.10. With3.10at 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 ⊆ R^nd be a bounded open cube,j j1, . . . , jn∈Z^nd withΛj : Λ∩Λj,1/∅(here,Λj,1 {|x−jk| ≤1/2,1≤k≤n}), then for everyf∈L²Λ_j,

E

f,1E0,E0η H_Λⁿ

f

≤ 8

c²sηf². 3.12

Proof. For everyj∈Z^d,we defineu_j :R^nd → Rby

u_j

x₁, . . . , x_n :ⁿ

k1

u x_k−j

3.13

and setω_j ωl_{l /j}.Fix a component ofj,sayj₁,then we get a decomposition Vⁿ

ω, x₁, . . . , x_n

ω_j1u_j1

x₁, . . . , x_n

l∈Z^d l /j1

ω_lu_l

x₁, . . . , x_n

3.14

of the random potentialVⁿω,and the same is true forH_Λⁿω:

H_Λⁿω −Δ_Λ

l∈Z^d l /j1

ω_lu_l1_Λω_j1u_j11_Λ:H_Λⁿ ω_j1

ω_j1u_j11_Λ. 3.15

By the covering conditionu1_−1/2,1/2^d ≥ con the single site-potentialu,we getuj1 ≥ c1_Λj, hence we can writef guj1,wheregx fx/uj1xalmost everywhere, sog ≤c⁻¹f.

By spectral calculus, _E0η

E0

dE

ϕ,IH−E−iη⁻¹ϕ

≥ π 4

ϕ,1E0,E0ηHϕ

, 3.16

for every self-adjointH,see13,3.9. The equalities and estimates in3.15and3.16allow us to put the problem into a form, where the results of spectral averaging,11, Section 3, apply

E

f,1E0,E0η H_Λⁿ

f E

Rdμ_j1 ω_j1

g, u_j11E0,E0ηH_Λⁿ ω_j1

ω_j1u_j1 u_j1g

≤ 4 πE

Rdμ_j1

ω_j1^E0η

E0

dEI

g, u_j1H_Λⁿω ω_j1u_j1−E−iη−1

u_j1g

≤ 8

c²f²sη. 3.17

Proof ofTheorem 3.1. ByH.A.2andH.A.3, we get aP-almost sure boundVⁿω ≤ |Vⁿ|, then Lemma 3.2implies that the restrictionsH_Λⁿω andH_Λ,Nⁿ ω ofHⁿω to a bounded open cube with Dirichlet or Neumann boundary conditions define self-adjoint operators with compact resolventP-almost sure. LetJ :{j∈ Z^nd : Λj,1∩Λ/∅}and forj ∈J setΛj : Λj,1∩Λ.ThenΛ : Λ\ ∪_j∈JΛ_j has Lebesgue measure 0,so by15, XIII.15, Propositions 3 and 4, we have

−Δ_Λ≥ −Δ_Λ,N ≥ −Δ_Λ\Λ,N

j∈J

−Δ_Λj,N

. 3.18

So withH_i,Λⁿ

j,N defined inLemma 3.2, we getP-almost sure:

H_Λⁿω≥H_Λ,Nⁿ :

j∈J

H_i,Λⁿ _j,N− |Vⁿ|. 3.19

By spectral calculus, Trace

1E0,E0η

H_Λⁿω

≤e^E0ηTrace

1E0,E0η

H_Λⁿω

e^−HΛnω

. 3.20

Letϕkω_k∈Nbe the orthogonal basis ofL²Λconsisting out of eigenvectors ofH_Λⁿωto eigenvaluesμ_kωand letMω:{k∈N:μ_kω∈E0, E₀η},then

Trace

1_E0,E0η

H_Λⁿω

e^−HΛnω

k∈Mω

e^{−ϕkω,HΛnωϕkω}

≤

k∈Mω

e^{−ϕkω,HΛ,Nn ϕkω}

≤

k∈Mω

ϕkω, e^−HΛ,Nn ϕkω Trace1E0,E0η

H_Λⁿω e^−HΛ,Nn ,

3.21

where the last estimate follows from Jensen’s inequality. Letφk,j_k∈Nbe an orthonormal basis ofL²Λjconsisting of eigenvectors ofH_i,Λⁿ _j,N to the eigenvaluesE_k,j,then

Trace

1_E0,E0η

H_Λⁿω e^−HΛ,Nn

k∈N

j∈J

φ_k,j,1_E0,E0η

H_Λⁿω φ_k,j

e^−Ek,j|Vn|. 3.22

Asφ_k,j∈L²Λjandφk,j ≤1,Lemma 3.3implies

E

φ_k,j,1_E0,E0η

H_Λⁿω φ_k,j

≤ 8

c²sη. 3.23

AsV_iⁿis nonnegative, the eigenvaluesE_k,jofH_i,Λⁿ _j,N −ΔΛj,NV_iⁿ|Λjare estimated from what follows by the eigenvalues of−ΔΛj,N.These are known explicitly, see15, page 266, which can be used to estimate

k∈N

j∈J

e^−Ek,j ≤CardJ e^π2

e^π2−1 ^nd

. 3.24

If the side-length ofΛis bigger than 1,then CardJ≤3^nd|Λ|,so when applying expectation value to the chain of inequalities3.20to3.24, it implies

E

Trace1_E0,E0η H_Λⁿ

≤e^E0η|Vn| 3e^π2

e^π2−1 ^nd

8

c²sη|Λ|. 3.25

Under the assumptionsH.A.2andH.A.3, we have

NE ENE,·1_Ω ENE,·, 3.26

hence by the Wegner estimate we can deduce regularity properties ofN from those of the conditioned measuresμj_j∈Zd via

0≤NEη−NE≤CWEηsη. 3.27

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