The Integrated Density of States for an Interacting Multiparticle Homogeneous Model and Applications to the Anderson Model

Abstract

For a system of 𝑛 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.

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 ℝ^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 states (IDS), then, so does the interacting 𝑛-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 ℝ^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 noninteracting 𝑛-particle Hamiltonian satisfies 𝐻^𝑛₀=−Δ+𝑉^𝑛_ext where the Laplacian −Δ on ℝ^𝑛𝑑 describes the free kinetic energy of the 𝑛 particles. As all the particles are in the same background, the potential 𝑉^𝑛_ext is of the form𝑉^𝑛_ext(𝑥₁,…,𝑥_𝑛)=^𝑛_𝑘=1𝑉¹(𝑥_𝑘). (1.1)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)‖(𝑉¹)₋𝜙‖≤𝛼‖Δ𝜙‖+𝛾(𝛼)‖𝜙‖, (1.2)(H.1.b)the operator 𝐻¹ admits an integrated density of states, say 𝑁₁, that is, if 𝐻¹_0.𝐿 denotes the Dirichlet restriction of 𝐻¹ to a cube Λ(0,𝐿) centered at 0 of side-length, 𝐿, then the following limit exists𝑁₁(𝐸)∶=lim_𝐿→+∞𝐿^−𝑑Trace(𝟏_{]−∞,𝐸]}(𝐻¹_0.𝐿)). (1.3) Assumption (H.1.a) implies essential self-adjointness of −Δ+𝑉¹ on 𝐶^∞₀(ℝ^𝑑) by [4, Theorem X.29]. Indeed,𝑉^𝑛_ext=(𝑉^𝑛_ext)₊−(𝑉^𝑛_ext)₋,(𝑉^𝑛_ext)_±(𝑥₁,…,𝑥_𝑛)∶=^𝑛_𝑗=1(𝑉¹)_±(𝑥_𝑗), (1.4)where

(i)(𝑉^𝑛_ext)₋ is infinitesimally −Δ-bounded, that is, (1.2) holds for the same constants and the Laplacian over ℝ^𝑛𝑑;(ii)(𝑉^𝑛_ext)₊ is nonnegative locally square integrable. The self-adjoint extensions of −Δ+𝑉¹ and −Δ+𝑉^𝑛_ext are again denoted by 𝐻¹ and 𝐻^𝑛₀; 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 interacting 𝑛-particle Hamiltonian is of the form𝐻^𝑛∶=−Δ+𝑉^𝑛_𝑖+𝑉^𝑛_ext, (1.5)where𝑉^𝑛_𝑖(𝑥₁,…,𝑥_𝑛)∶=_{1≤𝑘<𝑙≤𝑛}𝑉(𝑥_𝑘−𝑥_𝑙) (1.6)is a localized repulsive interaction potential generated by the particles; so we assume that

(H.2)𝑉∶ℝ^𝑑→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 both 𝑉¹ and 𝑉; we assume that

(H.3)the operator 𝑉^𝑛_𝑖(𝐻^𝑛₀−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); 𝐻^𝑛₀ is self-adjoint on 𝒟(𝐻^𝑛₀)⊆𝒟(−Δ), hence ‖𝑉^𝑛_𝑖(𝐻^𝑛₀−𝑖)⁻¹‖≤‖𝑉^𝑛_𝑖(−Δ−𝑖)⁻¹‖⋅‖(−Δ−𝑖)(𝐻^𝑛₀−𝑖)⁻¹‖, where ‖(−Δ−𝑖)(𝐻^𝑛₀−𝑖)⁻¹‖<∞ due to closed graph theorem and ‖𝑉^𝑛_𝑖(−Δ−𝑖)⁻¹‖<∞ for Coulomb and Yukawa's interaction potentials 𝑉^𝑛_𝑖; see [4, Theorem X.16].

2. The Integrated Density of States

We now compute the IDS for the 𝑛-particle model. Let Λ_𝐿=Λ(0,𝐿) be the cube in ℝ^𝑑 centered at 0 with side-length 𝐿 and write Λ^𝑛_𝐿=Λ_𝐿×⋯×Λ_𝐿 for the product of 𝑛 copies of Λ_𝐿. We denote the restriction of the interacting 𝑛-particle Hamiltonian 𝐻^𝑛 to Λ^𝑛_𝐿 with Dirichlet boundary conditions by 𝐻^𝑛_𝐿. Clearly assumptions (H.2) and (H.1.a) guarantee that 𝐻^𝑛_𝐿 is bounded from what follows with compact resolvent. Hence, for any 𝐸∈ℝ, one defines the normalized counting functions𝑁_𝐿(𝐸)∶=𝐿^−𝑛𝑑Trace(𝟏_{]−∞,𝐸]}(𝐻^𝑛_𝐿)). (2.1)As usual, 𝑁, the IDS of 𝐻^𝑛 is defined as the limit of 𝑁_𝐿(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(𝐻^𝑛_𝐿)]. 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 𝑁_𝐿(𝐸) is equivalent [5].

2.1. The IDS for the Noninteracting 𝑛-Particle System

Recall that, by assumption (H.1.b), the single particle model 𝐻¹ admits an IDS (see [5]) and a density of states measure denoted, respectively, by 𝑁₁ and 𝜈₁.

Let 𝐻^𝑛_0,𝐿 be the restriction of 𝐻^𝑛₀ to Λ^𝑛_𝐿 with Dirichlet boundary conditions. One has the following lemma.

Lemma 2.1. The IDS for the noninteracting 𝑛-particle Boltzmann model given by𝑁_ni(𝐸)∶=lim_𝐿→∞1𝐿^𝑛𝑑Trace(𝟏_{]−∞,𝐸]}(𝐻^𝑛_0,𝐿)) (2.2)exists and satisfies𝑁_ni=𝑁₁∗𝜈₁∗⋯∗𝜈₁. (2.3) 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 operator 𝐻^𝑛₀ is the sum of 𝑛 commuting Hamiltonians, each of which is unitarily equivalent to 𝐻¹; so is 𝐻^𝑛_0,𝐿, its restriction to the cube Λ^𝑛_𝐿. As the sum decomposition of 𝐻^𝑛₀ commutes with the restriction to Λ^𝑛_𝐿, the eigenvalues of 𝐻^𝑛_0,𝐿 are exactly the sum of 𝑛 eigenvalues of 𝐻¹ restricted to Λ_𝐿. This immediately yields thatTrace(𝟏_{]−∞,𝐸]}(𝐻^𝑛_0,𝐿))=(𝑁^𝐿₁∗𝜈^𝐿₁∗⋯∗𝜈^𝐿₁)(𝐸), (2.4)where 𝑁^𝐿₁(𝐸) is the eigenvalue counting function for 𝐻¹ restricted to Λ_𝐿, and 𝜈^𝐿₁ is its counting measure (i.e., 𝑑𝑁^𝐿₁). The normalized counting function and measure, 𝑁^𝐿₁ and 𝜈^𝐿₁, are defined as𝑁^𝐿₁=1𝐿^𝑑𝑁^𝐿₁,𝜈^𝐿₁=1𝐿^𝑑𝜈^𝐿₁. (2.5)The existence of the density of states of 𝐻¹ then exactly says that 𝑁^𝐿₁ and 𝜈^𝐿₁ converge, respectively, to 𝑁₁ and 𝜈₁. The convergence of 𝑁^𝐿₁∗𝜈^𝐿₁∗⋯∗𝜈^𝐿₁ 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 noninteracting 𝑛-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 Interacting 𝑛-Particle System

Let 𝐻^𝑛_𝐿 denote the restriction of 𝐻^𝑛 to the box Λ^𝑛_𝐿 with Dirichlet boundary conditions. Our main result is.

Theorem 2.2. Assume (H.1), (H.2), and (H.3) are satisfied. For any 𝜑∈𝐶^∞₀(ℝ), one has1𝐿^𝑛𝑑Trace[𝜑(𝐻^𝑛_𝐿)−𝜑(𝐻^𝑛_0,𝐿)]−−−−−^𝐿→∞→0. (2.6)As the density of states measure of 𝐻^𝑛 is defined by⟨𝜑,𝑑𝑁⟩=lim_𝐿→+∞1𝐿^𝑛𝑑Trace[𝜑(𝐻^𝑛_𝐿)], (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 interacting 𝑛-particle Boltzmann model 𝐻^𝑛 exists and coincides with that of the noninteracting model 𝐻^𝑛₀; hence, it satisfies𝑁=𝑁_ni=𝑁₁∗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 2.2 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⟨𝜑,𝑑𝑁^𝐹⟩=lim_𝐿→+∞𝑛!𝐿^𝑛𝑑Trace_⋀𝑛𝐿²_(Λ¹_𝐿)[𝜑(𝐻^𝑛_𝐿)], (2.9)where ⋀_𝑛𝐿²(Λ¹_𝐿) denotes 𝑛-fold antisymmetric tensor product of 𝐿²(Λ¹_𝐿);(ii) for the Bose statistics, the Bose integrated density of states⟨𝜑,𝑑𝑁^𝐵⟩=lim_𝐿→+∞𝑛!𝐿^𝑛𝑑Trace^𝑠_{⊕𝑛𝐿2(Λ1𝐿)}[𝜑(𝐻^𝑛_𝐿)], (2.10)where ⊕^𝑠_𝑛𝐿²(Λ¹_𝐿) denotes 𝑛-fold symmetric tensor product of 𝐿²(Λ¹_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 cube) in the free case (i.e., when the interaction vanishes). Consider the cube Λ¹_𝐿 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#_𝐿(𝐸)∶=#{eigenvaluesof𝐻^𝑛_0,𝐿on𝐿²(Λ^𝑛_𝐿)lessthan𝐸}=#{(𝑗₁,𝑗₂,…,𝑗_𝑛)∶𝐸_𝑗1(𝐿)+𝐸_𝑗2(𝐿)+⋯+𝐸_𝑗𝑛#(𝐿)≤𝐸},_𝐹𝐿(𝐸)∶=#{eigenvaluesof𝐻^𝑛_0,𝐿on_𝑛𝐿²(Λ¹_𝐿)lessthan𝐸}=#{(𝑗₁,𝑗₂,…,𝑗_𝑛)∶𝑗₁<𝑗₂<⋯<𝑗_𝑛,𝐸_𝑗1(𝐿)+⋯+𝐸_𝑗𝑛#(𝐿)≤𝐸},_𝐵𝐿(𝐸)∶=#{eigenvaluesof𝐻^𝑛_0,𝐿on⊕^𝑠_𝑛𝐿²(Λ¹_𝐿)lessthan𝐸}=#{(𝑗₁,𝑗₂,…,𝑗_𝑛)∶𝑗₁≤𝑗₂≤⋯≤𝑗_𝑛,𝐸_𝑗1(𝐿)+⋯+𝐸_𝑗𝑛(𝐿)≤𝐸}. (2.11)Hence,𝑛!#^𝐹_𝐿(𝐸)≤#_𝐿(𝐸)≤𝑛!#^𝐵_𝐿(𝐸). (2.12)Uniformly in 𝐿, the eigenvalues (𝐸_𝑗(𝐿))_𝑗≥1 are lower bounded by, say, −𝐶. Hence, if 𝐸_𝑗1(𝐿)+𝐸_𝑗2(𝐿)+⋯+𝐸_𝑗𝑛(𝐿)≤𝐸, then, for 𝑘=1,…,𝑛, one has 𝐸_𝑗𝑘(𝐿)≤𝐸+𝐶𝑛 so that 𝑗_𝑘≤𝑁^𝐿₁(𝐸+𝐶𝑛)=𝑁^𝐿₁(𝐸+𝐶𝑛)𝐿^𝑑. This implies that0≤#^𝐵_𝐿(𝐸)−#^𝐹_𝐿(𝐸)=#{(𝑗₁,𝑗₂,…,𝑗_𝑛𝑗);¹≤𝑗₂≤⋯≤𝑗_𝑛,∃𝑘<𝑙s.t.𝑗_𝑘=𝑗_𝑙𝐸_𝑗1(𝐿)+𝐸_𝑗2(𝐿)+⋯+𝐸_𝑗𝑛}≤(𝐿)≤𝐸_{0𝑥002𝑑𝑐}𝐶𝐿^{𝑑(𝑛−1)}. (2.13)Thus, dividing (2.12) and (2.13) by 𝐿^𝑛𝑑 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 has 𝑁=𝑁^𝐵=𝑁^𝐹.

Proof. We take some 𝑞>𝑛𝑑/2 and specify the appropriate choice later on. By assumptions (H.1.a) and (H.2), there exists 𝜁>0 such that−∞<−𝜁≤min(inf_𝐿≥1{inf[𝜎(𝐻^𝑛_0,𝐿)∪𝜎(𝐻^𝑛_𝐿)]},inf[𝜎(𝐻^𝑛₀)∪𝜎(𝐻^𝑛)]). (2.14)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∈𝐶^∞₀(ℝ) supported in (−𝜁−1,+∞). For such a function, let ^0𝑥0𝜑^02𝑑𝑐 be an almost analytic extension of the function 𝑥↦(𝑥+𝜆₀)^𝑞𝜑(𝑥)∈𝐶^∞₀(ℝ), that is, ^0𝑥0𝜑^02𝑑𝑐 satisfies
(i)^0𝑥0𝜑^02𝑑𝑐∈𝒮({𝑧∈ℂ∶|ℑ𝑧|<1},(ii) for any 𝑘∈ℕ, the family of functions (𝑥↦(𝜕𝜑^{0𝑥002𝑑𝑐}/𝜕𝑧)(𝑥+𝑖𝑦)|𝑦|^−𝑘)_0<|𝑦|<1 is bounded in 𝒮(ℝ). The functional calculus based on the Helffer-Sjöstrand formula implies𝜑(𝐻^𝑛_𝐿)−𝜑(𝐻^𝑛_0,𝐿𝑖)=2𝜋_ℂ𝜕𝜑^{0𝑥002𝑑𝑐}𝜕𝑧(𝑧)[(𝐻^𝑛_𝐿+𝜆₀)^−𝑞(𝐻^𝑛_𝐿−𝑧)⁻¹−(𝐻^𝑛_0,𝐿+𝜆₀)^−𝑞(𝐻^𝑛_0,𝐿−𝑧)⁻¹]𝑑𝑧𝑑𝑧. (2.15)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(𝐻^𝑛_𝐿+𝜆₀)^−𝑞(𝐻^𝑛_𝐿−𝑧)⁻¹−(𝐻^𝑛_0,𝐿+𝜆₀)^−𝑞(𝐻^𝑛_0,𝐿−𝑧)⁻¹=(𝐻^𝑛_0,𝐿+𝜆₀)^−𝑞[(𝐻^𝑛_𝐿−𝑧)⁻¹−(𝐻^𝑛_0,𝐿−𝑧)⁻¹]+[(𝐻^𝑛_𝐿+𝜆₀)^−𝑞−(𝐻^𝑛_0,𝐿+𝜆₀)^−𝑞](𝐻^𝑛_𝐿−𝑧)⁻¹=−(𝐻^𝑛_0,𝐿+𝜆₀)^−𝑞(𝐻^𝑛_0,𝐿−𝑧)⁻¹(𝑉^𝑛_𝑖)(𝐻^𝑛_𝐿−𝑧)⁻¹−^𝑞_𝑙=1(𝐻^𝑛_0,𝐿+𝜆₀)^{𝑙−𝑞−1}(𝑉^𝑛_𝑖)(𝐻^𝑛_𝐿+𝜆₀)^−𝑙(𝐻^𝑛_𝐿−𝑧)⁻¹. (2.16)Estimating the trace of (2.16), we choose 𝜀>0 and write𝑉^𝑛_𝑖=𝑉^𝑛_𝑖⋅𝟏_{|𝑉^𝑛_{𝑖|≤𝜀}}+𝑉^𝑛_𝑖⋅𝟏_{|𝑉^𝑛_𝑖|>𝜀} (2.17)and note that 𝑉^𝑛_𝑖⋅𝟏_{|𝑉^𝑛_{𝑖|≤𝜀}} is bounded by ‖𝑉^𝑛_𝑖⋅𝟏_{|𝑉^𝑛_{𝑖|≤𝜀}}‖≤𝜀. As 𝑉 is nonnegative, one hassupp(𝑉^𝑛_𝑖⋅𝟏_{|𝑉^𝑛_𝑖|>𝜀})⊆^𝑛𝑛_{𝑗=1𝑖=1𝑖≠𝑗}{(𝑥₁,…,𝑥_𝑛)∈ℝ^𝑛𝑑∶𝑉(𝑥_𝑖−𝑥_𝑗𝜀)≥𝑛(𝑛−1)}. (2.18)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 that𝜇({|𝑉^𝑛_𝑖|>𝜀}∩Λ^𝑛_𝐿)≤𝐶(𝑛,𝜀)𝐿^{(𝑛−1)𝑑}, (2.19)where 𝜇(⋅) denotes the Lebesgue measure. Using decomposition (2.17) of 𝑉^𝑛_𝑖, we obtain|Trace(𝐻^𝑛_0,𝐿+𝜆₀)^−𝑞(𝐻^𝑛_0,𝐿−𝑧)⁻¹(𝑉^𝑛_𝑖)(𝐻^𝑛_𝐿−𝑧)⁻¹|≤𝜀|ℑ𝑧|²Trace|(𝐻^𝑛_0,𝐿+𝜆₀)^−𝑞1|+|ℑ𝑧|‖(𝑉^𝑛_𝑖)(𝐻^𝑛_𝐿−𝑧)⁻¹‖⋅Trace|(𝐻^𝑛_0,𝐿+𝜆₀)^−𝑞𝟏_{|𝑉^𝑛_{𝑖|>𝜀}∩Λ}^𝑛_𝐿|≤𝜀|ℑ𝑧|²‖(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹‖^𝑞_𝒯𝑞+1|ℑ𝑧|²‖(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹‖_𝒯^𝑞−1_𝑞⋅‖(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹𝟏_{|𝑉^𝑛_{𝑖|>𝜀}∩Λ}^𝑛_𝐿‖_𝒯𝑞⋅‖(𝑉^𝑛_𝑖)(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹‖, (2.20)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|Trace(𝐻^𝑛_0,𝐿+𝜆₀)^{𝑙−𝑞−1}(𝑉^𝑛_𝑖)(𝐻^𝑛_𝐿+𝜆₀)^−𝑙(𝐻^𝑛_𝐿−𝑧)⁻¹|≤Trace|(𝐻^𝑛_𝐿+𝜆₀)^−𝑙(𝐻^𝑛_0,𝐿+𝜆₀)^{𝑙−𝑞−1}(𝑉^𝑛_𝑖)(𝐻^𝑛_𝐿−𝑧)⁻¹|≤‖(𝐻^𝑛_𝐿+𝜆₀)^−𝑙(𝐻^𝑛_0,𝐿+𝜆₀)^𝑙‖⋅Trace|(𝐻^𝑛_0,𝐿+𝜆₀)^−𝑞−1(𝑉^𝑛_𝑖)(𝐻^𝑛_𝐿−𝑧)⁻¹|≤𝐶|ℑ𝑧|‖(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹‖_𝒯^𝑞−1_𝑞⋅‖(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹𝟏_{|𝑉^𝑛_{𝑖|>𝜀}∩Λ}^𝑛_𝐿‖𝒯_𝑞⋅‖(𝑉^𝑛_𝑖)(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹‖𝜀+𝐶|ℑ𝑧|‖(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹‖^𝑞_𝒯𝑞. (2.21)We are now left with estimating ‖(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹‖_𝒯𝑞 and ‖(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹𝟏_{|𝑉^𝑛_{𝑖|>𝜀}∩Λ}^𝑛_𝐿‖_𝒯𝑞 for 𝑞 sufficiently large, depending on 𝑛𝑑. Therefore, we compute‖(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹𝟏_{|𝑉^𝑛_{𝑖|>𝜀}∩Λ}^𝑛_𝐿‖_𝒯𝑞≤‖(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹(−Δ_Λ^𝑛_𝐿+𝜆₀)^1/2‖_𝒯2𝑞⋅‖(−Δ_Λ^𝑛_𝐿+𝜆₀)^−(1/2)𝟏_{|𝑉^𝑛_{𝑖|>𝜀}∩Λ}^𝑛_𝐿‖_𝒯2𝑞, (2.22)where −Δ_Λ^𝑛_𝐿 is the Dirichlet Laplacian on Λ^𝑛_𝐿. We use the decomposition (1.4). As the Laplacians are positive, the infinitesimal −Δ-boundedness on (𝑉^𝑛_ext)₋, [4, Theorem X.18] and the definition of 𝛾 imply the following form bound:|⟨𝜙,(𝑉^𝑛_ext)₋1𝜙⟩|≤2⟨𝜙,−Δ_Λ^𝑛_𝐿𝜙⟩+𝛾‖𝜙‖². (2.23)As 𝜆₀>2𝛾+1, one has𝐻^𝑛_0,𝐿+𝜆₀≥−Δ_Λ^𝑛_𝐿+(𝑉^𝑛_ext)₋+𝜆₀≥12(−Δ_Λ^𝑛_𝐿−2𝛾+2𝜆₀1)≥2(−Δ_Λ^𝑛_𝐿+𝜆₀). (2.24)Thus, the operator 𝐻^𝑛_0,𝐿+𝜆₀ is invertible and(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹≤2(−Δ_Λ^𝑛_𝐿+𝜆₀)⁻¹. (2.25)Let (𝜇_𝑗)_𝑗 and (𝜙_𝑗)_𝑗, respectively, denote the eigenvalues and eigenfunctions of the Dirichlet Laplacian −Δ_Λ^𝑛_𝐿 (the index 𝑗 runs over (ℕ^𝑛𝑑)^∗). For 𝑞∈ℕ such that 2𝑞>𝑛𝑑, we compute‖(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹(−Δ_Λ^𝑛_𝐿+𝜆₀)^1/2‖_𝜏^2𝑞_2𝑞=_{𝑗∈ℕ𝑛𝑑}(𝜇_𝑗(−Δ_Λ^𝑛_𝐿)+𝜆₀)^𝑞⟨𝜙_𝑗,(𝐻^𝑛_0,𝐿+𝜆₀)⁻¹𝜙_𝑗⟩^2𝑞≤2^2𝑞_{𝑗∈ℕ𝑛𝑑}(𝜇_𝑗(−Δ_Λ^𝑛_𝐿)+𝜆₀)^𝑞⟨𝜙_𝑗,(−Δ_Λ^𝑛_𝐿+𝜆₀)⁻¹𝜙_𝑗⟩)^2𝑞=2^2𝑞_{𝑗∈ℕ𝑛𝑑}(𝜇_𝑗(−Δ_Λ^𝑛_𝐿)+𝜆₀)^−𝑞≤𝐶𝐿^𝑛𝑑. (2.26)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 𝐶_𝑞>0 such that, for any measurable subset Λ^′⊆Λ^𝑛_𝐿, one has‖(−Δ_Λ^𝑛_𝐿+𝜆₀)^−1/2𝟏_Λ′‖_𝒯^2𝑞_2𝑞≤𝐶_𝑞𝜇(Λ′). (2.27)Choosing Λ^′={|𝑉^𝑛_𝑖|>𝜀}∩Λ^𝑛_𝐿 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 thatTrace|(𝐻^𝑛_𝐿+𝜆₀)^−𝑞(𝐻^𝑛_𝐿−𝑧)⁻¹−(𝐻^𝑛_0,𝐿+𝜆₀)^−𝑞(𝐻^𝑛_0,𝐿−𝑧)⁻¹|𝜀≤𝑐(|ℑ𝑧|²𝐿^𝑛𝑑+1|ℑ𝑧|²𝐿^{𝑛𝑑−(𝑑/2𝑞)}+𝜀𝐿|ℑ𝑧|^𝑛𝑑+1𝐿|ℑ𝑧|^{𝑛𝑑−(𝑑/2𝑞)}). (2.28)By using this inequality in (2.15), we get (2.6) as ^0𝑥0𝜑^02𝑑𝑐 being almost analytic, 𝜕𝜑^{0𝑥002𝑑𝑐}(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 form𝑉¹(𝜔,𝑥)=_{𝑗∈ℤ𝑑}𝜔_𝑗𝑢(𝑥−𝑗), (3.1)with a family 𝜔_𝑗∶Ω→ℝ of random variables on (Ω,ℙ). This one-particle models leads us to the 𝑛-particle random “background” potential𝑉^𝑛(𝜔,𝑥₁,…,𝑥_𝑛)=^𝑛_𝑘=1𝑉¹(𝜔,𝑥_𝑘) (3.2)and the interacting 𝑛-particle Hamiltonian reads as𝐻^𝑛(𝜔)=−Δ+𝑉^𝑛_𝑖+𝑉^𝑛(𝜔). (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 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 𝑁_ni=𝑁 for both, the noninteracting and interacting 𝑛-particle system. Note that only translations along a “diagonal” vector (𝑗,𝑗,…,𝑗)∈ℤ^𝑛𝑑 leave 𝐻^𝑛(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 type𝔼(Trace𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^𝑛_Λ))≤𝐶_𝑊𝜂|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 recently (see, e.g., [11, 12]).

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

(H.A.2)The single-site potential 𝑢 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 interacting 𝑛-particle Anderson model, we can choose rather general probabilistic hypothesis like in [13]:

(H.A.3)(𝜔_𝑗∶Ω→ℝ)_{𝑗∈ℤ𝑑} is a family of bounded random variables on the probability space (Ω,ℙ). When 𝜇_𝑗 denotes the conditional probability measure for 𝜔_𝑗 at site 𝑗∈ℤ^𝑑 conditioned on all the other random variables (𝜔_𝑖)_𝑖≠𝑗, that is, for all 𝐴∈ℬ(ℝ),𝜇_𝑗(𝐴)=ℙ({𝜔_𝑗∈𝐴∣(𝜔_𝑖)_𝑖≠𝑗}), (3.5)then, a Wegner estimate à la [13] uses the quantity𝑠(𝜂)∶=sup_{𝑗∈ℤ𝑑}𝔼{sup_𝐸∈ℝ𝜇_𝑗([𝐸,𝐸+𝜂])} (3.6)and is stated as follows.

Theorem 3.1. Let us assume (H.A.2) and (H.A.3), and let Λ⊆ℝ^𝑛𝑑 be a bounded open cube of side length ≥1, let 𝐻^𝑛_Λ(𝜔) be the restriction of 𝐻^𝑛(𝜔) to Λ with Dirichlet boundary conditions. Then, there exists an increasing function𝐶_𝑊𝐸∶ℝ→[0,∞[₀↦𝐶_𝑊(𝐸₀), (3.7)such that for all 𝜂>0𝔼(Trace𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^𝑛_Λ))≤𝐶_𝑊(𝐸₀)𝑠(𝜂)|Λ|. (3.8)In order to prove Theorem 3.1, we prove two preparatory lemmas.

Lemma 3.2. Let Λ⊆ℝ^𝑛𝑑 be an open bounded cube, then the restrictions 𝐻^𝑛_𝑖,Λ and 𝐻^𝑛_𝑖,Λ,𝑁 of 𝐻^𝑛_𝑖=−Δ+𝑉^𝑛_𝑖 to Λ with Dirichlet or Neumann boundary conditions define self-adjoint operators with compact resolvent.Proof. 𝑉^𝑛_𝑖 is infinitesimally −Δ form bounded according to [4, Theorem X.18], so the infinitesimal form bound|⟨Ψ,𝑉^𝑛_𝑖Ψ⟩|≤𝜀‖∇Ψ‖²+𝑏_𝜀‖Ψ‖² (3.9)is true for Ψ∈𝐻¹(ℝ^𝑛𝑑), in particular (3.9) is true for Ψ∈𝒟(−Δ_Λ)=𝐻¹₀(Λ)⊆𝐻¹(ℝ^𝑛𝑑). Hence, the form sum defines via representation theorem a self-adjoint operator 𝐻^𝑛_𝑖,Λ=−Δ_Λ+𝑉^𝑛_𝑖|_Λ. The eigenvalues 𝜇_𝑘(−Δ_Λ) tend to infinity, so by the minimax principle and (3.9), we see that 𝐻^𝑛_𝑖,Λ has compact resolvent. The proof of⟨Ψ,𝑉^𝑛_𝑖Ψ⟩≤𝜀‖∇Ψ‖²+𝑐_𝜀‖Ψ‖²,Ψ∈𝐻¹(Λ) (3.10)uses the extension operator 𝐸_Λ′∶𝐻¹(Λ)→𝐻¹₀(Λ^′) to Λ^′∶={𝑥∈ℝ^𝑛𝑑∶dist(𝑥,Λ)<1}, which has the properties 𝐸_Λ′Ψ|_Λ=Ψ,‖𝐸_Λ′Ψ‖_𝐻¹≤𝑐₁‖Ψ‖_𝐻¹ and ‖𝐸_Λ′Ψ‖_𝐿²≤𝑐₂‖Ψ‖_𝐿²; see [14, Satz 5.6 and Folgerung 5.2]. For 𝐸_Λ′Ψ∈𝐻¹₀(Λ^′)⊆𝐻¹(ℝ^𝑛𝑑), we use (3.9), hence by 𝑉^𝑛_𝑖≥0 and the above properties of 𝐸_Λ′ we get for Ψ∈𝐻¹(Λ),0≤⟨Ψ,𝑉^𝑛_𝑖Ψ⟩≤⟨𝐸_Λ′Ψ,𝑉^𝑛_𝑖𝐸_Λ′Ψ⟩≤𝜀‖∇(𝐸_Λ′Ψ)‖²_𝐿2+𝑏_𝜀‖𝐸_Λ′Ψ‖²_𝐿2=𝜀‖(𝐸_Λ′Ψ)‖²_𝐻1+(𝑏_𝜀−𝜀)‖𝐸_Λ′Ψ‖²_𝐿2≤𝜀𝑐²₁‖∇Ψ‖²+(𝑐²₂(𝑏_𝜀−𝜀)+𝜀𝑐²₁)‖Ψ‖², (3.11)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 let Λ⊆ℝ^𝑛𝑑 be a bounded open cube, 𝐣=(𝑗₁,…,𝑗_𝑛)∈ℤ^𝑛𝑑 with Λ_𝐣∶=Λ∩Λ(𝐣,1)≠∅ (here, Λ(𝐣,1)={|𝑥−𝑗_𝑘|≤1/2,1≤𝑘≤𝑛}), then for every 𝑓∈𝐿²(Λ_𝐣),𝔼{⟨𝑓,𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^𝑛_Λ8)𝑓⟩}≤𝑐²𝑠(𝜂)‖𝑓‖². (3.12)Proof. For every 𝑗∈ℤ^𝑑, we define 𝑢_𝑗∶ℝ^𝑛𝑑→ℝ by𝑢_𝑗(𝑥₁,…,𝑥_𝑛)∶=^𝑛_𝑘=1𝑢(𝑥_𝑘−𝑗) (3.13)and set ^0𝑥0𝜔^02𝑑𝑐_𝑗=(𝜔_𝑙)_𝑙≠𝑗. Fix a component of 𝐣, say 𝑗₁, then we get a decomposition𝑉^𝑛(𝜔,𝑥₁,…,𝑥_𝑛)=𝜔_𝑗1𝑢_𝑗1(𝑥₁,…,𝑥_𝑛)+_{𝑙∈ℤ𝑑𝑙≠𝑗1}𝜔_𝑙𝑢_𝑙(𝑥₁,…,𝑥_𝑛) (3.14)of the random potential 𝑉^𝑛(𝜔), and the same is true for 𝐻^𝑛_Λ(𝜔):𝐻^𝑛_Λ(𝜔)=−Δ_Λ+_{𝑙∈ℤ𝑑𝑙≠𝑗1}𝜔_𝑙𝑢_𝑙𝟏_Λ+𝜔_𝑗1𝑢_𝑗1𝟏_Λ=∶𝐻^{0𝑥002𝑑𝑐𝑛}_Λ(𝜔^{0𝑥002𝑑𝑐}_𝑗1)+𝜔_𝑗1𝑢_𝑗1𝟏_Λ. (3.15)By the covering condition 𝑢𝟏_[−1/2,1/2]^𝑑≥𝑐 on the single site-potential 𝑢, we get 𝑢_𝑗1≥𝑐𝟏_Λ𝐣, hence we can write 𝑓=𝑔𝑢_𝑗1, where 𝑔(𝑥)=𝑓(𝑥)/𝑢_𝑗1(𝑥) almost everywhere, so ‖𝑔‖≤𝑐⁻¹‖𝑓‖. By spectral calculus,^𝐸0_𝐸^+𝜂₀𝑑𝐸⟨𝜑,ℑ(𝐻−𝐸−𝑖𝜂)⁻¹𝜋𝜑⟩≥4⟨𝜑,𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻)𝜑⟩, (3.16)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], apply𝔼⟨𝑓,𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^𝑛_Λ)𝑓⟩=𝔼_ℝ𝑑𝜇_𝑗1(𝜔_𝑗1)⟨𝑔,𝑢_𝑗1𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^{0𝑥002𝑑𝑐𝑛}_Λ(𝜔^{0𝑥002𝑑𝑐}_𝑗1)+𝜔_𝑗1𝑢_𝑗1)𝑢_𝑗1≤4𝑔⟩𝜋𝔼_ℝ𝑑𝜇_𝑗1(𝜔_𝑗1)^𝐸0_𝐸^+𝜂₀𝑑𝐸ℑ⟨𝑔,𝑢_𝑗1(𝐻^{0𝑥002𝑑𝑐𝑛}_Λ(𝜔^{0𝑥002𝑑𝑐})+𝜔_𝑗1𝑢_𝑗1−𝐸−𝑖𝜂)⁻¹𝑢_𝑗1≤8𝑔⟩𝑐²‖𝑓‖²𝑠(𝜂). (3.17)

Proof. By (H.A.2) and (H.A.3), we get a ℙ-almost sure bound ‖𝑉^𝑛(𝜔)‖≤|𝑉^𝑛|, then Lemma 3.2 implies that the restrictions 𝐻^𝑛_Λ(𝜔) and 𝐻^𝑛_Λ,𝑁(𝜔) of 𝐻^𝑛(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∈ℤ^𝑛𝑑∶Λ(𝐣,1)∩Λ≠∅} and for 𝐣∈𝐽 set Λ_𝐣∶=Λ(𝐣,1)∩Λ. Then Λ′∶=Λ⧵∪_𝐣∈𝐽Λ_𝐣 has Lebesgue measure 0, so by [15, 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 with 𝐻^𝑛_{𝑖,Λ𝐣,𝑁} defined in Lemma 3.2, we get ℙ-almost sure:𝐻^𝑛_Λ(𝜔)≥𝐻^𝑛_Λ,𝑁∶=_𝐣∈𝐽𝐻^𝑛_{𝑖,Λ𝐣,𝑁}−|𝑉^𝑛|. (3.19)By spectral calculus,Trace(𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^𝑛_Λ(𝜔)))≤𝑒^𝐸0+𝜂Trace(𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^𝑛_Λ(𝜔))𝑒^{−𝐻𝑛Λ(𝜔)}). (3.20)Let (𝜑_𝑘(𝜔))_𝑘∈ℕ be the orthogonal basis of 𝐿²(Λ) consisting out of eigenvectors of 𝐻^𝑛_Λ(𝜔) to eigenvalues 𝜇_𝑘(𝜔) and let 𝑀(𝜔)∶={𝑘∈ℕ∶𝜇_𝑘(𝜔)∈]𝐸₀,𝐸₀+𝜂]}, thenTrace(𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^𝑛_Λ(𝜔))𝑒^{−𝐻𝑛Λ(𝜔)})=_{𝑘∈𝑀(𝜔)}𝑒^{−⟨𝜑𝑘(𝜔),𝐻𝑛Λ(𝜔)𝜑𝑘(𝜔)⟩}≤_{𝑘∈𝑀(𝜔)}𝑒^{−⟨𝜑𝑘(𝜔),𝐻𝑛Λ,𝑁𝜑𝑘(𝜔)⟩}≤_{𝑘∈𝑀(𝜔)}⟨𝜑_𝑘(𝜔),𝑒^{−𝐻𝑛Λ,𝑁}𝜑_𝑘(𝜔)⟩=Trace(𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^𝑛_Λ(𝜔))𝑒^{−𝐻𝑛Λ,𝑁}), (3.21)where the last estimate follows from Jensen's inequality. Let (𝜙_𝑘,𝐣)_𝑘∈ℕ be an orthonormal basis of 𝐿²(Λ_𝐣) consisting of eigenvectors of 𝐻^𝑛_{𝑖,Λ𝐣,𝑁} to the eigenvalues 𝐸_𝑘,𝐣, thenTrace(𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^𝑛_Λ(𝜔))𝑒^{−𝐻𝑛Λ,𝑁})=_{𝑘∈ℕ𝐣∈𝐽}⟨𝜙_𝑘,𝐣,𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^𝑛_Λ(𝜔))𝜙_𝑘,𝐣⟩𝑒^{−𝐸𝑘,𝐣+|𝑉𝑛|}. (3.22)As 𝜙_𝑘,𝐣∈𝐿²(Λ_𝐣) and ‖𝜙_𝑘,𝐣‖≤1, Lemma 3.3 implies𝔼⟨𝜙_𝑘,𝐣,𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^𝑛_Λ(𝜔))𝜙_𝑘,𝐣8⟩≤𝑐²𝑠(𝜂). (3.23)As 𝑉^𝑛_𝑖 is nonnegative, the eigenvalues 𝐸_𝑘,𝐣 of 𝐻^𝑛_{𝑖,Λ𝐣,𝑁}=−Δ_Λ𝐣,𝑁+𝑉^𝑛_𝑖|_Λ𝐣 are estimated from what follows by the eigenvalues of −Δ_Λ𝐣,𝑁. These are known explicitly, see [15, page 266], which can be used to estimate_{𝑘∈ℕ𝐣∈𝐽}𝑒^{−𝐸𝑘,𝐣}𝑒≤Card(𝐽)(^𝜋2𝑒^𝜋2)−1^𝑛𝑑. (3.24)If the side-length of Λ is bigger than 1, then Card(𝐽)≤3^𝑛𝑑|Λ|, so when applying expectation value to the chain of inequalities (3.20) to (3.24), it implies𝔼(Trace𝟏_{]𝐸0,𝐸0+𝜂]}(𝐻^𝑛_Λ))≤𝑒^{𝐸0+𝜂+|𝑉𝑛|}(3𝑒^𝜋2𝑒^𝜋2)−1^𝑛𝑑8𝑐²𝑠(𝜂)|Λ|. (3.25)

Under the assumptions (H.A.2) and (H.A.3), we have𝑁(𝐸)=𝔼(𝑁(𝐸,⋅)𝟏_Ω′)=𝔼(𝑁(𝐸,⋅)), (3.26)hence by the Wegner estimate we can deduce regularity properties of 𝑁 from those of the conditioned measures (𝜇_𝑗)_{𝑗∈ℤ𝑑} via0≤𝑁(𝐸+𝜂)−𝑁(𝐸)≤𝐶_𝑊(𝐸+𝜂)𝑠(𝜂). (3.27)