Short-range quantum crystals with defects

# The reduced Hartree-Fock model for short-range quantum crystals with defects

Salma Lahbabi CNRS & Laboratoire de Mathématiques (UMR 8088), Université de Cergy-Pontoise
95000 Cergy-Pontoise Cedex, France
CERMICS, École Nationale des Ponts et Chaussées (Paristech)
& INRIA (Micmac Project), 6-8 Av. Blaise Pascal, 77455 Champs-sur-Marne, France
August 30, 2019
###### Abstract.

In this article, we consider quantum crystals with defects in the reduced Hartree-Fock framework. The nuclei are supposed to be classical particles arranged around a reference periodic configuration. The perturbation is assumed to be small in amplitude, but need not be localized in a specific region of space or have any spatial invariance. Assuming Yukawa interactions, we prove the existence of an electronic ground state, solution of the self-consistent field equation. Next, by studying precisely the decay properties of this solution for local defects, we are able to expand the density of states of the nonlinear Hamiltonian of a system with a random perturbation of Anderson-Bernoulli type, in the limit of low concentration of defects. One important step in the proof of our results is the analysis of the dielectric response of the crystal to an effective charge perturbation.

## 1. Introduction

In solid state physics and materials science, the presence of defects in materials induces many interesting properties, such as Anderson localization and leads to many applications such as doped semi-conductors The mathematical modeling and the numerical simulation of the electronic structure of these materials is a challenging task, as we are in the presence of infinitely many interacting particles.

The purpose of this paper is to construct the state of the quantum electrons of a mean-field crystal, in which the nuclei are classical particles arranged around a reference periodic configuration. We work with the assumption that the nuclear distribution is close to a chosen periodic arrangement locally, but the perturbation need not be localized in a specific region of space and it also need not have any spatial invariance. To our knowledge, this is the first result of this kind for Hartree-Fock type models for quantum crystals, with short-range interactions. By studying precisely the behavior of our solution, we are then able to expand the density of states of the Hamiltonian of the system in the presence of a random perturbation of Anderson-Bernoulli type, in the limit of low concentration of defects, that is when the Bernoulli parameter tends to zero. The state of the random crystal and the mean-field Hamiltonian were recently constructed in [8]. Our small- expansion is the nonlinear equivalent of a previous result by Klopp [19] in the linear case.

The mean-field model we consider in this paper is the reduced Hartree-Fock model [31], also called the Hartree model in the physics literature. It is obtained from the generalized Hartree-Fock model [25] by removing the exchange term. As the Coulomb interaction is long-range, it is a difficult mathematical question to describe infinite systems interacting through the Coulomb potential. In the following, we assume that all the particles interact through Yukawa potential of parameter . In fact, we can assume any reasonable short-range potential, but we concentrate on the Yukawa interaction in dimension for simplicity. We consider systems composed of infinitely many classical nuclei distributed over the whole space and infinitely many electrons.

We start by recalling the definition of the reduced Hartree-Fock (rHF) model for a finite system composed of a set of nuclei having a density of charge and electrons. The electrons are described by the -body wave-function (called a Slater determinant)

 ψ(x1,⋯,xN)=1√N!det(φj(xi)),

where the functions satisfy . The rHF equations then read

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩Hφi=λiφiH=−12Δ+V−ΔV+m2V=∣∣Sd−1∣∣(ρψ−νnuc)∀1≤i≤N, (1)

where and are the smallest eigenvalues of the operator , assuming that . Here, is the Lebesgue measure of the unit sphere (, , ). The existence of a solution of (1) is due to Lieb and Simon [26].

In order to describe infinite systems, it is more convenient to reformulate the rHF problem in terms of the one-particle density matrix formalism [24]. In this formalism, the state of the electrons is described by the orthogonal projector of rank and the equations (1) can be recast as

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩γ=1(H≤ϵF)H=−12Δ+V−ΔV+m2V=∣∣Sd−1∣∣(ργ−νnuc), (2)

where formally and the Fermi level is any real number in the gap .

For infinite systems, the rHF equation is still given by (2), but is now an infinite rank operator as there are infinitely many electrons in the system. The operator needs to be locally trace class for the electronic density to be well-defined in .

The rHF equation (2) was solved for periodic nuclear densities

 νnuc=νper=∑k∈Rη(⋅−k)

by Catto, Le Bris and Lions in [10], and periodic nuclear densities with local perturbations

 νnuc=∑k∈Rη(⋅−k)+ν

were studied by Cancès, Deleurence and Lewin in [7]. We have denoted by the underlying discrete periodic lattice. The corresponding Hamiltonians are denoted by and . Stochastic distributions,

 νnuc(ω,⋅)=∑k∈Rη(⋅−k)+∑k∈Rqk(ω)χ(⋅−k)

for instance, were treated in [8].

Our present work follows on from [7, 6, 8]. We are going to solve the equation (2) in the particular case where

 νnuc=νper+ν, (3)

where is a periodic nuclear distribution so that the corresponding background crystal is an insulator (the mean-field Hamiltonian has a gap around ), and is a small enough arbitrary perturbation of the background crystal. The perturbation needs to be small in amplitude locally, but must not be local or have any spatial invariance.

The rHF model is an approximation of the -body Schrödinger model, for which there is no well-defined formulation for infinite systems so far. The only available result is the existence of the thermodynamic limit of the energy: the energy per unit volume of the system confined to a box, with suitable boundary conditions, converges when the size of the box grows to infinity. The first theorem of this form for Coulomb interacting systems is due to Lieb and Lebowitz in [22]. In this latter work, nuclei are considered as quantum particle and rotational invariance plays a crucial role. For quantum systems in which the nuclei are classical particles, the thermodynamic limit was proved for perfect crystals by Fefferman [12] (a recent proof has been proposed in [17]) and for stationary stochastic systems by Blanc and Lewin [4]. Similar results for Yukawa interacting systems are simpler than for the Coulomb case and follow from the work of Ruelle and Fisher [13] for perfect crystals and Veniaminov [32] for stationary stochastic systems. Unfortunately, very little is known about the limiting quantum state in both cases.

For (orbital-free) Thomas-Fermi like theories, the periodic model was studied in [26, 9], the case of crystals with local defects was studied in [5] and stochastic systems were investigated in [3]. To the best of our knowledge, the only works dealing with systems with arbitrary distributed nuclei are [9, 2] for Thomas-Fermi type models.

As mentioned before, our work is the first one to consider this kind of systems in the framework of Hartree-Fock type models. Our results concern small perturbations of perfect crystals interacting through short-range Yukawa potential. Extending these results to more general geometries and for the long-range Coulomb interaction are important questions that we hope to address in the future.

After having found solutions of (2) for any (small enough) , we study the properties of this solution for local perturbations . This enables us to investigate small random perturbations of perfect crystals. Precisely, we consider nuclear distributions

 νnuc(ω,x)=νper(x)+∑k∈Rqk(ω)χ(x−k),

where are i.i.d. Bernoulli variables of parameter and is a compactly supported function which is small enough in . We are interested in the properties of the system in the limit of low concentration of defects, that is when the parameter goes to zero. We prove that the density of states of the mean-field Hamiltonian , which describes the collective behavior of the electrons, admits an expansion of the form

 np=n0+J∑j=1μjpj+O(pJ+1). (4)

Here, is the density of states of the unperturbed Hamiltonian and is a function of the spectral shift function for the pair of operators and , the latter being the mean-field Hamiltonian of the system with only one local defect constructed in [7]. We give in Theorem 2.7 a precise meaning of .

In [19], Klopp considers the empirical linear Anderson-Bernoulli model

 H=−12Δ+V0+VwithV(ω,x)=∑k∈Rqk(ω)η(x−k),

where is a linear periodic potential and an exponentially decaying potential. He proves that the density of states of the Hamiltonian admits an asymptotic expansion similar to (4). The case where is distributed following a Poisson law instead of Bernoulli is dealt with in [20]. Our proof of (4) follows the same lines as the one of Klopp. The main difficulty here is to understand the decay properties of the mean-field potential solution of the self-consistent equations (2). For this reason, we dedicate an important part of this paper to the study of these decay properties. In Theorem 2.3 below, we show that for a compactly supported perturbation , the difference decays faster than any polynomial far from the support of the perturbation . Moreover, we show that the potential generated by two defects that are far enough is close to the sum of the potentials generated by each defect alone.

The article is organized as follow. In Section 2, we present the main results of the paper. We start by recalling the reduced Hartree-Fock model for perfect crystals and perfect crystals with local defects in Section 2.1. In Section 2.2, we state the existence of solutions to the self-consistent equations (2) for given by (3). We also explain that our solution is in some sense the minimizer of the energy of the system. We also prove a thermodynamic limit, namely, the ground state of the system with the perturbation confined to a box converges, when the size of the box goes to infinity, to the ground state of the system with the perturbation . In Section 2.3, we prove decay estimates for the mean-field density and potential. In Section 2.4, we present the expansion of the density of states of the mean-field Hamiltonian. The proofs of all these results are provided in Sections 456 and 7. In Section 3, we study the dielectric response of a perfect crystal to a variation of the effective charge distribution, which plays a key role in this paper.

Acknowledgement. I thoroughly thank Éric Cancès and Mathieu Lewin for their precious help and advices. The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013 Grant Agreement MNIQS no. 258023).

## 2. Statement of the main results

### 2.1. The rHF model for crystals with and without local defects

In defect-free materials, the nuclei and electrons are arranged according to a discrete periodic lattice of , in the sense that both the nuclear density and the electronic density are -periodic functions. For simplicity, we take in the following. The reduced Hartree-Fock model for perfect crystals has been rigorously derived from the reduced Hartree-Fock model for finite molecular systems by means of thermodynamic limit procedure in [10, 7] in the case of Coulomb interaction. The same results for Yukawa interaction are obtained with similar arguments. The self-consistent equation (2) then reads

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩γ0=1(Hper≤ϵF)Hper=−12Δ+Vper−ΔVper+m2Vper=∣∣Sd−1∣∣(ργ0−νper). (5)

It has been proved in [10, 7] that (5) admits a unique solution which is the unique minimizer of the periodic rHF energy functional.

Most of our results below hold only for insulators (or semi-conductors). We therefore make the assumption that

 Hper has a spectral gap around ϵF. (6)

The rHF model for crystals with local defects was introduced and studied in [7]. A solution of the rHF equation (2) is constructed using a variational method. One advantage of this method is that there is no need to assume that the perturbation is small in amplitude. The idea is to find a minimizer of the infinite energy of the system by minimizing the energy difference between the perturbed state and the perfect crystal. The ground state density matrix can thus be decomposed as

 γ=γ0+Qν, (7)

where is a minimizer of the energy functional

 Eν(Q)=Trγ0((Hper−ϵF)Q)+12Dm(ρQ−ν,ρQ−ν) (8)

on the convex set

 K={Q∗=Q,−γ0≤Q≤1−γ0,(−Δ+1)12Q∈S2(L2(Rd)),(−Δ+1)12Q±±(−Δ+1)12∈S1(L2(Rd))}, (9)

where , and . We use the notation to denote the Schatten class. In particular is the set of Hilbert-Schmidt operators. The second term of (8) accounts for the interaction energy and is defined for any charge densities by

 Dm(f,g) =∣∣Sd−1∣∣∫Rd¯¯¯¯¯¯¯¯¯¯¯ˆf(p)ˆg(p)|p|2+m2dp=∫Rd∫Rdf(x)Ym(x−y)g(y)dxdy,

where is the Fourier transform of . The Yukawa kernel , the inverse Fourier transform of , is given by

 Ym(x)=⎧⎪ ⎪⎨⎪ ⎪⎩m−1e−m|x|ifd=1,K0(m|x|)ifd=2,|x|−1e−m|x|ifd=3,

where is the modified Bessel function of the second type [23]. It has been proved in [7] that the energy functional (8) is convex and that all its minimizers share the same density . These minimizers are of the form

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩γ=1(H≤ϵF)+δH=−12Δ+V−ΔV+m2V=∣∣Sd−1∣∣(ργ−νper−ν), (10)

where . If is small enough in the -norm, then .

One of the purposes of this article is to find decay estimates of the potential solution of (10) that are necessary in the study of the Anderson-Bernoulli random perturbations of crystals.

### 2.2. Existence of ground states

In this section, we state our results concerning the electronic state of a perturbed crystal. The host crystal is characterized by a periodic nuclear density such that the gap assumption (6) holds. The perturbation is given by a distribution . The total nuclear distribution is then

 νnuc=νper+ν.

In Theorem 2.1 below, we show that if is small enough in the -norm, then the rHF equation (2) admits a solution . This solution is unique in a neighborhood of . The proof consists in formulating the problem in terms of the density and using a fixed point technique, in the spirit of [15].

###### Theorem 2.1 (Existence of a ground state).

There exists and such that for any satisfying , there is a unique solution to the self-consistent equation

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩γ=1(H≤ϵF)H=−12Δ+V−ΔV+m2V=∣∣Sd−1∣∣(ργ−ν−νper) (11)

satisfying

 ∥∥ργ−ργ0∥∥L2\rm unif ≤C∥ν∥L2\rm unif . (12)

We denote this solution by , the response electronic density by and the defect mean-field potential by .

For a local defect such that , equation (11) admits a unique solution which coincides with the ground state solution of (7) constructed in [7]. Indeed, the solution given in Theorem 2.1 is a solution of the defect problem (10). Moreover, in the proof of Theorem 2.1, we prove that has a gap around , thus necessarily in (10). As all the solutions of (10) share the same density, (10) (thus (11)) admits a unique solution.

The ground state constructed in Theorem 2.1 is in fact the unique minimizer of the "infinite" rHF energy functional. Indeed, following ideas of [16], we can define the relative energy of the system with nuclear distribution by subtracting the "infinite" energy of from the "infinite" energy of a test state :

 Erelν(γ):=Trγν((H−ϵF)(γ−γν))+12Dm(ργ−ργν,ργ−ργν).

This energy is well-defined for states such that is finite rank and smooth enough for instance, but one can extend it to states in a set similar to in (9). The minimum of the energy is attained for . Moreover, as has a gap around , is strictly convex and is its unique minimizer.

In the following theorem, we show that if we confine the defect to a box of finite size, then the ground state of the system defined by the theory of local defects presented in Section 2.1 converges, when the size of the box goes to infinity, to the ground state of the system with the defect defined in Theorem 2.1. We denote by .

###### Theorem 2.2 (Thermodynamic limit).

There exists such that for any satisfying , the sequence converges in to as .

### 2.3. Decay estimates

In this section, we prove some decay estimates of the mean-field potential and the mean-field density , which will be particularly important to understand the system in the presence of rare perturbations in the next section.

Theorem 2.3 below is crucial in the proof of Theorem 2.7. Indeed, we will need uniform decay estimates for compactly supported defects, with growing supports and uniform local norms.

###### Theorem 2.3 (Decay rate of the mean-field potential and density).

There exists and such that for any satisfying , we have for

 ∥Vν∥H2\rm unif (Rd∖CR(ν))+∥ρν∥L2\rm unif (Rd∖CR(ν))≤Ce−C′(logR)2∥ν∥L2\rm unif (Rd), (13)

where .

###### Remark 2.4.

Using the same techniques as in the proof of Theorem 2.3, we can prove (see [21]) that there exists and such that for any satisfying and , we have for

 ∥Vν∥H2(Rd∖CR(ν))+∥ρν∥L2(Rd∖CR(ν))≤Ce−C′(logR)2∥ν∥L2(Rd). (14)

Estimate (14) gives a decay rate of the solution of the rHF equation for crystals with local defects, far from the support of the defect. In particular, it shows that . This decay is due to the short-range character of the Yukawa interaction. In the Coulomb case, it has been proved in [6] that for anisotropic materials, .

The decay rate of and proved in Theorem 2.3 is faster than the decay of any polynomial, but is not exponential, which we think should be the optimal rate.

Proposition 2.5 below is an important intermediary result in the proof of Theorem 2.2. It says that the mean-field density and potential on a compact set depend mainly on the nuclear distribution in a neighborhood of this compact set.

###### Proposition 2.5 (The mean-field potential and density depend locally on ν).

There exists such that for any there exists such that for any satisfying and any we have

 ∥∥Vν−VνL∥∥H2\rm unif (B(0,L/4β))+∥∥ρν−ρνL∥∥L2\rm unif (B(0,L/4β))≤CLβ∥ν∥L2\rm unif ,

where .

In the same way, we obtain the following result which will be very useful in the proof of Theorem 2.7. We prove that the potential generated by two defects that are far enough is close to the sum of the potentials generated by each defect alone in the sense of

###### Proposition 2.6.

There exists such that for any , there exists such that for any satisfying and , we have

 ∥∥Vν1+ν2−Vν2∥∥H2\rm unif% (CR/4β(ν2))+∥∥ρν1+ν2−ρν2∥∥L2\rm unif (CR/4β(ν2)) ≤CRβ(∥ν1∥L2\rm unif +∥ν2∥L2\rm unif ).
###### Proof.

The proof is the same as the one of Proposition 2.5 with and . ∎

### 2.4. Asymptotic expansion of the density of states

In this section, we use our previous results to study a particular case of random materials. In the so-called statistically homogeneous materials, the particles are randomly distributed over the space with a certain spatial invariance. More precisely, the nuclear distribution (thus the electronic density) is stationary in the sense

 νnuc(τk(ω),x)=νnuc(ω,x+k),

where is an ergodic group action of on the probability set (see Figure 1).

One famous example of such distributions is the Anderson model

 νnuc(ω,x)=∑k∈Zdqk(ω)χ(x−k),

where, typically, and the ’s are i.i.d. random variables. The reduced Hartree-Fock model for statistically homogeneous materials was introduced in [8]. The state of the electrons is described by a random self-adjoint operator acting on such that almost surely. The rHF equation is then

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩γ(ω)=1(H(ω)≤ϵF)+δ(ω)H(ω)=−12Δ+V(ω,⋅)−ΔV(ω,⋅)+m2V(ω,⋅)=∣∣Sd−1∣∣(ργ(ω)−ν(ω,⋅))almost surely, (15)

where almost surely. The solutions of (15) turn out to be the minimizers of the energy functional

 E––νnuc(γ)=Tr–––((−12Δ−ϵF)γ)+D––m(ργ−νnuc,ργ−νnuc),

where and

 D––m(f,g)=E(∫Rd∫Γf(x)Ym(x−y)g(y)dxdy).

Here, denotes the semi-open unit cube. Thanks to the convexity of , it has been proved in [8] that the minimizers of share the same density. Therefore, the Hamiltonian solution of (15) is uniquely defined.

In this paper, we are interested in the particular case of random perturbation of perfect crystals

 νnuc(ω,x)=νper(x)+νp(ω,x)

in the limit of low concentration of defects. We restrict our study to Anderson-Bernoulli type perturbations, that is, we suppose that at each site of , there is a probability to see a local defect , independently of what is happening in the other sites. More precisely, we consider the probability space endowed with the measure and the ergodic group action . The defect distribution we consider is then given by

 νp(ω,x)=∑k∈Zdqk(ω)χ(x−k)

where is the coordinates of and with . The ’s are i.i.d. Bernoulli variables of parameter . If , then almost surely and (15) admits a unique solution. For almost every , this solution coincides with the solution of (11) constructed in Theorem 2.1. For convenience, we will from now on use the notation

 H0=Hper−ϵF,

where we recall that is the Fermi level. We introduce the mean-field Hamiltonian corresponding to the system with the defect

 Hp=H0+VνpwithVνp(ω,x)=Ym∗(ρνp−νp).

As is stationary with respect to the ergodic group and uniformly bounded in , then by [27, Theorem 5.20], there exists a deterministic positive measure , the density of states of , such that for any in the Schwartz space

 ∫Rφ(x)np(dx)=Tr–––(φ(Hp)).

For , we define the self-consistent operator corresponding to the system with the defects in

 HK=H0+VK,

where

 VK=Ym∗(ρK−νK),νK=∑k∈Kχ(⋅−k)andρK=ρνK.

If , we denote by the spectral shift function [33] for the pair of operators and . It is the tempered distribution in satisfying, for any ,

 Tr(φ(HK)−φ(H0))=∫RξK(x)φ′(x)dx=−∫Rξ′K(x)φ(x)dx.

In Theorem (2.7) below, we give the asymptotic expansion of the density of states in terms of powers of the Bernoulli parameter .

###### Theorem 2.7.

For such that and such that , we define the tempered distribution by

 μK(x)=−1|K|∑K′⊂K(−1)∣∣K∖K′∣∣ξ′K′(x).

There exists such that if , then

1. for , is a well-defined convergent series in .

2. for , there exists , independent of such that for any ,

where is the density of states of the unperturbed Hamiltonian and .

In Theorem 2.7, we only present the expansion of the density of states until the second order . The proof of the expansion up to any order should follow the same lines and techniques used here.

A result similar to Theorem 2.7 was obtained in [19] in the linear case. Materials with low concentration of defects were studied by Le Bris and Anantharaman [1]. in the framework of stochastic homogenization.

The proof of Theorem 2.7 follows essentially the proof of [19, Theorem 1.1]. It uses the decay of the potential related to each local defect. In [19, Theorem 1.1], the linear potential is assumed to decay exponentially. In our nonlinear model, the decay estimates established in Section 2.3 play a crucial role in the proof.

The rest of the paper is devoted to the proofs of the results presented in this section. In the next section, we study the dielectric response of the crystal to an effective charge perturbation. The results of Section 3 will be used in later sections.

## 3. Dielectric response for Yukawa interaction

In this section, we study the dielectric response of the electronic ground state of a crystal to a small effective charge perturbation . This means more precisely that we expand the formula

 Qf=1(H0+f∗Ym≤0)−1(H0≤0)

in powers of (for small enough) and state important properties of the first order term. The higher order term will be dealt with later in Lemma 4.1. For Coulomb interactions and local perturbation , where is the Coulomb space, this study has been carried out in [6] in dimension .

The results of this section can be used in the linear model or the mean-field framework. In the reduced Hartree-Fock model we consider in this paper, the effective charge perturbation is , where is the electronic density of the response of the crystal to the nuclear perturbation defined in Theorem 2.1. Expanding (formally) in powers of and using the resolvent formula leads to considering the following operator

 Q1,f=12iπ∮C1z−H0f∗Ym1z−H0dz,

where is a smooth curve in the complex plane enclosing the whole spectrum of below (see Figure 2).

By the residue Theorem, the operator does not depend on the particular curve chosen as above. We recall that is bounded with relative bound . Thus is bounded below by the Rellich-Kato theorem [28, Theorem X.12]. Theorem 3.1 below studies the properties of the dielectric response operator and the operator , which will play an important role in the resolution of the self-consistent equation (11). In particular, it gives the functional spaces on which and are well-defined for both local and extended charge densities. It also says that is local in the sense that its off-diagonal components decay faster than any polynomial. We consider , endowed with the scalar product

 ⟨f,g⟩H−1=1(2π)d∫Rd¯¯¯¯¯¯¯¯¯¯¯ˆf(p)ˆg(p)|p|2+m2dp.
###### Theorem 3.1 (Properties of the dielectric response).

We have

1. The operator

 L:H−1(Rd)→H−1(Rd)f↦−ρQ1,f,

is well-defined, bounded, non-negative and self-adjoint. Hence is invertible and bicontinuous.

2. The operator is bounded from to and is a well-defined, bounded operator from into itself.

3. The operator

 L:L2\rm unif (Rd)→L2\rm unif (Rd)f↦−ρQ1,f,

is well-defined and bounded. The operator is invertible on and its inverse is bounded.

4. There exist and such that for any such that , we have

 ∥∥∥1Γ+j11+L1Γ+k∥∥∥B≤Ce−C′(log|k−j|)2. (16)
###### Proof.

The proof consists in the following 6 steps. In the whole paper and are constants whose value might change from one line to the other.

#### Step 1

Proof of (i). The proof is similar to the one of [6, Proposition 2], with the Yukawa kernel , instead of the Coulomb kernel. In the Yukawa case, plays the role of the Coulomb space. The proof of [6, Proposition 2] can easily be adapted to our case. We skip the details for the sake of brevity.

#### Step 2

Proof of (ii). Let . Then and

 (17)

Therefore, by [6, Proposition 1], , where has been defined in (9), and . Arguing by duality, we have for any ,

 Tr(Q1,fW)=∫RdρQ1,fW. (18)

Besides, by the Kato-Seiler-Simon inequality [30, Theorem 4.1] for

 ∀p≥2,∥f(−i∇)g(x)∥S2≤(2π)−dp∥f∥Lp∥g∥Lp (19)

and the fact that

 (z−H0)−1(1−Δ) is uniformly bounded on % the contour C, (20)

we have

and

 ∣∣Tr(Q1,fW)∣∣=∣∣∣12iπ∮CTr(1z−H0Ym∗f1z−H0W)dz∣∣∣ ≤C∥Ym∗f∥L2∥W∥L2. (21)

The bound (20) follows from the following lemma.

###### Lemma 3.2.

Let . Then there exists , depending only on the -norm of , such that for any , we have

 ∥∥(−Δ+1)(−Δ+W−z)−1∥∥B≤C1+|z|d(z,σ(−Δ+W)).

In particular, if is a compact set of , then is uniformly bounded on .

The proof of Lemma 3.2 is elementary, it can be read in [21]. In view of (17), (18) and (21), it follows that

 ∣∣∣∫Rd(Lf)W∣∣∣ ≤C∥f∥H−1∥W∥L2.

We deduce that

 ∥Lf∥L2≤C∥f∥H−1.

We now prove that is bounded on . Let and such that . Then, . As is bounded from into itself, we have

 ∥f∥H−1≤C∥g∥H−1≤C∥g∥L2.

Therefore, as is continuous from to ,

 ∥f∥L2 =∥g−Lf∥L2≤∥g∥L2+∥Lf∥L2≤∥g∥L2+C∥f∥H−1≤C∥g∥L2,

which concludes the proof of (ii).

#### Step 3

Proof of the first part of (iii): is well-defined and bounded on . First, we consider a bounded operator and prove that is locally trace class. For and , we have by (20) and the Kato-Simon-Seiler inequality (19) that is trace class and that there exists independent of such that

 ∣∣∣Tr(χ1z−H0A1z−H0χ)∣∣∣ ≤∥∥∥χ1z−H0A1z−H0χ∥∥∥S1 ≤∥∥∥χ1z−H0∥∥∥S2∥A∥B∥∥∥1z−H0χ∥∥∥S2≤C∥A∥B∥χ∥2L2.

It follows that the operator is locally trace class and that its density is in . We now show that is in fact in