The Integrated Density of States for an Interacting Multiparticle Homogeneous Model and Applications to the Anderson Model
For a system of 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 -particle Hamiltonian. Moreover, this integrated density of states coincides with that of the free particle Hamiltonian. For the interacting -particle Anderson model, we prove regularity properties of the integrated density of states by establishing a Wegner estimate.
Recently, models describing interacting quantum particles in a random potential have been studied (see, e.g., [1–3]). We consider interacting particles moving in a “homogeneous” potential in the -dimensional configuration space 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 -particle Hamiltonian. The proof consists of two steps. First, we prove the claim for the noninteracting -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 -particle Anderson model in
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 -particle Hamiltonian satisfies where the Laplacian on describes the free kinetic energy of the particles. As all the particles are in the same background, the potential is of the formHence, the noninteracting -particle Hamiltonian is a sum of one-particle Hamiltonians On the one particle potential we assume that
(H.1.a) is locally square integrable and is an infinitesimally -bounded potential, that is, and for all there exists such that for all (H.1.b)the operator admits an integrated density of states, say that is, if denotes the Dirichlet restriction of to a cube centered at 0 of side-length, then the following limit exists Assumption (H.1.a) implies essential self-adjointness of on by [4, Theorem X.29]. Indeed,where
(i) is infinitesimally -bounded, that is, (1.2) holds for the same constants and the Laplacian over (ii) is nonnegative locally square integrable. The self-adjoint extensions of and are again denoted by and 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., ).
In the definition of the density of states, we could also have considered the case of Neumann or other boundary conditions.
The interacting -particle Hamiltonian is of the formwhereis a localized repulsive interaction potential generated by the particles; so we assume that
(H.2) is measurable nonnegative locally square integrable and tends to at infinity. The standard repulsive interaction in three-dimensional space is of course the Coulomb interaction In some cases, due to screening, it must be replaced by the Yukawa's interaction
Finally, we make one more assumption on both and we assume that
(H.3)the operator is bounded. Assumption (H.3) is satisfied in the case of the Coulomb and Yukawa potential for those satisfying (H.1.a); is self-adjoint on hence where due to closed graph theorem and for Coulomb and Yukawa's interaction potentials see [4, Theorem X.16].
2. The Integrated Density of States
We now compute the IDS for the -particle model. Let be the cube in centered at with side-length and write for the product of copies of We denote the restriction of the interacting -particle Hamiltonian to with Dirichlet boundary conditions by Clearly assumptions (H.2) and (H.1.a) guarantee that is bounded from what follows with compact resolvent. Hence, for any one defines the normalized counting functionsAs usual, the IDS of is defined as the limit of when Equivalently, one can define the density of states measure applied to a test function as the limit of 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 is equivalent .
2.1. The IDS for the Noninteracting -Particle System
Recall that, by assumption (H.1.b), the single particle model admits an IDS (see ) and a density of states measure denoted, respectively, by and
Let be the restriction of to with Dirichlet boundary conditions. One has the following lemma.
Lemma 2.1. The IDS for the noninteracting -particle Boltzmann model given byexists and satisfies 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 this results from assumption (H.1.a). When the field is not bounded from what follows, one will need some estimate on the decay of and 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 is the sum of commuting Hamiltonians, each of which is unitarily equivalent to so is its restriction to the cube As the sum decomposition of commutes with the restriction to the eigenvalues of are exactly the sum of eigenvalues of restricted to This immediately yields thatwhere is the eigenvalue counting function for restricted to and is its counting measure (i.e., ). The normalized counting function and measure, and are defined asThe existence of the density of states of then exactly says that and converge, respectively, to and The convergence of to 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 -body Hamiltonian. Moreover, in the case when the two limits coincide for the one-body Hamiltonian, they also coincide for the noninteracting -body Hamiltonian. Using Dirichlet-Neumann bracketing, one then sees that the integrated densities of states for both the one-body and noninteracting -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 -Particle System
Let denote the restriction of to the box with Dirichlet boundary conditions. Our main result is.
Theorem 2.2. Assume (H.1), (H.2), and (H.3) are satisfied. For any one hasAs the density of states measure of is defined bywe immediately get the following corollary.
Corollary 2.3. Assume (H.1), (H.2), and (H.3) are satisfied. The IDS for the interacting -particle Boltzmann model exists and coincides with that of the noninteracting model hence, it satisfiesNote that, in view of the remark concluding Section 2.1, we see that the integrated density of states of the interacting -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 stateswhere denotes -fold antisymmetric tensor product of (ii) for the Bose statistics, the Bose integrated density of stateswhere denotes -fold symmetric tensor product of 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 and let be the eigenvalue of the single particle Hamiltonian repeated according to multiplicity. The three counting functions are then given byHence,Uniformly in the eigenvalues are lower bounded by, say, Hence, if then, for one has so that This implies thatThus, dividing (2.12) and (2.13) by and taking the limit 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
Proof. We take some and specify the appropriate choice later on.
By assumptions (H.1.a) and (H.2), there exists such thatLet be given by (1.2) for
By (2.14), we only need to prove (2.6) for supported in For such a function, let be an almost analytic extension of the function that is, satisfies
(i)(ii) for any the family of functions is bounded in The functional calculus based on the Helffer-Sjöstrand formula impliesIn 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 asEstimating the trace of (2.16), we choose and writeand note that is bounded by As is nonnegative, one hasAs, by assumption (H.2), tends to at infinity, (2.18) implies that there exists (independent of ) such thatwhere denotes the Lebesgue measure. Using decomposition (2.17) of we obtainwhere denotes the th Schatten class norm (see ) and we used Hölder's inequality. In the same way, the cyclicity of the trace yieldsWe are now left with estimating and for sufficiently large, depending on Therefore, we computewhere is the Dirichlet Laplacian on We use the decomposition (1.4). As the Laplacians are positive, the infinitesimal -boundedness on [4, Theorem X.18] and the definition of imply the following form bound:As one hasThus, the operator is invertible andLet and respectively, denote the eigenvalues and eigenfunctions of the Dirichlet Laplacian (the index runs over ). For such that we computeThe last estimate is a direct computation using the explicit form of the Dirichlet eigenvalues.
By [6, Lemma 2.2], we know that, for such that there exists such that, for any measurable subset one hasChoosing and taking (2.19) into account, then by combining estimates (2.20)–(2.27), we get that there exists depending only on (and the bound in assumption (H.3)), such thatBy using this inequality in (2.15), we get (2.6) as being almost analytic, vanishes to any order in as 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, The one particle Anderson potential is of the formwith a family of random variables on This one-particle models leads us to the -particle random “background” potentialand the interacting -particle Hamiltonian reads asFor 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 Its discontinuity set is countable. By [9, pp. 311f], for almost every 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 -almost sure integrated density of states for both, the noninteracting and interacting -particle system. Note that only translations along a “diagonal” vector leave invariant. Hence, for an application of ergodic theorems (as in the one particle case) for the proof of existence and -almost sure constancy of 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., ). Usually, it comes into play through a Wegner estimate, that is, an estimate of the type
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 is nonnegative, compactly supported, and that there is some such that for
For the proof of a Wegner estimate in the interacting -particle Anderson model, we can choose rather general probabilistic hypothesis like in :
(H.A.3) is a family of bounded random variables on the probability space When denotes the conditional probability measure for at site conditioned on all the other random variables that is, for all then, a Wegner estimate à la  uses the quantityand is stated as follows.
Theorem 3.1. Let us assume (H.A.2) and (H.A.3), and let be a bounded open cube of side length let be the restriction of to with Dirichlet boundary conditions. Then, there exists an increasing functionsuch that for all In order to prove Theorem 3.1, we prove two preparatory lemmas.
Lemma 3.2. Let be an open bounded cube, then the restrictions and of to with Dirichlet or Neumann boundary conditions define self-adjoint operators with compact resolvent.Proof. is infinitesimally form bounded according to [4, Theorem X.18], so the infinitesimal form boundis true for in particular (3.9) is true for Hence, the form sum defines via representation theorem a self-adjoint operator The eigenvalues tend to infinity, so by the minimax principle and (3.9), we see that has compact resolvent. The proof ofuses the extension operator to which has the properties and see [14, Satz 5.6 and Folgerung 5.2]. For we use (3.9), hence by and the above properties of we get for 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 be a bounded open cube, with (here, ), then for every Proof. For every we define byand set Fix a component of say then we get a decompositionof the random potential and the same is true for :By the covering condition on the single site-potential we get hence we can write where almost everywhere, so By spectral calculus,for every self-adjoint see , (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
Proof. By (H.A.2) and (H.A.3), we get a -almost sure bound then Lemma 3.2 implies that the restrictions and of to a bounded open cube with Dirichlet or Neumann boundary conditions define self-adjoint operators with compact resolvent -almost sure. Let and for set Then has Lebesgue measure so by [15, XIII.15, Propositions 3 and 4], we haveSo with defined in Lemma 3.2, we get -almost sure:By spectral calculus,Let be the orthogonal basis of consisting out of eigenvectors of to eigenvalues and let thenwhere the last estimate follows from Jensen's inequality. Let be an orthonormal basis of consisting of eigenvectors of to the eigenvalues thenAs and Lemma 3.3 impliesAs is nonnegative, the eigenvalues of are estimated from what follows by the eigenvalues of These are known explicitly, see [15, page 266], which can be used to estimateIf the side-length of is bigger than then so when applying expectation value to the chain of inequalities (3.20) to (3.24), it implies
Under the assumptions (H.A.2) and (H.A.3), we havehence by the Wegner estimate we can deduce regularity properties of from those of the conditioned measures via
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