Analyticity of the One-Particle Density Matrix

It is proved that the one-particle density matrix γ(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (x, y)$$\end{document} for multi-particle Coulombic systems is real analytic away from the nuclei and from the diagonal x=y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x = y$$\end{document}.


Introduction
The objective of the paper is to study analytic properties of the one-particle density matrix for the molecule, consisting of N electrons and N 0 nuclei described by the following Schrödinger operator: where R l ∈ R 3 and Z l > 0, l = 1, 2, . . . , N 0 , are the positions and the charges, respectively, of N 0 nuclei, and x j ∈ R 3 , j = 1, 2, . . . , N are positions of N electrons. The notation Δ k is used for the Laplacian w.r.t. the variable x k . The positions of the nuclei are assumed to be fixed, and as a result the very last term in (1.1) is constant. Thus, in what follows we drop this term and instead of (1.1) we study the operator This operator acts on the Hilbert space L 2 (R 3N ), and it is self-adjoint on the domain D(H) = D(H (0) ) = H 2 (R 3N ), since V is infinitesimally H (0) -bounded, see, e.g., [20,Theorem X.16]. Let ψ = ψ(x) be an eigenfunction of the operator H with an eigenvalue E ∈ R, i.e., with obvious modifications if j = 1 or j = N . The one-particle density matrix is defined as the functioñ see [10]. This function is one of the key objects in the multi-particle quantum mechanics, see, e.g., [5,6,18,19] for details and further references. If one assumes that all N particles are fermions (resp. bosons), i.e., that the function ψ is antisymmetric (resp. symmetric) under the permutations x j ↔ x k , then the definition (1.5) simplifies: where we have denotedx =x 1 , so x = (x 1 ,x). Our objective is to study the regularity properties of the functionγ(x, y) in the general form (1.5) without any symmetry assumptions. Regularity properties of solutions of elliptic equations are a classical and widely studied subject. For instance, it immediately follows from the general theory, see, e.g., [13], that any local solution of (1.4) is real analytic away from the singularities of the potential (1.3). In his famous paper [16], T. Kato showed that a local solution is locally Lipschitz with "cusps" at the points of particle coalescence. Further regularity results include [8,11,12]. We cite the most recent paper [8] for further references.
As far as the one-particle density matrix (1.6) is concerned, in the analytic literature a special attention has been paid to the one-particle densitỹ ρ(x) =γ(x, x). It was shown in [9] that in spite of the nonsmoothness of ψ, the densityρ(x) remains smooth as long as x = R l , l = 1, 2, . . . , N 0 , because of the averaging inx. Moreover, the same authors prove in [10] thatρ is in fact real analytic away from the nuclei, see also [14] for an alternative proof. Paper [10] also claims that the proofs therein imply the analyticity ofγ(x, y) for all x, y away from the nuclei. However, the methods of [10] do not suffice to substantiate this claim. The objective of the current paper is to bridge this gap and prove the real analyticity for the one-particle density matrixγ(x, y) for all x = y away from the nuclei. We emphasize that the condition x = y is not just an annoying technical restriction-the functionγ(x, y) genuinely cannot be infinitely smooth on the diagonal. We have no analytic proof of this fact, but we can present an indirect justification. As shown in [21], the eigenvalues λ k (Γ), k = 1, 2, . . . , of the non-negative compact operator Γ : L 2 (R 3 ) → L 2 (R 3 ) with kernelγ(x, y) decay at the rate of k −8/3 as k → ∞. In general, for integral operators the rate of decay is dictated by the smoothness of the kernel: the smoother the kernel is, the faster the eigenvalues decrease, see bibliography in [21]. If the functionγ(x, y) were infinitely differentiable for all x and y including the diagonal x = y (but excluding the nuclei), then the eigenvalues of Γ would decay faster than any negative power of their number, which is not the case. To justify the non-smoothness for x = y, we also mention a calculation presented in [2] (see also [3]) that suggests thatγ(x, y) −γ(x, x) should behave as |x − y| 5 for x close to y. In fact, order 5 of this homogeneous singularity is consistent with the k −8/3 -decay of the eigenvalues, see again [21] and bibliography therein. As the anonymous referee has pointed out to the authors, the non-analyticity of the one-particle matrix on the diagonal is not only of purely theoretical interest but also has important practical consequences for the electronic structure calculations as it imposes limits upon the accuracy of electronic properties computed with finite basis sets, see, e.g., [4].
One should also say that the real analyticity of the densityρ(x) =γ(x, x), x = 0, established in [10], means that the density matrixγ(x, y) is analytic along the diagonal x = y as a function of one variable x, which does not contradict the non-smoothness ofγ(x, y) on the diagonal as a function of the two variables x and y.
Before we state the main result note that regularity of each of the terms in (1.5) can be studied individually. Furthermore, it suffices to establish the real analyticity of the function (1.6) The remaining terms in the sum (1.5) are handled by relabeling the variables. Thus, from now on we use the terms one-particle density matrix and oneparticle density for the functions (1.6) and ρ(x) = γ(x, x), respectively. The next theorem constitutes our main result.
As mentioned above, the eigenfunction ψ(x) loses smoothness at the particle coalescence points. Therefore, a direct differentiation of (1.6) w.r.t. x and y under the integral will not produce the required analyticity. In order to circumvent this difficulty, we use the property that the eigenfunction preserves smoothness even at the coalescence points if one replaces the standard derivatives by cleverly chosen directional ones. For example, the function ψ is infinitely smooth in the variable x 1 + x 2 + · · · x N , as long as x j = R l for each j = 1, 2, . . . , N, l = 1, 2, . . . , N 0 . In other words, it is infinitely differentiable with respect to the directional derivative Such regularity follows from the fact that the potential (1.3) is smooth w.r.t. D on the same domain. In particular, This approach was successfully used in [10] (or even in the earlier paper [9]) in the study of the electron density ρ(x) = γ(x, x). To illustrate the use of the directional derivatives in the study of ρ(x), below we give a simplified example. Assume that N 0 = 1 and that R 1 = 0. For further simplification instead of the function γ(x, x), we consider only a part of it. Precisely, let ζ ∈ C ∞ 0 (R 3 ) be a real-valued function such that ζ(t) = 0 for |t| > ε/2 with some ε > 0. Let us show that the integral The presence of the cut-off functions in (1.8) means that all the particles are within a ε/2 distance from the first particle. Thus, on the domain of integration all variables are separated from 0: |x j | ≥ |x| − |x − x j | > ε/2, j = 2, 3, . . . , N. Rewrite F making the change of variables x j = w j + x, j = 2, 3, . . . , N, under the integral: Differentiating the integral w.r.t. x, we get: Now it is clear that by virtue of smoothness of ψ w.r.t. the derivative D, this relation can be differentiated arbitrarily many times, thereby proving that F ∈ C ∞ for all x : |x| > ε.
The complete proof of real analyticity of ρ(x) in [10] is more involved. In particular, it requires the study of various cut-off functions that keep some of the particles "close" to each other, but separate from the rest of them (we call this group of particles the cluster associated with the given cut-off ). Adaptation of the above argument to such cases leads to the introduction of the cluster derivatives (i.e., directional derivatives involving only the particles in a cluster), and it is far from straightforward.
As in [10], in the current paper our starting point is again a careful analysis of the smoothness properties of the eigenfunction ψ with respect to the cluster derivatives. However, we find the argument in [10] somewhat condensed and sketchy in places. Thus, we provide our own proofs that contain more detail and at the same time, as we believe, are sometimes simpler than in [10].
To obtain bounds for the derivatives of γ(x, y), we use again cluster derivatives, but the method of [10] is ineffective if applied directly. It has to be reworked taking into account the presence of two variables (i.e., x, y) instead of one. At the heart of our approach is the concept of an extended cut-off function that depends on the variables (x, y) ∈ R 3 × R 3 andx ∈R 3N −3 . Any such function Φ(x, y,x) has two clusters associated with it, whose properties are linked to each other (see Subsect. 4.2). This enables us to apply the cluster derivatives method to integrals of the form: (1.9) At the last stage, we construct a partition of unity on R 3N +3 which consists of extended cut-offs, and split γ(x, y) in the sum of terms of the form (1.9). Estimating each of them individually, we get the desired real analyticity. The paper is organized as follows. In Sect. 2, we state Theorem 2.2, involving more general interactions between particles, that implies Theorem 1.1 as a special case. This step allows to include other physically meaningful potentials, such as, for example, the Yukawa potential. An important conclusion of this section is that the claimed analyticity of the function γ(x, y) follows from appropriate L 2 -bounds on the derivatives of γ(x, y), enunciated in Theorem 2.3. The rest of the paper is devoted to the proof of Theorem 2.3.
Section 3 is concerned with the study of the directional derivatives of the eigenfunction ψ. The main objective is to establish suitable L 2 -estimates for higher-order derivatives of ψ on the open sets, separating different clusters of variables. Here our argument follows that of [10] with some simplifications. In Sect. 4, we study in detail properties of smooth cut-off functions including the extended cut-offs Φ = Φ(x, y,x), x, y ∈ R 3 ,x ∈ R 3N −3 , and clusters associated with them. In Sect. 5, we put together the results of Sects. 3 and 4 to estimate the derivatives of integrals of the form (1.9) with extended cut-offs Φ. These estimates are used to prove Theorem 2.3 with the help of a partition of unity that consists of extended cut-offs. This completes the proof of Theorem 2.2, and hence that of the main result, Theorem 1.1. Appendix contains some elementary combinatorial formulas that are used throughout the proof.
We conclude the introduction with some general notational conventions.
Constants. By C or c with or without indices, we denote various positive constants whose exact value is of no importance.
Coordinates. As mentioned earlier, we use the following standard notation for the coordinates: Clusters. Let R = {1, 2, . . . , N}. An index set P ⊂ R is called cluster. The cluster R is called maximal. We denote |P| = card P, P c = R\P, P * = P\{1}. If P = ∅, then |P| = 0 and P c = R.
0 , then the derivative ∂ m x is defined in the standard way: This notation extends to x ∈ R d with an arbitrary dimension d ≥ 1 in the obvious way. Denote also A central role is played by the following directional derivatives. For a cluster P and each m = (m , m , m ) ∈ N 3 0 , we define the cluster derivatives (1.10) These operations can be viewed as partial derivatives w.r.t. the variable k∈P x k . Let P = {P 1 , P 2 , . . . , P M } be a cluster set, and let m = (m 1 , m 2 , . . . , m M ), m k ∈ N 3 0 , k = 1, 2, . . . , M. Then, we denote Supports. For any smooth function f = f (x), we define the set supp 0 f = {x : f (x) = 0}. Note that this set is open and its closure supp 0 f gives the standard definition of the support supp f . For supp 0 f , we immediately get the useful property that supp 0 (fg) = supp 0 f ∩ supp 0 g. (1.11) Furthermore, for any m ∈ N 3N 0 , |m| = 1, we have

Main Theorem
The main theorem 1.1 is derived from the following result that holds for more general potentials than (1.3).
and for every ε > 0, we have for all m ∈ N 3 0 with some positive constant A 0 = A 0 (ε). The condition (2.2) implies that the functions V k,l and W k,j are real analytic on R 3 \{0}. Instead of the potential V C defined in (1.3), we consider the potential Thus, the potential (1.3) is a special case of (2.3). Working with more general potentials allows one to include into consideration other physically meaningful interactions, such as, e.g., the Yukawa potential. This generalization was pointed out in [10]. We need the following elementary elliptic regularity fact, which we give with a proof, since it is quite short.
The constant C depends on N and N 0 only.
For v ∈ H 1 , Hv ∈ L 2 , it follows from (2.4) that Consequently, in view of the straightforward bound the function v is H 2 , and hence, (2.4) implies that for all δ > 0. Together with (2.6) and (2.7), this leads to the bound Taking δ = (2C 1 ) −1 , we easily derive (2.5) with a suitable constant C > 0.
Note that the estimate (2.8) shows that the potential Theorem 1.1 is a consequence of the following result. For the sake of simplicity, we prove this theorem only for the case of a single atom, i.e., for N 0 = 1. The general case requires only obvious modifications. Without loss of generality, we set R 1 = 0. Thus, (2.3) rewrites as: 9) and the stated analyticity of γ(x, y) will be proved on the set This result is derived from the following L 2 -bound on the set The derivation of Theorem 2.2 from Theorem 2.3 is based on the following elementary lemma.

12)
for all s ∈ N d 0 , with some positive constant B. Then, f is real analytic on Ω.
Proof. Let x 0 ∈ Ω, and let r > 0 be such that for all s ∈ N d 0 and for each x ∈ B(x 0 , r), with some positive constants C and R, possibly depending on x 0 and r. According to [17,Proposition 2.2.10], this would imply the required analyticity.
Let β ∈ C ∞ 0 (R d ) be a function supported on B(x 0 , 2r) and such that β = 1 on B(x 0 , r). Denote with a constant C depending on l. Now it follows from (2.12) that By (7.2), the right-hand side does not exceed Consequently, This bound leads to (2.13) with explicitly given constants C and R. The proof is now complete.
Proof of Theorem 2.2. According to Theorem 2.3 and Lemma 2.4, the function γ(x, y) is real analytic on D ε for all ε > 0. Consequently, it is real analytic on as required.
The rest of the paper is focused on the proof of Theorem 2.3.

More Notation
Here we introduce some important sets in R 3N and R 3N −3 . For ε ≥ 0 introduce 14) The set X P (ε), ε > 0, separates the points x k and x j labeled by the clusters P and P c , respectively. Note that X P (ε) = X P c (ε).
Define also the sets separating x k 's from the origin: for |P| > 0. (2.15) It is also convenient to introduce a similar notation involving only the variablê x: For the cluster sets (2.17) Now we introduce the standard cut-off functions with which we work. Let Now we define two radially symmetric functions ζ, θ ∈ C ∞ (R 3 ) as follows:

Regularity of the Eigenfunctions
In this section, we establish estimates for the derivatives D m P ψ of the eigenfunction ψ, see (1.10) for the definition of the cluster derivatives. Our argument is an expanded version of the approach suggested in [10], which, in turn, was inspired by the proof of analyticity for solutions of elliptic equations with analytic coefficients, see, e.g., the classical monograph [13,Section 7.5].
The key point of our argument is the regularity of the functions D m P ψ for all m ∈ N 3M 0 on the domain U P (ε) with arbitrary positive ε. As before, in the estimates below we denote by C, c with or without indices positive constants whose exact value is of no importance. For constants that are important for subsequent results, we use the notation L or A with indices. The letter L (resp. A) is used when the constant is independent of (resp. dependent on) ε.
We begin the proof of the required property with studying the regularity of the potential (2.9).

Regularity of the Potential (2.9)
The next assertion is a more detailed variant of [10, Part 2 of Lemma A.3] adapted to the potential (2.9). Lemma 3.1. Let V be as defined in (2.9), and let P = {P 1 , P 2 , . . . , P M } be an arbitrary cluster set. Then, for all m ∈ N 3M Proof. Without loss of generality, we may assume that m = (m 1 , m 2 , . . . , m M ) with all |m j | ≥ 1. Indeed, suppose that m 1 = 0 and represent Repeating, if necessary, this procedure we can eliminate all zero components of m, and the clusters, attached to them. Thus, we assume henceforth that For the clusters P 1 , P 2 , . . . , P M introduce two disjoint sets of index pairs: If |m| = 1, then a direct differentiation gives the formula This function is C ∞ on U Ps (ε), and further differentiation gives the same formula for all |m| ≥ 1. Similarly, Consequently, By the definition (2.17), for x ∈ U P (ε) we have |x j | > ε for all j ∈ ∩ s P s , and |x j − x k | > ε for all (j, k) ∈ ∩ s M s . Thus, the functions above are C ∞ on U P (ε), and, by the definition (2.9), it follows from (2.2) that This bound coincides with (3.1).
Now we proceed to the study of the derivatives D m P ψ.

Regularity of the Derivatives
. . , P M } be a cluster set, and consider the function u m = D m P ψ with some m ∈ N 3M 0 . As an eigenfunction of H, the function ψ is H 2 (R 3N ), and, by elliptic regularity, it is smooth (even analytic) on the set Our objective is to show that the function ψ has derivatives u m of all orders |m| ≥ 0 on the larger set U P (0) ⊃ S, and that u m ∈ H 2 (U P (ε)) for all ε > 0.
Let us begin with a formal calculation. Since H E ψ = 0, by Leibniz's formula, we obtain Thus, u m is a solution of the equation H E u m = f m . The next assertion gives this statement a precise meaning. First we observe that by Lemma 3.1, D s P V L ∞ (U P (ε)) < ∞ for every s : |s| ≥ 1, and all ε > 0. Therefore, if u m ∈ L 2 (U P (ε)) for all m : |m| ≤ p, then f m ∈ L 2 (U P (ε)) for all |m| ≤ p + 1.

Lemma 3.2.
Suppose that u m ∈ L 2 (U P (ε)) for some ε > 0 and all m ∈ N 3M 0 such that |m| ≤ p with some p ∈ N 0 . Then, u m is a weak solution of the equation for all η ∈ C ∞ 0 (U P (ε)) and all m : |m| ≤ p.
Proof. As noted before the lemma, f m ∈ L 2 (U P (ε)) for all |m| ≤ p + 1, so that both sides of (3.3) are finite. Throughout the proof we use the fact that D s P V ∈ C ∞ on U P (ε) for all s : |s| ≥ 1, see Lemma 3.1.
We prove the identity (3.3) by induction. First note that (3.3) holds for m = 0, since ψ is an eigenfunction and f 0 = 0. Suppose that it holds for all m : |m| ≤ k, with some k ≤ p − 1. We need to show that this implies (3.3) for m + l, where l ∈ N 3M 0 : |l| = 1. As u m+l = D l P u m , we can integrate by parts, Integrating by parts and using definition of f m (see (3.2)), we get for the first integral on the right-hand side that Standard calculations involving binomial coefficients show that Substituting this in (3.4), we obtain that Proof. For brevity throughout the proof, we use the notation H α ε = H α (U P (ε)), α = 1, 2, L 2 ε = L 2 (U P (ε)). The claim holds for m = 0, since ψ ∈ H 2 (R 3N ) is an eigenfunction and f 0 = 0. Suppose that it holds for all m : |m| ≤ p ∈ N 0 . We need to show that this implies that u m+l ∈ H 2 ε , for all ε > 0, where l ∈ N 3M 0 : |l| = 1 and |m| = p.

Eigenfunction Estimates
Apart from the qualitative fact of smoothness of u m = D m P ψ, now we need to establish explicit estimates for u m . As before, we denote H E = H − E with an arbitrary E ∈ R.
so that the required bound holds.
Assume now that |m| = 2. Without loss of generality assume that all clusters P s ∈ P, s = 1, 2, . . . , M, are distinct. Let ξ be the smooth function defined in (2.18). For arbitrary ε, δ > 0 define the cut-off Then, supp 0 η ⊂ U P (ε) and η = 1 on U P (ε + δ). It is also clear that with some positive constants C k independent of ε and δ, where the maximum is taken over all sets P of distinct clusters. Estimate, using the bound (2.5): with constants independent of ε, δ. Multiplying by δ 2 , we get the required estimate.
Let E be an eigenvalue of H and ψ be the associated eigenfunction. Now we use Lemma 3.4 for the function v = u m = D m P ψ ∈ H 2 U P (ε) , ε > 0.

Corollary 3.5.
There exists a constant L 2 > 0 independent of the cluster set P and of the parameters ε > 0, δ ∈ (0, 1), such that for all m ∈ N 3M 0 , k, l ∈ N 3N 0 , |k| + |l| ≤ 2, we have Proof. Apply Lemma 3.4 to the function v = u m and estimate Now we use the bound (3.5) to obtain estimates for the function u m with arbitrary m ∈ N 3M 0 . Let A 0 , L 2 and L 3 be the constants featuring in (3.1), (3.5) and (7.3), respectively. Define Thus, defined constant depends on the eigenvalue E and ε > 0, but is independent of the cluster set P and of δ ∈ (0, 1].

Lemma 3.6.
Let the constant A 1 be as defined in (3.6). Then, for all m ∈ N 3M 0 , all k ∈ N 3N 0 , |k| ≤ 1, and all ε > 0 and δ > 0 such that δ(|m| + 1) ≤ 1, we have Proof. The formula (3.7) holds for m = 0. Indeed, since δ ≤ 1, we get Further proof is by induction. As before, we use the notation u m = D m P ψ. Suppose that (3.7) holds for all m ∈ N 3M 0 such that |m| ≤ p with some p. Our task is to deduce from this that (3.7) holds for all m, such that |m| = p + 1. Precisely, we need to show that if |m| = p and l ∈ N 3M 0 is such that |l| = 1, then for all δ > 0 such that (p + 2)δ ≤ 1.
Since |l| + |k| = 1 + |k| ≤ 2, it follows from (3.5) that (3.9) By the induction hypothesis, the second term in the brackets on the right-hand side satisfies the bound (3.10) Let us estimate the first term on the right-hand side of (3.9). First we find suitable bounds for the norms of the functions u m−s , 0 ≤ s ≤ m, |s| ≥ 1, featuring in the definition of the function f m , see (3.2). Denote q = |s|. Since |m − s| ≤ p, we can use the induction assumption to obtain for allδ such that (p − q + 1)δ ≤ 1. In particular, the valueδ = (p + 1)(p − q + 1) −1 δ satisfies the latter requirement, because (p + 1)δ ≤ 1. Thus, For the derivatives of V , we use (3.1), so that Using the definition of f m , see (3.2), and putting together the two previous estimates, we obtain In view of (7.4), the right-hand side coincides with Estimate the coefficient p q , using (7. 3): where we have taken into account that (p+1)δ ≤ 1. By (3.6), we have A 0 A −1 1 ≤ 1/2, so that the sum on the right-hand side does not exceed 1. Since δ ≤ 1, we can now conclude that Substituting this bound together with (3.10) in (3.9), we arrive at the estimate By the definition (3.6), the factor L 2 (1 + L 3 A 0 ) does not exceed A 1 . This leads to the bound (3.8), and hence proves the lemma. (1 + |m|) |m| .

Admissible Cut-Off Functions
Let {f jk }, 1 ≤ j, k ≤ N , be a set of functions such that each of them is one of the functions ζ, θ or ∂ l x θ, l ∈ N 3 0 , |l| = 1, and f jk = f kj . We work with the smooth functions of the form: We call such functions admissible cut-off functions or simply admissible cutoffs. For any such function φ we also introduce the following "partial" products. For an arbitrary cluster Furthermore, for any two clusters P, S ⊂ R, such that S ∩ P = ∅, we define Note that It is straightforward to see that for any cluster P the function φ(x) can be represented as follows: Following [10], we associate with the function φ a cluster Q(φ) defined next. It follows from the above definition that Q(φ) always contains index 1. Note also that the notion of being linked defines an equivalence relation on R, and the cluster Q(φ) is nothing but the equivalence class of index 1.
For example, if N = 4 and with some l ∈ N 3 0 , |l| = 1, then Q(φ) = {1, 2, 3}. Let P = Q(φ). If P c is not empty, i.e., P = R, then by the definition of P, we always have f jk (x) = θ(x) for all j ∈ P and k ∈ P c , and hence the representation (4.4) holds with (4.5) The notion of associated cluster is useful because of its connection with the support of the cut-off φ. This is clear from the next two lemmata. Recall that the sets X P , T P are defined in (2.14) and (2.16), respectively.
Proof. If P c = ∅, then, by definition, X P = R 3N , and hence (4.6) is trivial. Suppose that P c is non-empty. The inclusion (4.6) immediately follows from the representation (4.4), formula (4.5) and the definition of the function θ.
To summarize in words, on the support of the admissible cut-off φ the variables x j , indexed by j ∈ P = Q(φ), are "close" to each other and "far" from the remaining variables.
Let φ be of the form (4.1), and let P = Q(φ). For each l ∈ N 3 0 , the function D l P φ has the form: where each φ (l) s,r is an admissible cut-off of the form: Proof. The representation (4.9) immediately follows from the definition (4.5). It is clear from (4.10) that φ In what follows a special role is played by the factorization (4.4) with P = {1}, so that (4.11) We call the functions ω and κ the canonical factors of φ. In the next corollary, we find the canonical factors for the cut-offs φ (l) s,r defined in (4.10). s,r can be represented as follows: and for all s ∈ P * .
Proof. The claim is an immediate consequence of (4.10).

Extended Cut-Offs
Now we are ready to introduce the cut-off functions with which we work when estimating the derivatives of the one-particle density matrix γ(x, y). These cutoffs are functions of 3N + 3 variables and they are defined as follows. We say that two admissible cut-offs φ = φ(x 1 ,x) and μ = μ(x 1 ,x) are coupled to each other if they share the same canonical factor κ = κ( where ω is defined as in (4.11) and τ (x 1 ,x) = μ(x 1 ,x; {1}, R * ). Out of two coupled cut-offs φ, μ we construct a new function of 3N + 3 variables: (4.14) We call such Φ an extended cut-off. It is clear that every extended cut-off defines a pair of coupled admissible φ and μ uniquely. We say that the pair φ, μ and the extended cut-off Φ are associated with each other. The representations (4.14) and identity (1.11) give the equality From now on we denote P = Q(φ) and S = Q(μ). Below we list some useful properties of the extended cut-offs Φ and associated admissible φ, μ. Due to the nature of the definition (4.14), in all statements involving the functions φ, μ and Φ, the pairs {φ, P} and {μ, S} can be interchanged.
Due to Lemma 4.6 from now on we may assume that P * ⊂ S c . Lemma 4.7. Let φ and μ be coupled admissible cut-offs, and let P * ⊂ S c . Then, (4.16) and Proof. Since R * = P * ∪ P c , the function τ (x 1 ,x) = μ(x 1 ,x; {1}, R * ) factorizes as follows: As P * ⊂ S c , we have which leads to (4.16).
If P c = ∅, i.e., P = R, then the inclusion P * ⊂ S c implies that S c = P * . By Lemma 4.2, As X S = X S c , the claimed result follows.
In the next lemma, we collect all the information on the supports of the coupled φ, μ and associated extended cut-off Φ that we need in the next section. (4.20) If (x, y) ∈ D ε , then Proof. The second inclusion in (4.20) follows from the inclusion (4.6) applied to function μ and from the relation (4.17). The condition P * ⊂ S c is equivalent to S * ⊂ P c . Thus, the first inclusion in (4.20) follows from the inclusion (4.6) and the relation (4.17) applied to the function φ.
Similarly to the cluster derivatives (4.8) of the admissible cut-offs, now we need to investigate the cluster derivatives of the extended cut-offs. As before, let Φ(x, y,x) be an extended cut-off associated with the coupled admissible cut-offs φ, μ, and let P = Q(φ), S = Q(μ). It will be sufficient to confine our attention to the derivatives D l P w.r.t. the variable x for the cluster P. Assume that P c = ∅. Let φ (l) s,r , s ∈ P, r ∈ P c , be the admissible cut-offs defined in (4.10), and let ω (l) s,r and κ (l) s,r be their canonical factors detailed in (4.12) and (4.13), respectively. Using the factor τ from the canonical factorization μ(x) = τ (x 1 ,x)κ(x) we define a new admissible cut-off It is clear that φ Now we can describe the cluster derivatives of Φ. The notation D l P Φ(x, y,x) means taking the lth P-cluster derivative w.r.t. the variable x = (x,x). Lemma 4.9. Let Φ = Φ(x, y,x) be an extended cutoff associated with the coupled admissible φ and μ and assume that P * ⊂ S c . Let l ∈ N 3 0 be an arbitrary multi-index such that |l| = 1.

Estimating the Density Matrix
The proof of Theorem 2.3 which is given in Sect. 6, uses a partition of unity that consists of extended cut-off functions, i.e., functions of the form (4.14). Thus, the objective of this section is to estimate the derivatives of the function with an extended cut-off Φ on the set D ε : Theorem 5.1. Let Φ be an extended cut-off. Then, for all m, k ∈ N 3 0 we have The proof of Theorem 5.1 is conducted by induction, and hence, we have to involve a more general object than (5.1). Precisely, for cluster sets P =  We estimate the function γ k,m and its derivatives on the set D ε , ε > 0, defined in (2.10) with the help of Corollary 3.7 by reducing the estimates to the integrals D k P ψ L 2 (U P (ε)) and D m S ψ L 2 (U S (ε)) . To this end we assume that the admissible cut-offs φ and μ associated with Φ satisfy the following support conditions: and Recall that the sets X, T, T with various subscripts are defined in (2.14), (2.15), (2.16), (2.17). For brevity, throughout the proofs below for an arbitrary cluster set Q we use the notation T Q = T Q (ε/2), T Q = T Q (ε/2) and X Q = X Q (ε(4N ) −1 ).

Lemma 5.2.
Suppose that Φ is of the form (4.14) and that (5.4) and (5.5) hold. Then, there exists a constant A 3 , independent of the cluster sets P, S, and of the cut-off Φ, such that Proof. Let so that |ω|, |τ |, |κ|, |φ|, |μ| ≤ C a . Therefore, Now, using (5.5), we can estimate: By Hölder's inequality and by (5.4), the right-hand side does not exceed Since X P ∩ T P ⊂ U P (ε(4N ) −1 ) (see the definition (2.17)), and a similar inclusion holds for the cluster set S, by Corollary 3.7, the right-hand side does not exceed This implies the required bound.
Let the functions φ s,r be as defined in (4.10) and (4.22), respectively. As in the previous section, we use the notation P = Q(φ) and S = Q(μ). In the next lemma, we show how the derivatives of γ k,m w.r.t. the variable x transform into directional derivatives under the integral (5.2). Lemma 5.3. Suppose that (5.4) and (5.5) hold. Assume that P * ⊂ S c . Then, Furthermore, Let l ∈ N 3 0 be such that |l| = 1. If P c = ∅, then ∂ l x γ k,m (x, y; P, S; Φ) = γ (l,k),m x, y; {P, P}, S; Φ + γ k,(l,m) x, y; P, {P * , S}; Φ , (5.8) and both sides are square-integrable in (x, y) ∈ D ε . If P c = ∅, then for all s ∈ P, r ∈ P c we have and for every (x, y) ∈ D ε . Furthermore, the formula holds: Proof. According to (4.20) and the assumption (5.4), we have which coincides with (5.6). Moreover, by (4.21) and (5.5), which implies (5.7) for all (x, y) ∈ D ε . Thus, by Lemma 5.2 the terms on the right-hand side of (5.8) and two first terms on the right-hand side of (5.11) are square-integrable in (x, y) ∈ D ε . Let P c = ∅. Then, the inclusions (5.9) and (5.10) are consequences of (4.23) and (5.4), (5.5). Again by Lemma 5.2, the third term on the right-hand side of (5.11) is square-integrable in (x, y) ∈ D ε , as claimed.

and
The proof of this proposition is by induction. Lemma 5.2 provides the base step. Now we need to establish the induction step: Lemma 5.5. Suppose that for every extended cut-off Φ and every pair P = {P 1 , P 2 , . . . , P M } and S = {S 1 , S 2 , . . . , S K } of cluster sets such that Φ satisfies (5.4) and (5.5), the bound (5.13) holds for all multi-indices k ∈ N 3M 0 , m ∈ N 3K 0 , and all k, m ∈ N 3 0 , such that |k| ≤ p, |m| ≤ n with some p, n ∈ N 0 . Then, for such extended cut-offs Φ and cluster sets P, S the bound (5.13) holds for all k, m, such that |k| ≤ p + 1, |m| ≤ n.
If P c = ∅, then the only difference in the proof is that instead of (5.11) we use (5.8).

Proof of Proposition 5.4.
Step 1. Proof of (5.13) for all k and m = 0. According to Lemma 5.2, the required bound holds for k = m = 0. Thus, using Lemma 5.5, by induction we conclude that (5.13) holds for all k ∈ N 3 0 and m = 0, as claimed.
Step 2. Proof of (5.13) for k = 0 and all m. Using the symmetry property (5.3) and Step 1, we conclude that (5.13) holds for all m ∈ N 3 0 and k = 0. Step 3. Using Step 2 and Lemma 5.5, by induction we conclude that (5.13) holds for all k, m ∈ N 3 0 , as required.
Proof of Theorem 5.1. Recall that in the case m = 0, k = 0 we take P = S = ∅, so that the conditions (5.4) and (5.5) are automatically satisfied. Thus, the required bound follows directly from (5.13).

It is clear that
For every cluster S ⊂ R * define It is clear that so that Υ⊂Ξ, S⊂R * Φ Υ,S (x, y,x) = 1.
Since each function Φ Υ,S is an extended cut-off, we can use Theorem 5.1 for each term, which leads to (2.11), as required.
We say that k ≤ s for k, s ∈ N d 0 if k j ≤ s j , j = 1, 2, . . . , d. In this case we define Note the useful identity l≤k |l|=p k l = |k| p , ∀p ≤ |k|. (7.4) It follows by comparing the coefficients of the term t p in the expansions of both sides of the equality (1 + t) k1 (1 + t) k2 · · · (1 + t) k d = (1 + t) |k| , t ∈ R.
This simple argument is found in [15, Proposition 2.1].