Analyticity of the one-particle density matrix

It is proved that the one-particle density matrix $\gamma(x, y)$ for multi-particle systems is analytic away from the nuclei and from the diagonal $x = y$.


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 ∆ 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. [17,Theorem X.16].
Let ψ = ψ(x) be an eigenfunction of the operator H with an eigenvalue E ∈ R, i.e.
Regularity properties of solutions of elliptic equations is a classical and widely studied subject. For instance, it immediately follows from the general theory, see e.g. [10], that any local solution of (1.4) is real analytic away from the singularities of the potential (1.3). In his famous paper [13] T. Kato showed that a local solution is locally Lipschitz with "cusps" at the points of particle coalescence. Further regularity results include [8], [9], [5]. We cite the most recent paper [5] 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 densityρ(x) =γ(x, x). It was shown in [6], 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 [7] thatρ is in fact real analytic away from the nuclei, see also [11] for an alternative proof. Paper [7] also claims that the proofs therein imply the analyticity ofγ(x, y) for all x, y away from the nuclei. However, the methods of [7] 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 [18], 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 [18]. 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 heuristic argument presented in [2] 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 [18] and bibliography therein.
One should also say that the real analyticity of the densityρ(x) =γ(x, x), x = 0, established in [7], 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 of γ(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 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 one-particle density for the functions (1.6) and ρ(x) = γ(x, x) respectively.
The next theorem constitutes our main result.
Theorem 1.1. Let the function γ(x, y), (x, y) ∈ R 3 × R 3 , be defined by (1.6). Then γ(x, y) is real analytic as a function of variables x and y on the set 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 [7] (or even in the earlier paper [6]) 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 [7] 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 [7], 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 [7] 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 [7].
To obtain bounds for the derivatives of γ(x, y) we use again cluster derivatives, but the method of [7] 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 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.
Sect. 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 [7] 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 Sect. 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. The 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: x = (x 1 , x 2 , . . . , x N ), where x j ∈ R 3 , j = 1, 2, . . . , N. Very often it is convenient to represent x in the form 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.
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 Let and for every ε > 0, we have 2) for all l = 1, 2, . . . , N 0 , k, j = 1, 2, . . . , N 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 The Coulomb potentials V k,l (x) = −Z l |x| −1 and W k,j (x) = (2|x|) −1 satisfy (2.1) in view of the classical Hardy's inequality, see e.g. [17, The Uncertainty Principle Lemma, p. 169]. Furthermore, the bounds (2.2) can be deduced from the estimates for harmonic functions, established, e.g. in [4,Theorem 7,p. 29]. 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 [7].
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.

Note that the estimate (2.8) shows that the potential (2.3) is infinitesimally
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 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.

Lemma 2.4.
Let Ω ⊂ R d be an open set, and let Let f ∈ C ∞ (Ω) be a function such that Then f is real analytic on Ω. Proof. Let x 0 ∈ Ω, and let r > 0 be such that B(x 0 , 2r) ⊂ Ω. We aim to prove that for each x ∈ B(x 0 , r), with some positive constants C and R, possibly depending on x 0 . According to [14, 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 For l > d/4 we can estimate with a constant C depending on l. Now it follows from (2.12) that By (7.2), the right-hand side does not exceed According to (7.5), 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.
2.2. More notation. Here we introduce some important sets in R 3N and R 3N −3 . For ε ≥ 0 introduce 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: It is also convenient to introduce a similar notation involving only the variablex: (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 [7], which, in turn, was inspired by the proof of analyticity for solutions of elliptic equations with analytic coefficients, see e.g. the classical monograph [10, 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).
3.1. Regularity of the potential (2.9). The next assertion is a more detailed variant of [7, Part 2 of Lemma A.3] adapted to the potential (2.9).
Lemma 3.1. Let V be as defined in (2.9) with N 0 = 1, and let P = {P 1 , P 2 , . . . , P M } be an arbitrary collection of clusters. Then for all m ∈ N 3M 0 , |m| ≥ 1, the function D m P V is C ∞ on U P (ε), and the bound 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 |m j | ≥ 1, j = 1, 2, . . . , M.
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, These functions 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.

3.2.
Regularity of the derivatives D m P ψ. As before, let ψ ∈ H 2 (R 3N ) be an eigenfunction of the operator H, with the eigenvalue E ∈ R, i.e.
. . , 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 S = {x k = 0, x k = x j : j, k = 1, 2, . . . , N}. 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.
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, using (3.3) for |m| ≤ k: 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 into (3.4), we obtain that which coincides with (3.3) for m + l. Now by induction we conclude that (3.3) holds for all m : |m| ≤ p, as claimed.
Theorem 3.3. Let E be an eigenvalue of H and let ψ be the associated eigenfunction.
For each ε > 0 the function u m = D m P ψ belongs to H 2 (U P (ε)) for all m ∈ N 3M 0 . Proof. For brevity throughout the proof we use the notation H α ε = H α (U P (ε)), α = 1, 2, 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. Since u m ∈ H 2 ε , we have u m+l ∈ H 1 ε ⊂ L 2 ε for all ε > 0. Thus, by Lemma 3.2, u m+l satisfies (3.3) with f m+l ∈ L 2 ε . In order to show that u m+l ∈ H 2 ε , for all ε > 0, we apply Lemma 2.1. To this end let η 1 ∈ C ∞ (R 3N ) be a function such that η 1 (x) = 0 for x ∈ R 3N \ U P (ε/2) and η 1 (x) = 1 for x ∈ U P (ε). Thus, by (3.3), Since u m+l ∈ H 1 ε/2 , the right-hand side belongs to L 2 (R 3N ). Therefore, H(u m+l η) ∈ L 2 (R 3N ), and by Lemma 2.1, u m+l η 1 ∈ H 2 (R 3N ). As a consequence, u m+l ∈ H 2 ε , as required. Now, by induction, u m ∈ H 2 ε for all m ∈ N 3M 0 . 3.3. 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.
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 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| 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 By the induction hypothesis, the second term in the brackets on the right-hand side satisfies the bound 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 u m−s L 2 (U P (ε+(p+1)δ)) ≤ A p−q+1 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.

Cut-off functions and associated clusters
4.1. 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 cut-offs. For any such function φ we also introduce the following "partial" products. For an arbitrary cluster P ⊂ R = {1, 2, . . . , N} define Furthermore, for any two clusters P, S ⊂ R, such that S ∩ P = ∅, we define Note that D l P φ(x; P) = D l P φ(x; P c ) = 0, for all l ∈ N 3 0 , |l| ≥ 1. It is straightforward to see that for any cluster P the function φ(x) can be represented as follows: φ(x) = φ(x; P)φ(x; P c )φ(x; P, P c ). (4.4) Following [7], we associate with the function φ a cluster Q(φ) defined next. Definition 4.1. For an admissible cut-off φ, let I(φ) ⊂ {(j, k) ∈ R × R : j = k} be the set such that (j, k) ∈ I(φ), iff f jk = θ. We say that two indices j, k ∈ R, are φ-linked to each other if either j = k, or (j, k) ∈ I(φ), or there exists a sequence of pairwise distinct indices j 1 , j 2 , . . . , j s , 1 ≤ s ≤ N − 2, all distinct from j and k, such that (j, j 1 ), (j s , k) ∈ I(φ) and (j p , j p+1 ) ∈ I(φ) for all p = 1, 2, . . . , s − 1.
The cluster Q(φ) is defined as the set of all indices that are φ-linked to index 1.
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 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 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. holds.
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. Moreover, for all x 1 : |x 1 | > ε.
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 we have used the factorization (4.4) and property (4.3). By the definition (4.2), 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 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

12)
and for all s ∈ P * .
Proof. The claim is an immediate consequence of (4.10).

4.2.
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 cut-offs 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 to each other. The representations (4.14) and identity (1.11) give the equality supp Φ(x, y, · ) = supp φ(x, · ) ∩ supp µ(y, · ), ∀(x, y) ∈ R 3 × R 3 . 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).
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.
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 s,r are coupled to each other. Introduce the associated extended cut-off: 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.
Assume that P c = ∅. By (4.23) and (4.18), for all (x, y) ∈ D ε . Since the factor µ y,x; {1}, P c does not depend on x j with j ∈ P * , we have {1}, P c . Using (4.9) and (4.25) we get as required.

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 : The constant A depends on ε > 0, but does not depend on Φ.
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: 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 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). If P c = ∅, then for all s ∈ P, r ∈ P c we have supp φ (l) s,r ⊂ X P , supp µ (l) s,r ⊂ X S , (5.9) and supp φ (l) s,r (x, · ) ∩ supp µ (l) s,r (y, · ) ⊂ T P * ∩ T S * , (5.10) for every (x, y) ∈ D ε . Furthermore, the formula holds: and both sides are square-integrable in (x, y) ∈ D ε .
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 .
Changing the variables back tox, we rewrite the right-hand side as In the case P c = ∅, we have D l P Φ(x, y,x) = 0 by Lemma 4.9, so the sum coincides with the right-hand side of (5.8).
If P c = ∅, then by (4.24) the sum (5.12) coincides with the right-hand side of (5.11) again. This completes the proof.
Proposition 5.4. Let Φ be an extended cut-off, and let P, S be a pair of cluster sets such that (5.4) and (5.5) hold. Then for all m, k ∈ N 3 0 we have where the constant A 3 is as in Lemma 5.2 and A = 2A 3 + N 2 .
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 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.

It is clear that
As explained in Sect. 2, Theorem 2.3 implies Theorem 2.2, and hence Theorem 1.1.
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 k s = k! s!(k − s)! .
This simple argument is found in [