Ugo Boscain
email: ugo.boscain@polytechnique.edu

Jean-Paul Gauthier
email: gauthier@univ-tln.fr

Francesco Rossi
email: francesco.rossi@lsis.org

Mario Sigalotti
email: mario.sigalotti@inria.fr

Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems

Keywords: Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems

published or not. The documents may come

Introduction

In this paper we consider a closed quantum system of the form

i ψ(t) = H(u(t))ψ(t) = (H 0 + u 1 (t)H 1 + • • • + u m (t)H m )ψ(t), (1)
where ψ(•) describes the state of the system evolving in the unit sphere S of a finite-or infinite-dimensional complex Hilbert space H. The control u(•) = (u 1 (•), . . . , u m (•)) takes values in a subset U of R m and represents external fields. The Hamiltonian H(u) is a self-adjoint operator on H for every u ∈ U . System (1) is exactly (respectively, approximately) controllable if every point of S can be steered to (respectively, steered arbitrarily close to) any other point of S, by an admissible trajectory of [START_REF] Adami | Controllability of the Schrödinger equation via intersection of eigenvalues[END_REF].

When the dimension of H is finite, the exact controllability of (1) has been characterized in [START_REF] Albertini | Notions of controllability for bilinear multilevel quantum systems[END_REF] in terms of the Lie algebra generated by {H(u) | u ∈ U }. In the infinite-dimensional case, if the controlled Hamiltonians H 1 , . . . , H m are bounded, exact controllability can be ruled out by functional analysis arguments ( [START_REF] Ball | Controllability for distributed bilinear systems[END_REF][START_REF] Illner | Limitations on the control of Schrödinger equations[END_REF][START_REF] Turinici | On the controllability of bilinear quantum systems[END_REF]). Sufficient conditions for approximate controllability have been obtained by proving exact controllability of restrictions of [START_REF] Adami | Controllability of the Schrödinger equation via intersection of eigenvalues[END_REF] to spaces where the controlled Hamiltonians are unbounded ( [START_REF] Beauchard | Local controllability of a 1-D Schrödinger equation[END_REF][START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF][START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF]). Other sufficient conditions for approximate controllability have been obtained by control-Lyapunov arguments ( [START_REF] Beauchard | Semi-global weak stabilization of bilinear Schrödinger equations[END_REF][START_REF] Mirrahimi | Lyapunov control of a quantum particle in a decaying potential[END_REF][START_REF] Vahagn Nersesyan | Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications[END_REF][START_REF] Nersesyan | Global exact controllability in infinite time of Schrödinger equation[END_REF]) and Lie-Galerkin techniques ( [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF][START_REF] Boscain | Multi-input schrdinger equation: controllability, tracking, and application to the quantum angular momentum[END_REF][START_REF] Boussaid | Weakly-coupled systems in quantum control[END_REF][START_REF] Chambrion | Periodic excitations of bilinear quantum systems[END_REF][START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF]).

Both in the finite-and the infinite-dimensional case, checking the above-mentioned controllability criteria is not an easy task. Typical conditions require that the eigenvalues of H 0 are non-resonant (e.g., all gaps are different or rationally independent) and that the controlled Hamiltonians "sufficiently couple" the eigenstates of H 0 . Hence many efforts were made to find easily checkable sufficient conditions for controllability of [START_REF] Adami | Controllability of the Schrödinger equation via intersection of eigenvalues[END_REF].

Notice that most of the conditions mentioned above are obtained for single-input systems (m = 1). An alternative technique fully exploiting the multi-input framework uses adiabatic theory to obtain approximate descriptions of the evolution of (1) for slowly varying control functions u(•) [START_REF] Adami | Controllability of the Schrödinger equation via intersection of eigenvalues[END_REF][START_REF] Boscain | Adiabatic control of the Schroedinger equation via conical intersections of the eigenvalues[END_REF][START_REF] Guérin | Control of quantum dynamics by laser pulses: Adiabatic Floquet theory[END_REF]. Adiabatic methods work when the spectrum exhibits eigenvalue intersections. In [START_REF] Boscain | Adiabatic control of the Schroedinger equation via conical intersections of the eigenvalues[END_REF], in the case m = 2, it is shown how to exploit the existence of conical intersections (see Figure 1 and Definition 5) between every pair of subsequent eigenvalues to induce an approximate population transfer from any eigenstate to any other eigenstate or any nontrivial superposition of eigenstates (without controlling the relative phases). This kind of partial controllability is named spread controllability in [START_REF] Boscain | Adiabatic control of the Schroedinger equation via conical intersections of the eigenvalues[END_REF].

In this paper we study the whole controllability implications of the conditions ensuring spread controllability, namely the existence of conical intersections between every pair of subsequent eigenvalues. A relevant advantage of these conditions is that they consist in qualitative structural properties of the spectrum of H(u) as a function of u ∈ U . This might be useful when the explicit expression of the Hamiltonian is not known, but one has information about its spectrum (as it happens in many experimental situations).

In the following we say that the spectrum of H(•) is conically connected if all eigenvalue intersections are conical, each pair of subsequent eigenvalues is connected by a conical intersections such that all other eigenvalues are simple (see Figure 2). A notable property of conical connectedness is that it is a structurally stable property for m = 2 (when restricted to real Hamiltonians) and for m = 3. This structural stability dates back to the 1920s ( [START_REF] Born | Beweis des adiabatensatzes[END_REF][START_REF] Neumann | überdas Verhalten von Eigenwerten bei Adiabatischen Prozessen[END_REF]) and is discussed in more details in Section 2.1 (see Remark 6).

The main results of the paper about the relations between conically connected spectra and controllability are the following:

• if H is finite-dimensional and the spectrum of H(•) is conically connected then Lie{H(u) | u ∈ U } is equal to u(n) if the trace of H(u)
is nonzero for some u ∈ U or su(n) otherwise. In particular (1) is exactly controllable and the same is true for its lift in U(n) or SU(n);

• if H is infinite-dimensional and the spectrum of H(•) is conically connected then ( 1) is approximately controllable. (For a counterpart of the finite-dimensional lifted-system controllability, see Remark 16.)

Motivated by the exact/approximate dichotomy in the controllability of finite-/infinite-dimensional systems, we investigate in the last part of the paper the equivalence between exact and approximate controllability. We have already seen that exact controllability cannot hold when dim(H) = ∞, since we assume H 1 and H 2 to be bounded. When dim(H) < ∞ we prove that exact and approximate controllability are indeed equivalent, both for (1) and its lift on U(n) or SU(n). This last result holds in the more general setting where H(u) depends on u in a possibly nonlinear way.

The structure of the paper is the following. In Section 2 we introduce the basic definitions related to controllability and conical intersections and we prove the finite-dimensional exact controllability of a system exhibiting a conically connected spectrum and of its lift in U(n) or SU(n) (Theorem 8). In Section 3 we prove that an infinite-dimensional system having a conically connected spectrum is approximately controllable (Theorem 13). Finally, in Section 4 we prove the equivalence between approximate and exact controllability for finite-dimensional closed quantum mechanical systems.

2 Conical intersections and exact controllability in finite dimension

Basic definitions and facts

In this section we introduce some definitions and recall some basic facts about control systems evolving on finite-dimensional manifolds. We first define approximate and exact controllability for a smooth control system

q(t) = f (q(t), u(t)) (Σ) defined on a connected manifold M with controls u(•) taking values in U ⊂ R m .
Definition 1

• The reachable set A q 0 from a point q 0 ∈ M for (Σ) is the set of points q 1 ∈ M such that there exist a time T ≥ 0 and a L ∞ control u : [0, T ] → U for which the solution of the Cauchy problem q(t) = f (q(t), u(t)) starting from q(0) = q 0 is well defined on [0, T ] and satisfies q(T ) = q 1 .

• The system (Σ) is said to be exactly controllable if for every q 0 ∈ M we have A q 0 = M .

• The system (Σ) is said to be approximately controllable if for every q 0 ∈ M we have that A q 0 is dense in M .

A relevant class of control systems for our discussion is given by right-invariant control systems on Lie groups, namely, systems for which M is a connected Lie group and each vector field f (•, u), u ∈ U , is right-invariant. Lemma 3 below is a classical result concerning right-invariant control systems on compact Lie groups (see, e.g., [START_REF] Assoudi | On subsemigroups of semisimple Lie groups[END_REF] and [23, p. 155]).

Definition 2 Let (Σ) be a right-invariant control system and denote by e the identity of the group M . Let Lie{f (e, u) | u ∈ U } be the Lie algebra generated by {f (e, u) | u ∈ U }, i.e., the smallest subalgebra of the Lie algebra of M containing {f (e, u) | u ∈ U }. The orbit G of (Σ) is the connected subgroup of M whose Lie algebra is Lie{f (e, u) | u ∈ U }.

Lemma 3 Let M be a connected compact Lie group and consider a right-invariant control system (Σ) on M . The following conditions are equivalent:

• (Σ) is exactly controllable; • the orbit G of (Σ) is equal to M ; • Lie{f (e, u) | u ∈ U } is the Lie algebra of M .
The last condition is usually referred to as the Lie-bracket generating condition.

A general controlled closed quantum system evolving in a finite-dimensional Hilbert space can be written as

i ψ(t) = H(u(t))ψ(t), (2)
where ψ : [0, T ] → S 2n-1 ⊂ C n denotes the state of the system and H(u) is a Hermitian matrix smoothly depending on u ∈ U ⊂ R m . From now on let us take n ≥ 2, otherwise the controllability problem is trivial. Naturally associated with ( 2) is its lift on the unitary group U(n),

i ġ(t) = H(u(t))g(t), (3)
which is right-invariant and permits to write the solution ψ(•) of ( 2) starting from ψ 0 as ψ(t) = g(t)ψ 0 where g(•) is the solution of (3) starting from the identity. Lemma 3 implies that (3) is controllable in U(n) if and only if the Lie algebra generated by {iH(u

) | u ∈ U } is equal to u(n). If the trace of each matrix H(u), u ∈ U , is zero, then (3) is well posed in SU(n) and its exact controllability in SU(n) is equivalent to the condition Lie{iH(u) | u ∈ U } = su(n).
In order to deduce the controllability properties of (2) from those of (3) one has to turn towards the classification of transitive actions of subgroups of U(n) onto S 2n-1 ⊂ C n . As a consequence, system (2) is exactly controllable if and only if

Lie{iH(u) | u ∈ U } ⊇ su(n) if n is odd an algebra conjugate to sp(n/2) if n is even. (4)
(See [START_REF] Domenico | Introduction to quantum control and dynamics[END_REF].) Of special interests for this paper are closed control-affine quantum system driven by m external fields, satisfying the following assumption:

(A) Let m ≥ 2 and U be an open and connected subset of R m . We assume that H(•) is control-affine, i.e., it has the form

H(u) = H 0 + u 1 H 1 + • • • + u m H m .
In the following, under assumption (A), we focus on the controllability of the system

i ψ(t) = (H 0 + u 1 (t)H 1 + • • • + u m (t)H m )ψ(t), ψ(t) ∈ S 2n-1 , (5)
and its lift

i ġ(t) = (H 0 + u 1 (t)H 1 + • • • + u m (t)H m )g(t), g(t) ∈ U(n). ( 6
)
Remark 4 Let us briefly discuss the role of the assumptions listed in hypotheses (A).

The affine structure of H with respect to the control is natural in quantum control ( [START_REF] Domenico | Introduction to quantum control and dynamics[END_REF]) and allows the application of the controllability criteria we are using in the following (see Proposition 11). Moreover, the connectedness of U is required in order to apply adiabatic techniques in the whole set of control parameters.

A crucial hypothesis that we shall use to prove exact controllability of (6) (and hence, in particular, of ( 5)) is the existence of conical intersections (in the space of controls) between consecutive energy levels, and the fact that these conical intersections occur at distinct points in the space of controls. More precisely: Definition 5 Let (A) be satisfied. Let Σ(u) = {λ 1 (u), . . . , λ n (u)} be the spectrum of H(u), where the eigenvalues λ 1 (u) ≤ • • • ≤ λ n (u) are counted according to their multiplicities. We say that ū ∈ U is a conical intersection between the eigenvalues λ j and λ j+1 if λ j (ū) = λ j+1 (ū) has multiplicity two and there exists a constant c > 0 such that for any unit vector v ∈ R m and t > 0 small enough we have

λ j+1 (ū + tv) -λ j (ū + tv) > ct . (7)
See Figure 1 for the picture of a conical intersection. Notice that the hypothesis m ≥ 2 guarantees that conical intersections do not disconnect U . This is crucial in the arguments below (see, in particular, Lemma 9.) Remark 6 Conical intersections are not pathological phenomena. On the contrary, they happen to be generic for m = 3 or for m = 2, when restricted to real Hamiltonians, in the following sense.

Let us first consider the case m = 2. Let sym(n) be the set of all n × n symmetric real matrices. Then, generically with respect to the pair (H 1 , H 2 ) in sym(n) × sym(n) (i.e., for all (H 1 , H 2 ) in an open and dense subset of sym(n) × sym(n)), for each u = (u 1 , u 2 ) ∈ R 2 and λ ∈ R such that λ is a multiple eigenvalue of H 0 +u 1 H 1 +u 2 H 2 , the eigenvalue intersection u is conical. Moreover, each conical intersection u is structurally stable, in the sense that small perturbations of H 0 , H 1 and H 2 give rise, in a neighborhood of u, to conical intersections for the perturbed H. See Section 3 for a version of this result in infinite dimension and [START_REF] Boscain | Adiabatic control of the Schroedinger equation via conical intersections of the eigenvalues[END_REF] for more details.

In the case m = 3, let Herm(n) be the space of n × n Hermitian matrices. Then, generically with respect to the triple

(H 1 , H 2 , H 3 ) in Herm(n) 3 , for each u = (u 1 , u 2 , u 3 ) ∈ R 3 and λ ∈ R such that λ is a multiple eigenvalue of H 0 + u 1 H 1 + u 2 H 2 + u 3 H 3 ,
the eigenvalue intersection u is conical. Structural stability also holds, in the same sense as above. See [START_REF] Chittaro | Adiabatic control of quantum control systems with three inputs[END_REF] for more details and a discussion on the infinitedimensional counterpart of these properties.

The following definition identifies the Hamiltonians for which we can guarantee exact controllability from qualitative properties of their spectra. Roughly speaking we require all their eigenvalues to be connected by conical intersections and the conical intersections to occur at different points in the space of controls. Definition 7 Let (A) be satisfied. We say that the spectrum Σ(•) of H(•) is conically connected if all eigenvalue intersections are conical and for every j = 1, . . . , n -1, there exists a conical intersection ūj ∈ U between the eigenvalues λ j , λ j+1 , with λ l (ū j ) simple if l = j, j + 1.

See Figure 2 for a conically connected spectrum.

Conical connectedness implies exact controllability

The main result of Section 2 is the following theorem.

Theorem 8 Let (A) be satisfied and assume that the spectrum Σ(•) of H(•) is conically connected. Then the Lie algebra generated by {iH(u) | u ∈ U } is either u(n) or su(n) (in the case H 0 , . . . , H m ∈ su(n)). Hence, system (6) is either exactly controllable in U(n) or well-posed and exactly controllable in SU(n).

The proof of the theorem is based on the following lemma.

U such that if n j=1 α j λ j (ū) = 0 with (α 1 , . . . , α n ) ∈ Q n and ū ∈ Ū then α 1 = α 2 = • • • = α n . Proof. For every α = (α 1 , . . . , α n ) ∈ Q n define U α = {u ∈ U | n j=1 α j λ j (u) = 0}.
Let Ū be the complement in U of the union of all U α such that α j = α k for some j, k ∈ {1, . . . , m}.

Notice that, by definition of conical intersection and since m ≥ 2, {u ∈ U | Σ(u) is simple} is connected. Thanks to the analyticity of the spectrum in {u ∈ U | Σ(u) is simple}, either U α = U or U α has empty interior. The proof is completed by showing that if

U α = U then α 1 = • • • = α n .
Indeed, in this case U α has empty interior for every α such that α j = α k for some j, k ∈ {1, . . . , m} and the countable union of subsets of R m with empty interior and zero measure has empty interior and zero measure.

Assume that U α = U . Consider j ∈ {1, . . . , n-1} and an analytic path γ : R → U such that γ(0) = ūj , γ(0) = 0, where ūj ∈ U is a conical intersection between the eigenvalues λ j , and λ j+1 , with λ l (ū j ) simple if l = j, j + 1.

Since U α = U , we have for every t ∈ R, n l=1 α l λ l (γ(t)) = 0.

By analytic dependence of the spectrum along γ in a neighbourhood of γ(0) [START_REF] Rellich | Perturbation theory of eigenvalue problems[END_REF], the functions

t → λ j (γ(t)) if t < 0 λ j+1 (γ(t)) if t ≥ 0, t → λ j+1 (γ(t)) if t < 0 λ j (γ(t)) if t ≥ 0,
and t → λ l (γ(t)), l = j, j + 1, are analytic in a neighborhood of 0. Hence,

α j+1 λ j (γ(t)) + α j λ j+1 (γ(t)) + l =j,j+1 α l λ l (γ(t)) = 0
for t in a neighborhood of 0. Then

(α j -α j+1 )(λ j (γ(t)) -λ j+1 (γ(t))) = 0
for t in a neighborhood of 0. By definition of conical intersection it must be α j = α j+1 . Since j is arbitrary, we deduce that α 1 = • • • = α n concluding the proof.

Remark 10

The lemma fails to hold if m = 1, i.e., for single input systems. Consider for instance n = 3, H 0 = diag(0, 1, 2) and H 1 = diag(1, 1, 0). Then the eigenvalues of H(u) are u, u + 1 and 2. The spectrum is conically connected, but clearly Ū = ∅. Notice that Lie(iH 0 , iH 1 ) is made only by diagonal matrices and therefore {iH 0 , iH 1 } does not generate u(n). Hence, this example also shows that Theorem 8 does not hold if we remove the hypothesis m ≥ 2.

The proof of Theorem 8 is based on the following adaptation of a controllability criteria for single-input quantum control systems appeared in [10, Proposition 3.1]. The proof can be obtained following exactly the same arguments as in [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF].

Proposition 11 Let A 0 , A 1 , . . . , A m be skew-Hermitian n × n matrices. Denote by λ 1 , . . . , λ n the eigenvalues of A 0 , repeated according to their multiplicities and let φ 1 , . . . , φ n be an orthonormal basis of associated eigenvectors. Let

S 0 = {(j, k) ∈ {1, . . . , n} 2 | ∃ l ∈ {1, . . . , m} such that φ j , A l φ k = 0}.
Assume that there exists S ⊆ S 0 such that the graph having 1, . . . , n as nodes and S as set of edges is connected. Assume, moreover, that for every (j, k) ∈ S and (r, s) ∈ S 0 \ {(j, k)} we have λ j -λ k = λ r -λ s . Then Lie(A 0 , . . . , A m ) = su(n) if A 0 , . . . , A m ∈ su(n) and Lie(A 0 , . . . , A m ) = u(n) otherwise.

With a Hamiltonian H(•) as in assumption (A ∞ ) we can associate the control system

i ψ(t) = (H 0 + u 1 (t)H 1 + • • • + u m (t)H m )ψ(t), ψ(t) ∈ S, (8)
where S is the unit sphere of H. Existence of solutions of ( 8) for u of class L ∞ and H 1 , . . . , H m bounded is classical (see [START_REF] Pazy | Semigroups Of Linear Operators And Applications To Partial Differential Equations[END_REF]).

A typical case for which (A ∞ ) is satisfied is when H 0 = -∆ + V , where ∆ is the Laplacian on a domain Ω ⊂ R d (with suitable boundary conditions if Ω = R d ), V is a regular enough real-valued potential bounded from below, H = L 2 (Ω, C), and H 1 , . . . , H m are multiplication operators by L ∞ real-valued functions.

Conical connectedness implies approximate controllability in infinite dimension

The main technical assumption of this section is the following.

(B) The spectrum of H 0 is discrete without accumulation points and each eigenvalue has finite multiplicity.

Under assumptions (A ∞ ) and (B) the spectrum of H(u), u ∈ U , with eigenvalues repeated according to their multiplicities, can be described by Σ ∞ (u) = {λ j (u)} j∈N with λ j (u) ≤ λ j+1 (u) for every j ∈ N and each λ j (•) continuos on U . In analogy with Definition 7, we say that Σ(•) is conically connected if all eigenvalue intersections λ j = λ j+1 , j ∈ N, are conical (the definition of conical intersection extends trivially to this case) and for every j ∈ N there exists a conical intersection ūj ∈ U between the eigenvalues λ j , λ j+1 , with λ l (ū j ) simple if l = j, j + 1.

Remark 12

Recall from [START_REF] Boscain | Adiabatic control of the Schroedinger equation via conical intersections of the eigenvalues[END_REF] that conical intersections are generic in the case m = 2 in the reference case where

H = L 2 (Ω, C), H 0 = -∆ + V 0 : D(H 0 ) = H 2 (Ω, C) ∩ H 1 0 (Ω, C) → L 2 (Ω, C), H 1 = V 1 , H 2 = V 2 ,
with Ω a bounded domain of R d and V j ∈ C 0 (Ω, R) for j = 0, 1, 2. Indeed, generically with respect to the pair

(V 1 , V 2 ) in C 0 (Ω, R) × C 0 (Ω, R) (i.e., for all (V 1 , V 2 ) in a countable intersection of open and dense subsets of C 0 (Ω, R) × C 0 (Ω, R)), for each u ∈ R 2 and λ ∈ R such that λ is a multiple eigenvalue of H 0 + u 1 H 1 + u 2 H 2 ,
the eigenvalue intersection u is conical. Moreover, each conical intersection u is structurally stable, in the sense that small perturbations of V 0 , V 1 and V 2 give rise, in a neighbourhood of u, to conical intersections for the perturbed H.

The main purpose of this section is to extend Theorem 8 to the infinite-dimensional case, as follows.

Theorem 13 Let hypotheses (A ∞ ) and (B) be satisfied. If the spectrum Σ(•) is conically connected then ( 8) is approximately controllable.

The proof of Theorem 13 follows the same pattern as the one of Theorem 8. The first step is the following straightforward generalisation of Lemma 9.

Lemma 14 Let hypotheses (A ∞ ) and (B) be satisfied and assume that the spectrum Σ(•) is conically connected. Then there exists Ū ⊂ U which is dense and with zero-measure complement in U such that for each N ∈ N,

N j=1 α j λ j (ū) = 0 with (α 1 , . . . , α N ) ∈ Q N and ū ∈ Ū implies α 1 = α 2 = • • • = α N = 0.
In particular the spectrum of H(ū) for ū ∈ Ū as in Lemma 14 is such that two spectral gaps λ k (ū) -λ j (ū) and λ r (ū) -λ s (ū) are different if (k, j) = (r, s) and k = j, r = s.

In the infinite-dimensional case, the role of Proposition 11 is played by the following corollary of [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF]Theorem 2.6].

Proposition 15 Let hypotheses (A ∞ ) and (B) be satisfied. Assume that there exists

ū ∈ U such that λ k (ū) -λ j (ū) = λ r (ū) -λ s (ū) if (k, j) = (r, s), (k, j), (r, s) ∈ N 2 \ {(l, l) | l ∈ N}. Denote by (φ j (ū)) j∈N a Hilbert basis of eigenvectors of H(ū) and let S = {(j, k) ∈ N 2 | φ j (ū), H l φ k (ū) = 0 for some l = 1, . . . , m}.
If the graph having N as set of nodes and S as set of edges is connected then (8) is approximately controllable in S.

The proof of Theorem 13 is then concluded as follows: Lemma 14 guarantees the existence of ū such that the spectral gaps of Σ(ū) are all different; this allows to deduce the conclusion from Proposition 15 provided that no proper linear subspace of H spanned by eigenvectors of H(ū) is invariant for [START_REF] Beauchard | Semi-global weak stabilization of bilinear Schrödinger equations[END_REF]. As in the finite-dimensional case, this can be be proved by adiabatic methods, deducing from [30, Corollary 2.5] (or [START_REF] Boscain | Adiabatic control of the Schroedinger equation via conical intersections of the eigenvalues[END_REF]Proposition 3.4]) that for every pair of eigenvectors of H(ū) there exists and admissible trajectory of (8) connecting them with arbitrary precision.

Remark 16 Following [START_REF] Boscain | Multi-input schrdinger equation: controllability, tracking, and application to the quantum angular momentum[END_REF], a stronger version of Proposition 15, and hence of Theorem 13, could be stated, namely: under the same hypotheses, for every l ∈ N, ψ 1 , . . . , ψ l ∈ S, ε > 0, and every unitary transformation Υ of H, there exists a control function u : [0, T ] → U such that, for every j = 1, . . . , l the solution of (8) having ψ j as initial conditions arrives in a ε-neighborhood of Υ(ψ j ) at time T . Notice that this is the natural counterpart of controllability of the lift of (5) in the group of unitary transformations proved in Section 2.

4 Equivalence between exact and approximate controllability for finite-dimensional systems

In the previous sections we have seen several sufficient conditions for controllability, which is exact in the finite-dimensional case and approximate in the infinitedimensional one.

Our aim is to show that in the finite-dimensional case approximate controllability always yields exact controllability for systems of the type

i ψ(t) = H(u(t))ψ(t), ψ(t) ∈ S 2n-1 , u(t) ∈ U ⊂ R m , (9)
or

i ġ(t) = H(u(t))g(t), g(t) ∈ G , u(t) ∈ U ⊂ R m , (10)
where G denotes the group SU(n) if the trace of H(u) is zero for every u ∈ U and U(n) otherwise.

More precisely, we have the following.

Theorem 17 System ( 9) is approximately controllable if and only it is exactly controllable. The same holds for system [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF].

Remarks on Theorem 17

The proof of Theorem 17 is based on some results in representation theory, recalled in the following section. The statement of Theorem 17 for the lifted problem in SU(n) is folklore. Indeed, the proof follows from the following 1942 result by Smith [29, note on p. 312], as detailed below.

Theorem 18 ([29]) If a dense subgroup Ĝ of a simple Lie group G of dimension larger than 1 contains an analytic arc, then Ĝ = G.

Proof of Theorem 17 in the case G = SU(n). Let [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] be approximately controllable in SU(n). Then, the orbit from the identity is a dense subgroup Ĝ of SU(n). Any trajectory of [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] with constant u is an analytic arc, contained in Ĝ . Then Ĝ = SU(n), i.e., the orbit is the whole group. Lemma 3 yields that the accessible set coincides with G , i.e., that system (10) is exactly controllable.

Notice that the argument does not apply for G = U(n), since U(n) is not simple. Moreover, the equivalence between approximate and exact controllability on the sphere does not follow from the result on the lifted system. It is well-known, indeed, that approximate/exact controllability on the group and on the sphere are not equivalent since, as already recalled, if the Lie algebra generated by {iH(u) | u ∈ U } is equal to sp(n/2) then ( 9) is exactly controllable, while [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] is not (even approximately).

Some facts from group-representation theory

In this section, we recall the two basic main facts from representation theory that are needed in order to prove Theorem 17. We consider a finite-dimensional representation of a Lie group G, X : G → L(h), where h is a finite dimensional complex Hilbert space and L(h) denotes the space of endomorphisms of h.

Theorem 19 below is stated by Dixmier in [START_REF] Dixmier | Les C * -algèbres et leurs représentations[END_REF]. We need it for Lie groups, although it holds more generally for locally compact topological groups.

We recall that the intersection of the kernels of all unitary irreducible finitedimensional representations of a group G is a subgroup of G. Then, G is said to be injectable in a compact group1 if this subgroup is reduced to the identity of G. The second key fact that we need is due to Weil (see [33, p. 66]).

Proposition 20 ([19] 13.1.8) Let G = G 1 × G 2 be the Cartesian product of two locally compact topological groups, and let X be an irreducible representation of G. Define the representation X 1 of G 1 as X 1 (g 1 ) := X(g 1 , e) and the representation X 2 of G 2 as X 2 (g 2 ) := X(e, g 2 ). If X 1 and X 2 lie in a semisimple class of representations, then X is equivalent to the tensor product

X 1 ⊗ X 2 with X 1 , X 2 irreducible representations of G 1 , G 2 , respectively.
We would need to specify what a semisimple class of representations is, see [33, p. 65]. For our purpose, however, it is enough to recall that any class of bounded representation is semisimple (see, e.g., [33, p. 70]).

Remark 21

We finally recall some elementary properties for unitary representations of R p . First recall that each irreducible unitary representation is a character, namely, a representation of the type χ ξ (x) := e iξ•x for some ξ ∈ R p (see, e.g., [4, 6.1]). As a consequence we have that, for p ≥ 1, unitary irreducible representations of R p are not faithful.

Proof of Theorem 17

It is clear that exact controllability implies approximate controllability. The proof that approximate controllability implies exact controllability is based on the following two results.

Proposition 22 Let G be a connected Lie subgroup of U (n). If the inclusion repre- sentation  : G → U (n) is irreducible, then G is compact. Proof.
Observe that the inclusion  :

G → U (n) is a faithful (by definition) representation of G over C n , since U (n) ⊂ L(C n ) = gl(n, C).
Then, the kernel of  is reduced to {e}, and thus G is injectable in a compact group.

Applying Theorem 19, we have that G = R p × K with p ≥ 0 and K a compact group. Remark that  is unitary, hence bounded. As already recalled, the class of bounded representations of G is semisimple. Then we can apply Proposition 20, that gives us two irreducible bounded representations X 1 : R p → L(C m 1 ) and X 2 : K → L(C m 2 ) such that  is equivalent to X 1 ⊗ X 2 .

Since R p is abelian and X 1 is irreducible, then m 1 = 1. Bounded irreducible (one dimensional) continuous representations of R p must be unitary. Hence X 1 is a character of R p .

Since  is faithful, then X 1 and X 2 are faithful too. In conclusion, X 1 is a faithful irreducible unitary representation of R p . Then, thanks to Remark 21, we have that p = 0. Then G = K is compact.

Remark 23

The connectedness assumption in the statement of Proposition 22 is crucial: the groups SE(2, N ) in [START_REF] Boscain | Hypoelliptic diffusion and human vision: a semi-discrete new twist on the Petitot theory[END_REF] are counterexamples in the non-connected case.

Lemma 24 Let G be a subgroup of U (n) such that Gz is dense in S 2n-1 for every z ∈ S 2n-1 . Then  : G → U (n) is an irreducible representation of G.

Proof. Assume by contradiction that the inclusion is not irreducible, so that there exists a proper subspace h of C n which is invariant with respect to the action of G. Now take z ∈ h ∩ S 2n-1 and observe that Gz ⊂ h ∩ S 2n-1 . Thus Gz is not dense, leading to a contradiction.

We can now conclude the proof of Theorem 17.

Let G be the orbit of [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF], i.e., the subgroup of G whose Lie algebra is generated by {iH(u) | u ∈ U } (see Definition 2).

Assume that system ( 9) is approximately controllable. The reachable set from a point z ∈ S 2n-1 for ( 9) is contained in the orbit Gz. Hence, Gz is dense in S 2n-1 and Lemma 24 guarantees that the inclusion  : G → U (n) is an irreducible representation of G. We can then apply Proposition 22 and conclude that G is compact. In particular, Gz is compact in S 2n-1 for every z ∈ S 2n-1 . Finally, being Gz dense and compact in S 2n-1 , it coincides wit S 2n-1 , i.e., system (9) is exactly controllable.

Let now [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] be approximately controllable. Hence, G is dense in G . In particular, system (9) is also approximately controllable and, according to the argument above, G is compact. Hence G = G , i.e., by Lemma 3, ( 10) is exactly controllable. This concludes the proof of Theorem 17.

Remark 25 If the attainable set of system [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] is dense in any subgroup G of U (n) which acts transitively on S 2n-1 , then the same argument as above shows that ( 9) is exactly controllable in S 2n-1 and ( 10) is exactly controllable in G.

Figure 1 :

 1 Figure 1: A conical intersection when m = 2: the surfaces represent two eigenvalues of H(u 1 , u 2 ) as functions of u 1 and u 2 .

Figure 2 :

 2 Figure 2: A conically connected spectrum in the case m = 2.

  Theorem 19 ([19] 16.4.8) Let G be a connected, locally compact group. Then G is injectable in a compact group if and only if G = R p × K with p ≥ 0 and K a compact group.

The definition given here is not the most natural, since injectability in a compact group is related to the notion of compact group associated with a topological group that is defined via an universal property: For each topological group G there exists a compact group Σ and a continuous morphism α : G → Σ such that for any compact group Σ and continuous morphism α : G → Σ it exists a continuous morphism β : Σ → Σ such that α = β • α. We give here only the definition that fits better with our purposes. For such beautiful theory, see[19, 16.4].

Acknowledgements: This research has been supported by the European Research Council, ERC StG 2009 "GeCoMethods", contract number 239748.

Proof of Theorem 8. Applying Lemma 9 we deduce the existence of u 0 ∈ U such that if n j=1 α j λ j (u 0 ) = 0 with (α 1 , . . . , α n ) ∈ Q n then α 1 = • • • = α n . In particular, the spectrum of H(u 0 ) is simple and two spectral gaps λ j (u 0 )-λ k (u 0 ) and λ r (u 0 )-λ s (u 0 ) are different if (j, k) = (r, s) and j = k, r = s. Let φ 1 , . . . , φ n be an orthonormal basis of eigenvectors of H(u 0 ).

Let us conclude the proof by applying Proposition 11 to A 0 = iH(u 0 ), A j = iH j for j = 1, . . . , m: to this purpose, we are left to prove that the graph having 1, . . . , n as nodes and

as set of edges is connected.

Assume by contradiction that such graph is not connected. Then there exists a proper subspace V of C n generated by eigenvectors of H(u 0 ) which is invariant for the evolution of [START_REF] Beauchard | Local controllability of a 1-D Schrödinger equation[END_REF]. Without loss of generality V = span{φ 1 , . . . , φ r } with r < n.

Since the spectrum is conically connected, we can apply [30, Corollary 2.5] and deduce that there exists an admissible trajectory of (5) steering φ 1 to an arbitrary small neighbourhood of {e iθ φ n | θ ∈ R}. (See also [START_REF] Boscain | Adiabatic control of the Schroedinger equation via conical intersections of the eigenvalues[END_REF]Proposition 3.4] for a rephrasing in control terms of [30, Corollary 2.5], which deals with general adiabatic evolutions through conical intersections. The result is stated in [START_REF] Boscain | Adiabatic control of the Schroedinger equation via conical intersections of the eigenvalues[END_REF] in the case m = 2 for symmetric Hamiltonians but actualy holds in the general case.) The contradiction is reached, since V ∩ {e iθ φ n | θ ∈ R} = ∅.

Conical intersections and approximate controllability in infinite dimension

In this section we extend the controllability analysis of the previous section to systems of the form (5) evolving in infinite-dimensional spaces. Consider a separable infinite-dimensional complex Hilbert space H. In this section we make the following assumption:


MD J.-N A Argenson
email: jean-noel.argenson@ap-hm.fr

Keywords:

T he development of adverse local tissue reactions (ALTRs) in patients implanted with metal-on-metal (MoM) bearing sur-faces for hip arthroplasty may cause early failure or silent progression of the destructive reaction, making revision even more complex. Because of this early revision rate, the Articular Surface Replacement (ASR) system (DePuy Orthopaedics, Warsaw, IN, USA), both for resurfacing and for THA, was recalled in 2010 after 6 years of commercial use [START_REF] Whitwell | The articular surface replacement implant recall: A United Kingdom district hospital experience[END_REF].

This raises a few interesting questions: How do we screen nonrevised patients in order to detect ALTRs? Should we treat such lesions by revision based on risk stratification, as recently proposed in the consensus statement of American Academy of Orthopaedic Surgeons, American Association of Hip and Knee Surgeons, and the Hip Society [START_REF] Kwon | Risk stratification algorithm for managment of patients with metal-onmetal hip arthroplasty: consensus statement of the American Association of Hip and Knee Surgeons, the American Academy of Orthopaedic Surgeons, and the Hip Society[END_REF]? This screening currently includes a combi-nation of cross-sectional imaging, patient symptomatology and/or reported outcome, and metal ions level. The location and severity of ALTRs are important factors to be considered, as well as the interrelation with the patient outcomes.

In their current study, Malchau and colleagues examined the location of ALTRs, their severity, and the clinical consequences of using metal artifact reduction MRI and patient reported outcome scores on 288 individuals evaluated at a mean time of 6 years of either resurfacing or THA ASR. Nearly one-fourth of the nonrevised patients showed signs of moderate or severe ALTRs (pseudotumors), although they could not intraoperatively confirm this finding. Nevertheless, this incidence should be considered valid since all of the revised ASR implants showed solid pseudotumors on cross-sectional imaging, with MRI as the primary imaging modality [START_REF] Lainiala | Revision of metal-on-metal hip prostheses results in marked reduction of blood cobalt and chromium ion concentrations[END_REF]. location of ALTRs (anterior, posterior, or on both sides of the hip in axial images). However, the study authors did not examine patient MRIs for elevated metal ion levels. This is interesting because a previously published study [START_REF] Hailer | High metal ions levels after use of the ASR device correlate with development of pseudotumors and T cell activation[END_REF] found a correlation between the presence of pseduotumors on MRI and elevated metal ion levels. On the other hand, a recently published study [START_REF] Bayley | What are the predictors and prevalence of pseudotumor and elevated metal ions after large-diameter metal-on-metal THA?[END_REF] did not find a close correlation between ion levels and the presence of pseudotumors.

Malchau and colleagues assessed patient MRIs for the presence and

Where Do We Need To Go?

The current study describes a clear relation between the location of ALTRs and surgical approach, which could have an impact on onset of symptoms either by neurovascular effect or abductor damage; this may also play a role at the time of revision planning for evaluating the best soft tissue preserving approach according to surgeon practice.

The location and severity of ALTRs found in the present study deal with one single design, therefore, the results may not be generalized to all MoM hip arthroplasty designs. Because ALTRs may lead to catastrophic joint damage, it will be important to get a better understanding of the mechanism producing this adverse reaction in a variety of implants, as well as identify which patients may be experiencing ALTR, specifically if they are asymptomatic. A number of questions also remain regarding imaging: Which approach, ultrasound or MRI, affords the best balance of sensitivity, specificity, and cost-effectiveness? We know from this study and others [START_REF] Chang | Metal-on-metal total hip arthroplasty: Do symptoms correlate with MR imaging findings?[END_REF][START_REF] Macnair | Silent soft tissue pathology is common with a modern metal-onmetal hip arthroplasty[END_REF] that validated outcomes scores sometimes stay the same whether or not ALTR is present. Therefore, what is the best approach to using validated outcomes scores in patient assessment for this problem?

How Do We Get There?

Answering these questions will require large cohort studies with patients who were treated similarly, screened using the same methodological tools, and received the same implant. The authors of the present study do have the potential to drive such a study with the specific implant evaluated since, as of this writing, only 288 of the 1258 patients have been evaluated under their protocol. The involvement of the three other centers provide the possibility of using metal artifact reduction sequence protocol MRI, and will lead into a true worldwide multicenter prospective followup study. This contribution will be helpful, not only for the specific implant evaluated in the present study, but for all MoM bearing surfaces for hip arthroplasty, which may develop ALTRs.


Jean-Marie Barbaroux
email: barbarou@univ-tln.fr

Walter Farkas
email: farkas@math.ethz.ch

Bernard Helffer
email: bernard.helffer@math.u-psud.fr

Heinz Siedentop

ON THE HARTREE-FOCK EQUATIONS OF THE ELECTRON-POSITRON FIELD

We study the energy of relativistic electrons and positrons interacting via the second quantized Coulomb potential in the field of a nucleus of charge Z within the Hartree-Fock approximation. We show that the associated functional has a minimizer. In addition, all minimizers are purely electronic states, they are projections, and fulfill the no-pair Dirac-Fock equations.

Introduction

Heavy atoms should be described by relativistic quantum mechanics. It is commonly believed quantum electrodynamics (QED) yields such a description. Formally the Hamiltonian is given (Bjorken and Drell [START_REF] Bjorken | Relativistic Quantum Fields. International Series in Pure and Applied Physics[END_REF]Formula 15.28]) as [START_REF] Avron | The index of a pair of projections[END_REF] H = dx : ψ * (x) α • 1 i ∇ -√ αA(x) + mβ -αZ |x| ψ(x) : + α 2 dx dy :ψ * (x)ψ(x): :ψ * (y)ψ(y):

|x-y| + 1 8π R 3 : B(x) 2 + . A (x) 2 : dx ,
where the normal ordering denoted by colons is with respect to a given choice of the one-electron space. However, it is not clear how this expression can be selfadjointly realized as a positive operator. To simplify matters we omit the energy of the transverse radiation field coupled to the current, i.e., we set B = 0 and A = 0 in the above expression. We will also regularize the Coulomb interaction of the electron-positron field by normal ordering it completely. Both assumptions are simplifications. The first one can be justified by the physical wisdom that the presence of self-generated magnetic field is physically known to be small compared to the relativistic effects in heavy atoms. Moreover, the inclusion poses serious technical problems that we presently cannot solve. The second assumption ignores the vacuum polarization effects which are also small comared to the relativistic effects and would contribute to the Lamb shift only.

Based on an interesting observation of Chaix, Iracane, and Lions [START_REF] Chaix | From quantum electrodynamics to mean-field theory: I. The Bogoliubov-Dirac-Fock formalism[END_REF][START_REF] Chaix | From quantum electrodynamics to mean-field theory: II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation[END_REF], Bach et al. [START_REF] Volker | On the stability of the relativistic electron-positron field[END_REF][START_REF] Volker | Stability of matter for the Hartree-Fock functional of the relativistic electron-positron field[END_REF] showed positivity for the corresponding quadratic form without any constraints on the charge of the state, if the one-electron subspace is appropriately (Furry picture) chosen. In particular they showed that the vacuum state (particle number equal to zero) has energy zero. Date: March 23, 2004. Key words and phrases. Nopair atomic Dirac-Fock equations, electron-positron field, quasifree states, stability.

We thank K. Yajima for critical reading of a preliminary version of the manuscript and D. Hundertmark for useful and enjoyable discussions concerning various aspects of this paper. One of us, JMB, thanks B. Grébert for useful discussions concerning Section 3. Financial support of the Ministère de l'Education Nationale, de la Recherche et de la Technologie through ACI Blanche, of the Bayerisch-Französisches Hochschulzentrum, of the European Union through the IHP network "Analysis and Quantum", contract HPRN-CT-2002-00277, are gratefully acknowledged. Part of the work was done while three of us, JMB, BH, and HS, were visiting the Mittag-Leffler Institute, its support is also gratefully acknowledged.

In order to do so, they proved that the positivity of H on generalized Hartree-Fock states (quasifree states with finite free kinetic energy) is equivalent to the positivity of the Hartree-Fock functional (see the Definition given in ( 23)), a functional on density matrices γ, where the charge is given as tr γ, which thus is unrestricted.

For describing atoms one needs, however, to restrict to states with prescribed charge q. In order to implement this problem, we subtract from the energy the rest mass m, which -as we will see -will allow us to relax the constraint tr γ = q to tr γ ≤ q.

Given an electron space by the positive spectral subspace of a Dirac operator with a mean-field potential, we show that there exists a minimizer of the associated Hartree-Fock functional in a suitable set, if q ≤ Z (Theorem 3.9). Moreover, if q is a positive integer, all minimizers are projections of maximal rank. In particular, minimizers are purely electronic and are projections (Theorem 4.3). Finally, we show that the eigenfunctions of any minimizer γ fulfill the no-pair Dirac-Fock equations with the q first (positive) eigenvalues (Theorem 4.5).

According to Mittleman [START_REF] Mittleman | Theory of relativistic effects on atoms: Configuration-space Hamiltonian[END_REF], the most stable, i.e., highest ground state energy is the physical ground state. Thus one should maximize among the allowed oneelectron subspaces, yielding a max-min variational problem. The resulting Euler equation should, on a heuristic level, give the Dirac-Fock equations which were treated by Esteban, Séré, and Paturel [START_REF] Esteban | Solutions of the Dirac-Fock equations for atoms and molecules[END_REF][START_REF] Paturel | Solutions of the Dirac-Fock equations without projector[END_REF][START_REF] Esteban | Nonrelativistic limit of the Dirac-Fock equations[END_REF]. There are indications that this latter question might be answered affirmatively only under additional hypotheses: In a recent work [START_REF] Barbaroux | Some connections between Dirac-Fock and electron-positron Hartree-Fock[END_REF] an atom with total charge q -Z ≤ 0 is considered. If the ground state of the noninteracting problem with N = q electrons corresponds to a closed shell, then for small interaction maximizing over one-electron subspace yields the Dirac-Fock equations in the non-relativistic limit. However, if the noninteracting problem corresponds to an open shell, then, in the same limit, the max-min procedure does not yield a solution of the Dirac-Fock equations with self-consistent projector as considered by Esteban and Séré [START_REF] Esteban | Solutions of the Dirac-Fock equations for atoms and molecules[END_REF]. While this result is perturbative, it indicates on the one hand side that the Dirac-Fock equations and the Mittleman principle might agree in the case of filled shells whereas in the unfilled shell case it might give different results which raises the question which procedure is physically relevant, a problem that we have to leave open at this point.

We add a short guide through the paper for the orientation of the reader: Section 2 contains some basic material. We define the set of density matrices that will be allowed. There will be two types of density matrices, the charge density matrices γ for the electron-positron field and the density matrices δ giving the screening of the one-particle Dirac operator that defines the electron subspace. In addition this section contains some basic estimates on the direct and exchange energy. Section 3 contains the actual minimization. We first show that the elimination of positrons lowers the energy (Lemma 3.1); next we show that the density matrices can be restricted to finite rank (Lemma 3.3), and the minimization under the constraint tr γ ≤ q ≤ Z gives a minimizer -if existing -with charge equal to q (Lemma 3.7). This allows us to show the existence of minimizers (Theorem 3.9). In Section 4 we investigate the minimizers. They turn out to be projections that fulfill the no-pair Dirac-Fock equations (Theorem 4.5). In the last section we give an outlook with respect to the above mentioned program of Mittleman. We derive the corresponding Euler equation (Theorem 5.1). However, we are not able to show that there is a maximizer.

Definition of the Problem

A single relativistic electron or positron in the field of a nucleus of charge Z can be described by the Coulomb-Dirac operator (2)

D Z := α • 1 i ∇ + mβ -α Z |x| ,
where α is the Sommerfeld fine structure constant.

The operator D Z is self-adjointly realized in

H := L 2 (R 3 ) ⊗ C 4 and essentially self-adjoint on C ∞ 0 (R 3 \ {0}) ⊗ C 4 , if αZ ∈ (- √ 3/2, √ 3/2).
Here, we will even assume [START_REF] Bach | Error bound for the Hartree-Fock energy of atoms and molecules[END_REF] αZ ∈ [0, √ 3/2) and α ≥ 0 , throughout the paper. The domain of

D Z is H 1 (G)
where

G := R 3 × {1, 2, 3, 4} .
(Landgren and Rejto [START_REF] Landgren | An application of the maximum principle to the study of essential selfadjointness of Dirac operators[END_REF], Theorem 2.1). For Z = 0, D 0 is just the free Dirac operator.

In the present paper, we consider a larger class of Dirac operators, namely Dirac-Fock operators. They are Hamiltonians for a relativistic particle in a mean field created by other particles. For that purpose, we will consider operators with an additional mean field potential ( 4)

W (δ) = ϕ (δ) -X (δ) ,
where ϕ (δ) and X (δ) will be defined below.

For p ∈ [1, ∞), we denote by S p (H) = {A ∈ B(H) | tr |A| p < ∞},
and by S ∞ (H) the space of compact operators on H. Definition 2.1. We denote by F the Banach space of all self-adjoint operators δ on H such that the norm δ F := tr

( |D 0 | 1/2 |δ| |D 0 | 1/2 ) is finite.
For a given element δ ∈ F , we denote by (λ n ) the sequence of its eigenvalues and by (ξ n ) a corresponding orthonormal basis of eigenvectors; the associated integral kernel δ(x, y) is

δ(x, y) := n λ n ξ n (x)ξ n (y) . ( 5
)
(It is convenient to introduce the notation x = (x, s) for an element of G and dx for the product of the Lebesgue measure dx on R 3 with the counting measure in {1, 2, 3, 4}.) Associated with δ is its one-particle density [START_REF] Barbaroux | Some connections between Dirac-Fock and electron-positron Hartree-Fock[END_REF] ρ δ (x) :=

4 s=1 n λ n |ξ n (x)| 2 , its electric potential (7) ϕ (δ) (x) = R 3 ρ δ (y) |x -y| dy ,
and its exchange operator

X (δ) (8) ψ → δ(x, y)ψ(y) |x -y| dy .
The difference of these two operators is the mean field potential W (δ) defined in [START_REF] Volker | Stability of matter for the Hartree-Fock functional of the relativistic electron-positron field[END_REF]. Next, we define for the given δ the Coulomb-Dirac operator associated to δ as (9) D (δ) := D Z + αW (δ) .

As shown in the Appendix A (Lemmata A.7 and A.8), the operator W (δ) is bounded implying that D (δ) is self-adjoint with the same domain as the Coulomb-Dirac operator D Z which for αZ ∈ [0, √ 3/2) is identical with the domain of D 0 . Moreover (see Lemma A.9), W (δ) is relatively compact with respect to D 0 which implies The one-electron states are vectors in H + = Λ + H where Λ + is an orthogonal projection on H, whereas one-positron states are charge conjugated states in Λ -H, where Λ -:= 1 -Λ + ( [START_REF] Thaller | The Dirac Equation[END_REF]). We will take Λ + to be the projection Λ (δ) + onto the positive spectral subspace of the Dirac-Fock operator D (δ) , i.e., [START_REF] Chaix | From quantum electrodynamics to mean-field theory: II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation[END_REF] Λ + := Λ (δ)

σ ess (D (δ) ) = σ ess (D 0 ) = (-∞, -m] ∪ [m, ∞).
+ := χ [0,∞) (D (δ)
) , where χ I denotes the characteristic function of the set I. Thus, the choice of δ fixes the definition of the spaces of electrons and positrons.

The Coulomb scalar product is

(11) D[ρ, σ] := 1 2 R 3 dx R 3 dy ρ(x)σ(y) |x -y| .
The exchange scalar product for γ, γ

′ ∈ F is (12) E[γ, γ ′ ] := 1 2 dx dy γ(x, y)γ ′ (x, y) |x -y| . Lemma 2.2. Assume that γ, γ ′ ∈ F . Then |D[ρ γ , ρ γ ′ ]| ≤ π 4 γ 1 tr ( √ -∆|γ ′ |) , (13)
|E[γ, γ ′ ]| ≤ D[ρ |γ| , ρ |γ ′ | ] . ( 14
)
Proof. Expanding γ and γ ′ in their respective bases of eigenfunctions (see ( 5)), we get by the Cauchy-Schwarz inequality [START_REF] Landgren | An application of the maximum principle to the study of essential selfadjointness of Dirac operators[END_REF]

γ(x, y)γ ′ (x, y) |x -y| dx dy = µ λ µ ν λ ′ ν ξ µ (x)ξ µ (y)ξ ′ ν (x)ξ ′ ν (y) |x -y| dxdy ≤ µ |λ µ ||ξ µ (x)| 2 ν |λ ′ ν ||ξ ′ ν (y)| 2 |x -y| dxdy = R 3 R 3 ρ |γ| (x)ρ |γ ′ | (y) |x -y| dx dy ,
which shows that it suffices to prove [START_REF] Esteban | Solutions of the Dirac-Fock equations for atoms and molecules[END_REF].

To this end we remark that by Kato's inequality

(16) dx|ξ µ (x)| 2 dy |ξ ′ ν (y)| 2 |x -y| ≤ π 2 (ξ ′ ν , |∇|ξ ′ ν ) .
The claimed inequality follows now by multiplication with |λ µ λ ′ ν | and summation over µ and ν.

We will also need the following result of Bach et al ( [START_REF] Volker | On the stability of the relativistic electron-positron field[END_REF], Equation (30)) :

Lemma 2.3. If γ ∈ F , then E[γ, γ] ≤ π 4 tr γ * √ -∆ γ .
Given an operator A on the Hilbert space H the symbols

A ++ = Λ (δ) + AΛ (δ) + , A +-= Λ (δ) + AΛ (δ) -, A -+ = Λ (δ) -AΛ (δ) + , and A --= Λ (δ) -AΛ (δ)
-denote the matrix elements of the decomposition of A with respect to the splitting of the Hilbert space given by Λ Relativistic electrons and positrons are described by one-particle charge density matrices γ with certain additional properties reflecting the charge of the particle, and the fact that they are Fermions and thus obey the Pauli principle. Definition 2.4. Given δ ∈ F and q ∈ R + we define the following sets of oneparticle diagonal charge density matrices:

S (δ) := {γ ∈ F | -Λ (δ) -≤ γ ≤ Λ (δ) + , Λ (δ) -γΛ (δ) + = 0} , (17)
S (δ) q := {γ ∈ S (δ) | 0 ≤ tr γ ≤ q} , (18)
S (δ) ∂q := {γ ∈ S (δ) | tr γ = q} . ( 19
)
We note that all sets are closed subsets of F . Furthermore, the first two are convex. Note also that for γ ∈ S (δ) we have γ ++ ≥ 0 and γ --≤ 0 which follows directly from the definition. We also observe that :

γ 2 ++ ≤ γ ++ , (20)
γ 2 --≤ -γ --, (21)
which permits to get (in the case of diagonal density matrices)

tr (|D (δ) ||γ|) = tr (D (δ) γ) ≥ tr (|D (δ) |γ 2 ) . (22)
The elements of S (δ) are the one-particle (renormalized) charge density matrices γ of the electron-positron field. The trace is its total charge. Since we are interested in describing atoms we want to fix the charge of the electron-positron field to be q and minimize the energy over the set S (δ) ∂q . For technical reasons we will also use S (δ) q . We wish to point out that he derivation of the variational spaces of one-particle charge density matrices as done in [START_REF] Volker | On the stability of the relativistic electron-positron field[END_REF] does not give the extra condition Λ (δ) -γΛ (δ) + = 0 that appears in the definition of S (δ) . A formal calculation shows that if we do not assume that the one-particle density matrices have off diagonal terms equal zero, then in most cases, Lemma 3.1 and Theorem 4.3 and 4.5 do not hold. In particular, minimizers will contain electron-positron pairs.

The projections Λ

-can be physically interpreted as the one-particle density matrix of the Dirac sea. In particular, Λ

-is the Dirac sea under the influence of a nucleus of charge eZ and an electron-positron distribution given by δ.

For later purposes we also introduce (unrenormalized) density matrices as

Γ := Λ (δ) -+ γ ,
representing all electrons including those of the Dirac sea.

The unrenormalized density matrices are nonnegative expressing the fact that positrons occur in this picture only as 'holes' in the Dirac sea.

The energy of a system of electrons and positrons in Hartree-Fock approximation is given by the functional

(23) E : S (δ) → R γ → tr (D Z γ) + αQ[γ, γ] ,
where

(24) Q : S (δ) × S (δ) → R (γ, γ ′ ) → D[ρ γ , ρ γ ′ ] -E[γ, γ ′ ] .
As explained above we are primarily interested in the infimum of E| S (δ)

∂q

; for technical reasons we will also consider, for µ ∈ R,

(25) E µ : S (δ) → R γ → E(γ) -µ tr γ Lemma 2.5.
For any µ ∈ R, the energy functional E µ is well defined and continuous in the • F norm.

Proof. The lemma is an immediate consequence of the definition of the norm, Lemma 2.3 together with the fact that γ 2 ≤ |γ| and Inequalities ( 13) and ( 14).

Minimization of the Energy

3.1. Reduction to Electrons. Lemma 3.1. Assume q > 0, γ ∈ S (δ) ∂q , 0 ≤ δ ∈ F . Moreover, assume c α,Z,m,δ > πα(1/4 + max{tr δ, q}) .
Then there exists a nonnegative γ e ∈ S (δ)

∂q and R ∈ S (δ)
∂0 such that γ = γ e + R and E(γ e ) ≤ E(γ). In addition, equality can only occur if 0 ≤ γ.

Physically speaking this lemma says that it is favorable to have no positrons in the system and to restrict the minimization to electron states.

Proof. Using the spectral decomposition of γ, one can easily construct γ e and R such that

γ e ∈ S (δ) ∂q , R ∈ S (δ)
∂0 and γ = γ e + R. We also note that we can pick R = 0, if γ ≥ 0, and that we can pick R = 0, if γ ≥ 0. We have

E(γ e ) -E(γ) = -tr (D Z R) -αQ[R, R] -2αℜQ[R, γ e ] = -tr (D (δ) R) -αQ[R, R] -2αℜQ[R, γ e ] + 2αℜQ[R, δ] ≤ -tr (D (δ) R) + αE[R, R] -2αℜQ[R, γ e ] + 2αℜQ[R, δ] , (26)
where in the last inequality, we used positivity of

D[ρ R , ρ R ].
Let R + and R -be respectively the positive and negative part of R, i.e.

R = R + -R -with R ± ≥ 0 .
Using Lemma A.8 and the positivity of γ e yields

ℜQ[R + , γ e ] ≥ 0 and thus -ℜQ[R, γ e ] ≤ ℜQ[R -, γ e ] .
Similarly, we have

ℜQ[R, δ] ≤ ℜQ[R + , δ].
Therefore, from Inequality (26) we obtain

E(γ e ) -E(γ) ≤ -tr (D (δ) R) + αE[R, R] + 2αℜQ[R -, γ e ] + 2αℜQ[R + , δ] . (27)
From Lemma 2.3 and the definition of c α,Z,m,δ in (87), we get

(28) E[R, R] ≤ π 4 tr (|D 0 |R 2 ) ≤ π 4c α,Z,m,δ tr (|D (δ) |R 2 ) ≤ π 4c α,Z,m,δ tr (D (δ) R) ,
where the last inequality is a consequence of [START_REF] Thaller | The Dirac Equation[END_REF]. Using ( 14), [START_REF] Esteban | Solutions of the Dirac-Fock equations for atoms and molecules[END_REF], and, finally, the definition of c α,Z,m,δ in (87) we have

ℜQ[R -, γ e ] = D[ρ R-, ρ γe ] -ℜE[R -, γ e ] ≤ 2D[ρ R-, ρ γe ] ≤ π 2 tr γ e tr (|p|R -) ≤ π tr γ e 2c α,Z,m,δ tr (|D (δ) |R -) . ( 29
)
With exactly the same arguments as above, exchanging R -by R + and γ e by δ we get

(30) ℜQ[R + , δ] ≤ π tr δ 2c α,Z,m,δ tr (|D (δ) |R + ).
Since tr (|D (δ) ||R|) = tr (D (δ) R), we have, using ( 27)-(30)

(31) E(γ e ) -E(γ) ≤ -1 + πα 4c α,Z,m,δ + πα max{tr δ, q} c α,Z,m,δ tr (D (δ) R) ≤ 0 ,
under the hypothesis of the theorem. In addition we note that the last inequality is strict unless R = 0 or the prefactor vanishes.

Reduction to Density Matrices with Finite

Spectrum. Lemma 3.2. Assume 0 ≤ q, 0 ≤ δ ∈ F and 0 ≤ γ ∈ S (δ)
∂q . Then there exists a sequence of finite rank density matrices γ K ∈ S (δ)

∂q such that γ K -γ F → 0 as K → ∞. Proof. Let (ξ k ) k∈N be a complete set of eigenfunctions in H 1 (G) of γ associated with the eigenvalues λ k . Since γ ≥ 0, we have γ = Λ (δ) + γΛ (δ)
+ and, for all k, we have

ξ k ∈ Λ (δ) + H.
Assume that γ is not already of finite rank. Then, since γ is trace class, there exist infinitely many eigenvalues of γ in (0, 1). Let us pick λ n0 ∈ (0, 1), one of these eigenvalues.

Set

ǫ K := q - K k=1 λ k . Then ǫ K is a nonnegative monotone decreasing sequence tending to zero. Define γ K := K k=1 λ k |ξ k ξ k | + ǫ K |ξ n0 ξ n0 |.
We assume that K is chosen sufficiently large so that K ≥ n 0 and λ n0 + ǫ K < 1. Obviously each γ K is nonnegative, belongs to S (δ) ∂q and has finite rank. We now show that γ K → γ in F-norm as K → ∞. We have

γ -γ K = ∞ k=K+1 λ k |ξ k ξ k | -ǫ K |ξ n0 ξ n0 | . Thus, (32) γ -γ K F ≤ ∞ k=K+1 λ k tr (|D 0 ||ξ k ξ k |) + ǫ K (ξ n0 , |D 0 |ξ n0 ) .
The first in the right hand side tends to zero, since |D 0 ||γ| ∈ S 1 (H), and the second tends to zero, since ǫ K → 0.

The following is an immediate consequence of the continuity of E in the F -norm and the preceding density result. Equality in (33) holds only if γ is already a projection.

Lemma 3.3. Assume that q > 0. Then inf{E µ (γ) | 0 ≤ γ ∈ S (δ) ∂q } = inf{E µ (γ) | 0 ≤ γ ∈ S (δ) ∂q , rank(γ) < ∞}.
Proof. Suppose that γ is not a projection. Then γ has an eigenvalue λ ∈ (0, 1); we denote a corresponding normalized eigenvector by u and observe that it is in H 1 (G). Since tr (γ) ∈ N, there exists at least a second eigenvalue µ ∈ (0, 1); we denote a corresponding normalized eigenvector by v. We set (34) γ := γ + ǫS ,

where S := |u u| -|v v|. We get

(35) E(γ) -E(γ) = ǫ(tr (D Z S) + 2ℜQ[γ, S]) + ǫ 2 Q[S, S] .
By explicit computation and use of the Cauchy-Schwarz inequality, one can show that :

(36) Q[S, S] < 0 , if S is a difference of two orthogonal rank one projections. Now -depending on the sign of the coefficient linear in ǫ -we lower the energy by increasing or decreasing ǫ from zero, until one of the constraints 0 ≤ λ + ǫ, µ -ǫ ≤ 1 forbids any further increase or decrease of ǫ. This process leaves all the eigenvalues of γ unchanged except for µ and λ, one of which becomes either 0 or 1.

Since there are only finitely many eigenvalues of γ strictly between zero and one (even if they are counted according to their multiplicity), iterating this process eliminates all eigenvalues that are strictly between 0 and 1, i.e., we have found a density matrix P such that P 2 = P . Remark 3.5. Following the same method in the case of q ≥ 0, not necessarily integer, we can show that, given 0 ≤ γ ∈ S (δ) ∂q , there exists P equals a projection plus a rank one operator such that tr P = tr γ, P ≥ 0 and E( P ) ≤ E(γ), with equality only if γ is already a projection plus a rank one operator.

3.4.

Lower Bound on the Energy. In this subsection, we show that for sufficiently small fine structure constant α and atomic number Z, the energy is bounded from below. Theorem 3.6. Assume 0 ≤ δ ∈ F , and c α,Z,m,δ ≥ πα(1/4 + max{tr δ, q}) > 0.

Then, for all γ ∈ S (δ)

q , E(γ) ≥ 0.

Proof. By Lemma 3.1 we need to consider only positive γ's. In this case, [START_REF] Griesemer | A minimax principle for the eigenvalues in spectral gaps[END_REF] implies

Q[γ, γ] ≥ 0. Now, for f ∈ Λ (δ)
+ H, using the definition (87) of c α,Z,m,δ , Inequality [START_REF] Esteban | Solutions of the Dirac-Fock equations for atoms and molecules[END_REF] and the positivity of E (δ) , we obtain

(f, D Z f ) = (f, |D (δ) |f ) -α(f, W (δ) f ) ≥ c α,Z,m,δ (f, |p|f ) - π 4 α tr δ(f, |p|f ) ≥ 0 .
Thus, under our hypotheses, E(γ) ≥ tr (D Z γ) ≥ 0. Lemma 3.7. Assume 0 ≤ q, 0 ≤ δ ∈ F and suppose that, for all 0 ≤ p < q and all 0 ≤ γ ∈ S (δ)

p , the operator

Λ (δ) + D (γ) Λ (δ)
+ has infinitely many eigenvalues in (0, m).

Then inf E m (γ)|γ ∈ S (δ) ∂q , γ ≥ 0 = inf E m (γ)|γ ∈ S (δ) q , γ ≥ 0 . If in addition 0 ≤ γ is a minimizer of E m in S (δ)
q , it follows that tr γ = q.

Proof. That the left side bounds the right side from above is obvious. To prove the converse inequality, we assume 0 ≤ γ ∈ S (δ)

q , with tr γ < q. By Lemma 3.4 and Remark 3.5 we can assume that γ is a projection plus a rank one operator; in particular its range is finite dimensional. Since by assumption the discrete spectral subspace X of Λ

(δ) + D (γ) Λ (δ)
+ is infinite dimensional we can find u ∈ H + ∩ X ∩ γ(H) ⊥ with 0 < u ≤ 1 and define γ := γ + A with A := |u >< u|. We then get

E m (γ + A) -E m (γ) = tr (D Z A) + 2ℜQ(γ, A) -m u 2 = u, (D (γ) -m)u < 0 .
Thus, this construction yields a density matrix γ with strictly smaller energy and a trace that can be made bigger by min{1, q -tr γ}. Iteration of the construction yields the desired result. This proves both claims.

3.6. Existence of a Minimizer. We wish to show the existence of a minimizer by weak lower semi-continuity of the functional on a minimizing sequence and weak compactness. However, we are faced with the problem that we are minimizing on charge density matrices and the fact that the Coulomb potential is not relatively compact with respect to the relativistic energy |p|. The first problem has been addressed by Solovej [START_REF] Philip | Proof of the ionization conjecture in a reduced Hartree-Fock model[END_REF] in the non-relativistic context. To handle the second problem we will decompose the one-particle part of the energy.

Theorem 3.8. Assume 0 < c α,Z,m,δ , 0 ≤ δ ∈ F and q ∈ N. Furthermore, suppose that E m is bounded below on S (δ) ∂q and inf{E m (γ) | γ ∈ S (δ) q } = inf{E m (γ) | γ ∈ S (δ)
∂q , γ 2 = γ} . Then the energy functional E m | S (δ) q has a minimizer.

Proof. Let γ n be a minimizing sequence of orthogonal projections in S (δ) ∂q .

Step 1. Weak Limit of the Minimizing Sequence: Since E m (γ n ) and ϕ (δ) are bounded (Lemma A.7) (and thus also X (δ) , by Lemma A.8, and W (δ) ) there exists a constant C such that, for any p ≥ 1

C ≥ E m (γ n ) + αq W (δ) ≥ tr (|D (δ) |γ n ) ≥ c α,Z,m,δ tr (|D 0 |γ n ) = c α,Z,m,δ γ n F ≥ c α,Z,m,δ |D 0 | 1/2 γ n |D 0 | 1/2
p . Thus, according to Banach and Alaoglu, there exists, for all p > 1, γ ∞ with |D 0 | 1/2 γ ∞ |D 0 | 1/2 p < ∞ and a subsequence of γ n -also denoted by γ n -such that for all B with B q < ∞,

(37) tr (B|D 0 | 1/2 γ n |D 0 | 1/2 ) → tr (B|D 0 | 1/2 γ ∞ |D 0 | 1/2 ) ,
where 1/p + 1/q = 1. We denote γ n F,p :

= |D 0 | 1/2 γ n |D 0 | 1/2 p . Given p 1 ≥ p 2 ≥ 1
, it is always possible to extract a subsequence of γ n -again denoted by γ n -, such that it converges weakly in (F, . F,p1 ) and (F, . F,p2 ). Denote the weak limits respectively by γ

(B|D 0 | 1/2 γ (p1) ∞ |D 0 | 1/2 ) = lim n→∞ tr(B|D 0 | 1/2 γ n |D 0 | 1/2 ) = tr(B|D 0 | 1/2 γ (p2) ∞ |D 0 | 1/2 ) .
Thus, for all B ∈ S q1 (H), we have

tr (B|D 0 | 1/2 (γ (p1) ∞ -γ (p2) ∞ )|D 0 | 1/2 ) = 0 .
This yields γ (p1)

∞ = γ (p2)
∞ . Therefore, in the other parts of this proof, we will always assume we have chosen a subsequence of γ n such that, for the considered p, the weak-limits in (F, . F,p ) exist and coincide.

Step 2. Lower Semi-Continuity: We will now prove that taking the limit decreases the energy, i.e. lim inf

n→∞ E m (γ n ) ≥ E m (γ ∞ ). We define Λ m := χ [0,m] (D (δ) ) and Λ ′ m := Λ (δ)
+ -Λ m and split the energy functional. We treat the various terms separately

E m (γ n ) =T 1 (γ n ) + T 2 (γ n ) + T 3 (γ n ) + T 4 (γ n ) + T 5 (γ n ) := tr (Λ ′ m (D (δ) -m)Λ ′ m γ n ) + tr (Λ m (D (δ) -m)Λ m γ n ) -α tr (ϕ (δ) γ n ) + α tr (X (δ) γ n ) + αQ(γ n , γ n ) .
Step 2.1. Fix a basis (e ℓ ) ℓ∈N of H + , each element being in H 1 (G). Then

T 1 (γ n ) := tr Λ ′ m (D (δ) -m)Λ ′ m γ n = tr (Λ ′ m (D (δ) -m)Λ ′ m ) 1/2 γ n (Λ ′ m (D (δ) -m)Λ ′ m ) 1/2 = k (e k , (Λ ′ m (D (δ) -m)Λ ′ m ) 1/2 γ n (Λ ′ m (D (δ) -m)Λ ′ m ) 1/2 e k f k := ) = k tr |D 0 | -1/2 f k |D 0 | -1/2 f k H k := |D 0 | 1/2 γ n |D 0 | 1/2 .
Obviously, H k is a non-negative Hilbert-Schmidt operator. Thus, applying first Fatou's lemma and then using (37), we get (38

) lim inf n→∞ tr Λ ′ m (D (δ) -m)Λ ′ m γ n = lim inf n→∞ k tr (H k |D 0 | 1/2 γ n |D 0 | 1/2 ) ≥ k lim inf n→∞ tr (H k |D 0 | 1/2 γ n |D 0 | 1/2 ) = k tr (H k |D 0 | 1/2 γ ∞ |D 0 | 1/2 ) = tr Λ ′ m (D (δ) -m)Λ ′ m γ ∞ , which proves lim inf n→∞ T 1 (γ n ) ≥ T 1 (γ ∞ ).
Step 2.2. Because T 2 is continuous in the • F,2 -norm (Lemma A.13) the claim follows for T 2 .

Step 2.3. Since ϕ (δ) ∈ L 4 (R 3 ) (Lemma A.7), we have, by [START_REF] Simon | Trace Ideals and their Applications[END_REF]Theorem 4.1], that |D 0 | -1/2 ϕ (δ) |D 0 | -1/2 ∈ S 4 (H). By Hölder inequality, this implies that T 3 is continuous in the • F,4/3 -norm, and lim n→+∞ T 3 (γ n ) = T 3 (γ).

Step 2.4. We would like to prove

(39) lim inf n→∞ Q[γ n , γ n ] ≥ Q[γ ∞ , γ ∞ ] .
To that end, we will first show

(40) lim n→∞ γ n (x, y) = γ ∞ (x, y) ,
for a.e. (x, y) ∈ G 2 , and

(41) lim n→∞ γ n (x, x) = γ ∞ (x, x) ,
for a.e. x ∈ G. Now, (γ n ) is a bounded sequence in S 2 (H). Again, we can extract a subsequence such that γ n converges weakly to γ∞ in S 2 (H). Using (37), we get for all B ∈ S 2 (H)

tr (Bγ ∞ ) = lim n→∞ tr (Bγ n ) = lim n→∞ tr |D 0 | -1 2 B|D 0 | -1 2 ∈S2(H) |D 0 | 1 2 γ n |D 0 | 1 2 = tr |D 0 | -1 2 B|D 0 | -1 2 |D 0 | 1 2 γ ∞ |D 0 | 1 2
= tr (Bγ ∞ ) .

Thus γ∞ = γ ∞ . In particular we have

(42) γ n (•, •) → γ ∞ (•, •) , weakly in L 2 (G × G).
Using the spectral decomposition of the γ n 's, we may write each γ n (x, y) as

γ n (x, y) = q i=1 u (n) i (x)u (n) i (y) ,
where each sequence (u

(n) i ) n∈N (i = 1, . . . , q) is an orthonormal family in ∈ H 1/2 (G)∩ Λ (δ)
+ H. Since the sequence (tr (|p|γ n )) n∈N is bounded, it follows that, for each i ∈ {1, . . . , q}, the sequence (u

(n) i ) n∈N is bounded in H 1/2 (G).
Therefore, applying [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF]Theorem 16.1], for all χ ∈ C ∞ 0 (R 3 ) and i ∈ {1, . . . , q}, there exists a subsequence of (u

(n) i ) n∈N -also denoted (u (n) i ) n∈N -such that (χu (n) i ) converges strongly in L 2 (G).
Thus, after extraction of a subsequence of u

(n) 1 , u (n) 2 , . . . , u (n) q n∈N
, denoted again by u

(n) 1 , u (n) 2 , . . . , u (n) q n∈N
, we obtain, for all i = 1, . . . , q and for almost

every x ∈ G, (43) u
(n) i (x) → u (∞) i (x) .
Consequently, we obtain (44)

γ n (x, y) → β(x, y) := q i=1 u (∞) i (x) u (∞) i (y) ,
almost everywhere in G 2 . Now from ( 42) and (44) it follows by standard arguments that γ ∞ (x, y) = β(x, y) almost everywhere in G×G. Thus γ n (x, y) converges almost everywhere to γ ∞ (x, y). This together with (44) implies (40).

The above also immediately implies that (45)

γ ∞ = q i=1 |u (∞) i u (∞) i | ,
proving (41). Applying Fatou's lemma to the pointwise positive functions (x, y) → γ n (x, x)γ n (y, y) -|γ n (x, y)| 2 and using in addition (40) and (41) yields

lim inf n→∞ Q[γ n , γ n ] ≥ lim inf n→∞ γ n (x, x)γ n (y, y) -|γ n (x, y)| 2 |x -y| dx dy = γ ∞ (x, x)γ ∞ (y, y) -|γ ∞ (x, y)| 2 |x -y| dx dy = Q[γ ∞ , γ ∞ ] ,
which proves (39).

Step 2.5. Since ϕ (δ) ∈ L 4 (R 3 ) (Lemma A.7) and X (δ) ≤ ϕ (δ) (by Lemma A.8 and the positivity of X (δ) and δ), we have

|D (δ) | -1/2 X (δ) |D (δ) | -1/2 ∈ S 4 . Thus T 5 is continuous in the • F,4/3 -norm (by Hölder inequality). Therefore lim n→+∞ T 5 (γ n ) = T 5 (γ ∞ ) .
This concludes the proof of Theorem 3.8. Theorem 3.9. Assume 0 ≤ δ ∈ F , q ∈ N, q ≤ Z and πα(1/4 + max{tr δ, q}) < (d -4α tr δ). Therefore, by Theorem 3.6, the functional E m is bounded below on S (δ) q . Lemmata 3.1, 3.3 and 3.4 together with Remark 3.5 imply that it suffices to minimize over positive γ's in S (δ) q that can be written as a sum of a projection and a rank one operator. Moreover, Inequality (46) and Lemma A.12 permit to apply Lemma 3.7, which, together with the above and the fact that q is an integer yields

Then the functional E| S

(47) inf{E m (γ)|γ ∈ S (δ) q } = inf{E m (γ)|γ ∈ S (δ)
∂q , γ 2 = γ}. Thus, applying Theorem 3.8 and again Lemma 3.7 shows that E m | S (δ) q has a minimizer in S In this section, we will prove that all minimizers γ fulfill a no-pair Dirac-Fock equation. We first need to state preliminary results. The first was already used in the adiabatic theory (see Nenciu [START_REF] Nenciu | asymptotic perturbation theory for quantum mechanics: almost invariant subspaces and gauge invariant magnetic perturbation theory[END_REF] and references therein). Lemma 4.1 (Nenciu). Given an orthogonal projection P 0 , its orthogonal complement P ⊥ 0 = 1 -P 0 , a bounded operator A, and ǫ ∈ R, with 4|ǫ| A < 1, there exists an operator B ǫ with B ǫ ≤ 4 A 2 such that (48)

P ǫ = P 0 + ǫ P 0 AP ⊥ 0 + P ⊥ 0 A * P 0 + ǫ 2 B ǫ is an orthogonal projection.
Proof. We set (49)

P ǫ := 1 2πi |z-1|= 1 2 1 z -P 0 -ǫa dz,
where a := P 0 AP ⊥ 0 + P ⊥ 0 A * P 0 . We observe that a ≤ A . Therefore under the assumption 4ǫ A < 1, we obtain that σ(P 0 + ǫa) ⊂ (-1/4, 1/4) ∪ (3/4, 5/4). Thus, by the holomorphic functional calculus, P ǫ is the projector onto the eigenspace of (P 0 + ǫa) corresponding to (3/4, 5/4).

P ǫ = P 0 + ǫ 2πi |z-1|= 1 2 1 z -P 0 a 1 z -P 0 dz - ǫ 2 2πi |z-1|= 1 2 1 z -P 0 a 1 z -P 0 -ǫa a 1 z -P 0 dz . ( 50
)
Since P 0 is an orthogonal projection, there exists a basis (e j ) j∈N of H and I ⊂ N such that

P 0 = j∈I |e j e j |.
Note also for later purpose that

(51) 1 z -P 0 = 1 z -1 j∈I |e j e j | + 1 z j∈N\I |e j e j | = 1 z -1 P 0 + 1 z P ⊥ 0 .
Using Cauchy's residue Theorem and (51), we obtain

1 2iπ |z-1|= 1 2 1 z -P 0 a 1 z -P 0 dz = 1 2iπ |z-1|= 1 2 P ⊥ 0 aP 0 + P 0 aP ⊥ 0 dz z(z -1) = P ⊥ 0 aP 0 + P 0 aP ⊥ 0 = a . ( 52
)
This proves that the second summand in the right hand side of ( 50) is equal to ǫa. This leads us to introduce :

(53)

B ε := - 1 2πi |z-1|= 1 2 1 z -P 0 a 1 z -P 0 -ǫa a 1 z -P 0 dz .
Since a ≤ A and σ(P 0 + ǫa) ⊂ (-1/4, 1/4) ∪ (3/4, 5/4), we have

|z-1|= 1 2 1 z -P 0 a 1 z -P 0 -ǫa a 1 z -P 0 dz ≤ π a 2 sup |z-1|= 1 2 1 z -P 0 2 sup |z-1|= 1 2 1 z -P 0 -ǫa ≤ 16π A 2 . (54)
This, together with (50) and ( 52), proves Lemma 4.1.

In the case when γ is an orthogonal projection with range in H + , we apply Lemma 4.1 with P 0 = γ, P ǫ = γ ǫ (given by ( 48)). Proof. By construction, since Λ

(δ) + AΛ (δ) + = A, we have Λ (δ) -γ ǫ Λ (δ) -= Λ (δ) + γ ǫ Λ (δ) -= Λ (δ) -γ ǫ Λ (δ) + = 0 . Moreover, γ 2 ǫ = γ ǫ , thus -Λ (δ) -≤ γ ǫ ≤ Λ (δ) + and Λ (δ) -γ ǫ Λ (δ)
+ = 0. The trace condition is obviously fulfilled since tr γ ǫ depends continuously on the parameter ǫ. That D 0 γ ǫ is trace class follows immediately from the explicit expressions for the difference γ ǫ -γ in (50), from (51) and the assumptions on A.

Minimizers are Projections.

Theorem 4.3. Assume that 0 ≤ δ ∈ F , q ∈ N, and

0 < πα(1/4 + max{tr δ, q}) < c α,Z,m,δ . If γ is a minimizer of E| S (δ) ∂q , then γ = γ * = γ 2 = Λ (δ) + γΛ (δ) + .
Proof. The proof of Λ

(δ) + D (γ) Λ (δ) + , i.e., (55) γ, Λ (δ)
+ D (γ) Λ (δ) + = 0. Proof. Let A ∈ B(H) such that (56) D (δ) A * ∈ B(H) .
Then, for ǫ sufficiently close to zero, the projector (57)

γ ǫ := γ + ǫa + ǫ 2 B ǫ , with (58) a = γΛ (δ)
+ AΛ (δ)

+ γ ⊥ + γ ⊥ Λ (δ) + A * Λ (δ)
+ γ , and B ǫ given by (53) (with P 0 replaced by γ), belongs to S (δ)

∂q . Moreover (59) E(γ ǫ ) -E(γ) = ǫ{tr (D Z a)+2αℜQ[γ, a]}+ǫ 2 {tr (D Z B ǫ )+2αℜQ[γ, B ǫ ]+αQ[a+ǫB ǫ , a+ǫB ǫ ]}.
We want to show that the last term is o(ǫ). By ( 13), ( 14) and Lemmata A.7 and A.8, it is sufficient to show that there exists a constant c < ∞ such that, for all ǫ ∈ (-1, 1),

(60) max{ B ǫ 1 , a 1 , D (δ) B ǫ 1 , D (δ) a 1 } < c .
We first have

D (δ) a 1 ≤ D (δ) γΛ (δ) + AΛ (δ) + γ ⊥ 1 + D (δ) γ ⊥ Λ (δ) + A * Λ (δ) + γ 1 ≤ D (δ) γ 1 A + D (δ) γ A * γ 1 + D (δ) A * γ 1 < c , (61)
since γ ∈ F and (56) is assumed. We also have

D (δ) B ǫ 1 = 1 2π |z-1|= 1 2 D (δ) 1 z -γ a 1 z -γ -ǫa a 1 z -γ dz 1 ≤ 1 2π 2π 0 D (δ) 1 1 + 1 2 e iϕ -γ a 1 1 1 + 1 2 e iϕ -γ -ǫa a 1 1 + 1 2 e iϕ -γ dϕ ≤ c 1 ||a|| 2π 0 D (δ) 1 1 + 1 2 e iϕ -1 γ + 1 1 + 1 2 e iϕ (1 -γ) a 1 dϕ ≤ c 2 ||a|| D (δ) γ 1 a + D (δ) a 1 ,
where the constant c 2 is uniform in ǫ for ǫ close to zero. We also have used (51). Using (61), and γ ∈ F yields (62)

D (δ) B ǫ < c .
Similarly to the above, we show (63) a 1 < c and B ǫ 1 < c .

Inequalities (61), ( 62) and ( 63) yield (60). Since E(γ ǫ ) -E(γ) ≥ 0, whatever the sign of ǫ is, we conclude that the term linear in ǫ in (59) has to vanish

(64) tr (D Z a) + 2αℜQ[γ, a] = tr (D (γ) a) = 0 .
Thus, for all A satisfying (56), equalities (58) and ( 64) and the fact that [Λ

(δ) + , γ] = 0 (since γ is an orthonormal projection in S (δ) ∂q ) yield tr (γ ⊥ Λ (δ) + D (γ) Λ (δ) + γA) + tr (γΛ (δ) + D (γ) Λ (δ) + γ ⊥ A * ) = 0 .
Replacing A by iA, we first obtain :

tr (γ ⊥ Λ (δ) + D (γ) Λ (δ) + γA) = tr (γΛ (δ) + D (γ) Λ (δ) + γ ⊥ A * ) = 0 .
Since A can be taken in the set of rank one operators of the form |u v|, with u and v in

C ∞ 0 (R 3 ) ⊗ C 4 , we obtain γ ⊥ Λ (δ) + D (γ) Λ (δ) + γ = γΛ (δ) + D (γ) Λ (δ)
+ γ ⊥ = 0 , which yields (55). This result can be also written as follows Theorem 4.5. Assume that γ is an orthogonal projection minimizing E in S (δ) ∂q . Then there exist q normalized spinors ψ 1 , ..., ψ q ∈ Λ (δ)

+ (H) ∩ D(D Z ) such that (65) γ = q i=1 |ψ i ψ i | , and (66) Λ (δ)
+ D (γ) Λ (δ) + ψ i = ǫ i ψ i , for i = 1, . . . , q , with ǫ 1 , ..., ǫ q ∈ [0, 1].
Proof. The proof is immediate since the range of γ is finite dimensional reducing it to the simultaneous diagonalization of two commuting Hermitian matrices.

One may characterize the eigenvalues ǫ 1 , . . . , ǫ q more precisely following Bach et al [START_REF] Bach | Generalized Hartree-Fock theory and the Hubbard model[END_REF]: Theorem 4.6. [There are no unfilled shells in no-pair Dirac-Fock theory] Under the same assumptions of Theorem 4.5, ǫ 1 , ..., ǫ q are the q lowest eigenvalues of

Λ (δ) + D (δ) Λ (δ)
+ . Moreover, if ǫ q+1 denotes the (q + 1)-th eigenvalue (counting multiplicities) of the no-pair Dirac-Fock operator

Λ (δ) + D (γ) Λ (δ) + , we have, for all i = 1, • • • , q, the strict inequality ǫ i < ǫ q+1 .
Proof. Assume by contradiction that there exists a normalized eigenspinor v of Λ

(δ) + D (γ) Λ (δ)
+ with eigenvalue ǫ such that ǫ ≤ max{ǫ 1 , ..., ǫ q } and not in the range of γ. Then, for a normalized eigenvector u of γ with u, Λ

+ D (γ) Λ (δ) + u ≥ ǫ and for γ ′ := γ -|u u| + |v v|, we have E (δ) (γ ′ ) -E (δ) (γ) = v, D Z v -u, D Z u + 2αℜQ [γ, |v v| -|u u|] +Q [|v v| -|u u|, |v v| -|u u|] < v, D (γ) v -u, D (γ) u ≤ 0, (δ)
where -as in the proof of Lemma 3.4 -we used (36), with S = |v v| -|u u|. This gives a contradiction to the property that γ is the minimizer.

Outlook

In this final section, we first express the energy as a functional of the unrenormalized density matrix Γ and the Dirac sea Λ -. This has the advantage that the dependence of the energy on Λ -becomes explicit and that the constraining condition -Λ -≤ γ ≤ Λ + becomes 0 ≤ Γ ≤ 1, i.e. independent of Λ -. Throughout this section we will use the notation Λ := Λ -.

For a given q ∈ N, let us define the set

Υ q := (Γ, Λ) ∈ B(H) 2 | Γ, Λ orth. proj., (Γ -Λ) ∈ S 1 , tr (Γ -Λ) = q, D Z (Γ -Λ) ∈ S 1 , [D Z , Λ] ∈ B(H), [D Z , Γ] ∈ B(H)
and the following functional on Υ q :

E(Γ, Λ) := tr ((D Z -m)(Γ -Λ)) + αQ[Γ -Λ, Γ -Λ] ,
where

Q[•, •] is defined in (24). Note that if Λ = Λ (δ)
-for some δ ∈ F , and if (Γ -Λ) ∈ S (δ) , then we have E(Γ, Λ) = E(Γ -Λ).

Theorem 5.1. Assume that (Γ, Λ) ∈ Υ q is a critical point of E. Then, with γ := Γ -Λ, (67) [D (γ) , Γ] = [D (γ) , Λ] = 0.

Proof. For all ǫ ∈ R and A ∈ S 1 (H) such that : (68)

D Z A ∈ S 1 (H) , D Z A * ∈ S 1 (H) , 4|ǫ| A < 1 , 8qǫ 2 A 2 < 1 and ǫ 2 A 1 < 1 ,
we define (see Lemma 4.1) the orthogonal projector :

(69) Λ ǫ = 1 2πi |z-1|= 1 2 1 z -Λ -ǫa dz ,
where a = ΛAΛ ⊥ + Λ ⊥ A * Λ. From Lemma 4.1, we get the decomposition

(70) Λ ǫ = Λ + ǫa + ǫ 2 B ǫ ,
with P 0 replaced by Λ.

Let us prove that (Γ, Λ ǫ ) belongs to Υ q . We first show Γ -Λ ǫ ∈ S 1 (H). We have

(71) a 1 ≤ 2 A 1 < ∞ ,
and, as in the proof of (54), we get

(72) B ǫ 1 ≤ 16π A A 1 . Therefore, Γ -Λ ǫ = Γ -Λ + ǫa + ǫ 2 B ǫ ∈ S 1 (H). We next establish D Z Λ ǫ ∈ S 1 (H). Since (Γ, Λ) ∈ Υ q , [D Z , Λ] is a bounded operator and A, D Z A ∈ S 1 (H). Thus D Z ΛAΛ ⊥ 1 ≤ ΛD Z AΛ ⊥ 1 + [ D Z , Λ ]AΛ ⊥ 1 ≤ D Z A 1 + [ D Z , Λ ] A 1 < ∞ . (73)
Similarly, we can prove (74)

D Z Λ ⊥ A * Λ 1 ≤ D Z A * 1 + [ D Z , Λ ⊥ ]
A * 1 < ∞ , which implies, together with (73) that there exists a constant c such that (75) D Z a 1 < c .

Using again that [ D Z , Λ ] is bounded and Formula (51), valid with Λ instead of P 0 and Λ ⊥ instead of P ⊥ 0 , we get the existence of a constant c such that for all ǫ small enough (76)

D Z B ǫ 1 < c .
Inequalities (75) and (76) yield D Z Λ ǫ ∈ S 1 (H). Now, from (75), ( 76) and

[D Z , Λ ǫ ] = [D Z , Λ] + [D Z , ǫa + ǫ 2 B ǫ ], we obtain [ D Z , Λ ǫ ] ∈ B(H).
We finally prove that tr (Γ -Λ ǫ ) = q. For that purpose, we first note that, due to Effros [START_REF] Edward | Why the circle is connected: an introduction to quantized topology[END_REF] (see also Avron et al [START_REF] Avron | The index of a pair of projections[END_REF]Theorem 4.1]), and since from the above Γ -Λ ǫ ∈ S 1 (H), and both Γ and Λ ǫ are projections, we have tr (Γ -Λ ǫ ) ∈ Z. Furthermore, from (70), ( 71) and (72) we get

(Γ -Λ) -(Γ -Λ ǫ ) 1 = O(ǫ) .
Since tr (Γ -Λ ǫ ) is an integer and tr (Γ -Λ) = q, this yields, for ǫ small enough, tr (Γ -Λ ǫ ) = q. This concludes the proof that (Γ, Λ ǫ ) ∈ Υ q .

Let us now prove that for γ := Γ -Λ we have (77) [D (γ) , Λ] = 0 .

Since (Γ, Λ) is a critical point of E, for all A ∈ S 1 (H) satisfying (68), we have

(78) ∂E(Γ, Λ ǫ ) ∂ǫ ǫ=0 = 0 ,
where Λ ǫ is defined by (70). On the other hand, we have (79)

E(Γ, Λ ǫ ) = tr ((D Z -m)γ)+αQ[γ, γ] -ǫ tr ((D Z -m)a) + 2αℜQ[a, γ] + ǫ 2 tr ((D Z -m)B ǫ ) + αQ[a + ǫB ǫ , a + ǫB ǫ ] -2αℜQ[γ, B ǫ ] .
Inequalities (72) and (76) imply that there exists a constant c 1 such that for all ǫ small enough (80)

|tr ((D Z -m)B ǫ )| < c 1 .
Furthermore, using Lemma A.1, we have

tr (|D 0 | 1/2 |a + ǫB ǫ | |D 0 | 1/2 ) = tr (|a + ǫB ǫ | 1/2 |D 0 ||a + ǫB ǫ | 1/2 ) ≤ 1 d tr (|a + ǫB ǫ | 1/2 |D Z ||a + ǫB ǫ | 1/2 ) ≤ 1 d |D Z | |a + ǫB ǫ | 1 = 1 d D Z (a + ǫB ǫ ) 1 ,
which yields, together with (75) and ( 76), a + ǫB ǫ ∈ F . Thus Lemma 2.2 implies

|Q[a + ǫB ǫ , a + ǫB ǫ ]| ≤ 2D[a + ǫB ǫ , a + ǫB ǫ ] ≤ π 2 a + ǫB ǫ 1 tr (|D 0 ||a + ǫB ǫ |) ≤ π 2d a + ǫB ǫ 1 tr (|D Z ||a + ǫB ǫ |) ≤ π 2d a + ǫB ǫ 1 D Z (a + ǫB ǫ ) 1 (81)
According to (75) and (76), we conclude from (81) that there exists a constant c 2 such that for all ǫ small enough (82)

|Q[a + ǫB ǫ , a + ǫB ǫ ]| < c 2 .
Now we prove that there exists c 3 such that for all ǫ small enough (83)

|Q[γ, B ǫ ]| < c 3 .
We have, for W (Bǫ) being the mean field potential associated with B ǫ ∈ F , as defined in ( 4)

|Q[γ, B ǫ ]| = | tr (W (Bǫ) γ)| . Moreover | tr (W (Bǫ) γ)| ≤ W (Bǫ) γ 1 ≤ W (Bǫ) (D 0 ) -1 D 0 (D Z ) -1 D Z γ 1 . (84) Using Lemmata A.3 and A.4 with B ǫ instead of δ implies (85) W (Bǫ) (D 0 ) -1 ≤ 4 B ǫ 1 .
Since D 0 (D Z ) -1 is bounded, by using Inequalities (84), (85) and the fact that (Γ, Λ) ∈ Υ q , we obtain (83). Collecting (80), (82), and (83) yields, together with (79),

E(Γ, Λ ǫ ) = tr ((D Z -m)γ) +αQ[γ, γ]-ǫ tr ((D Z -m)a) + 2αℜQ[a, γ] + O(ǫ 2 ) = E(Γ, Λ) -ǫ tr ((D (γ) -m)a) + O(ǫ 2 ) = E(Γ, Λ) -ǫ tr (Λ ⊥ (D (γ) -m)ΛA) -ǫ tr (Λ(D (γ) -m)Λ ⊥ A * ) + O(ǫ 2 ) .
For A fixed as above, this result remains true for all ǫ small enough. Therefore, (78) implies tr Λ ⊥ (D (γ) -m)ΛA + tr Λ(D (γ) -m)Λ ⊥ A * = 0 .

As at the end of the proof of Theorem 4.4, we obtain [D (γ) , Λ] = 0. Finally, exchanging the roles of Γ and Λ in the above proof yields [D (γ) , Γ].

Appendix A. Some Spectral Properties of Screened Coulomb-Dirac Operators

We recall the following result:

Lemma A.1 (Brummelhuis et al [START_REF] Brummelhuis | Stability of the relativistic electron-positron field of atoms in Hartree-Fock approximation: Heavy elements[END_REF]). Let

d := (1/3)(1 -(αZ) 2 ) 1/2 ((4(αZ) 2 + 9) 1/2 -4αZ) and assume 0 ≤ αZ < √ 3/2. Then (86) |D Z | 2 ≥ d 2 |D 0 | 2 and |D Z | ≥ d|D 0 |.
We would like to compare |D 0 | and |D (δ) |.

Definition A.2. Given positive α, Z, and m and δ ∈ F , we define

(87) c α,Z,m,δ := sup{c ∈ R | |D (δ) | ≥ c|D 0 |} . Lemma A.3. If δ ∈ F , then, for all u ∈ H 1 (R 3 ) ⊗ C 4 , we have X (δ) u ≤ 2 δ 1 ∇u .
Proof. Using the spectral decomposition of δ, we have

X (δ) u 2 = δ(x, z) δ(x, y) u(z)u(y) |x -z||x -y| dx dy dz = i, j λ i λ j ξ i (x)ξ i (z) ξ j (x)ξ j (y)u(z)u(y) |x -z||x -y| dx dy dz = i, j λ i λ j ξ i (x)ξ j (x) ξ j (y)u(y) |x -y| dy ξ i (z)u(z) |x -z| dz dx ≤ i, j |λ i | |λ j | |ξ i (x)| |ξ j (x)| |ξ j (y)| 2 dy |u(y)| 2 |x -y| 2 dy dx ≤4 i, j |λ i | |λ j | |ξ i (x)| |ξ j (x)| ∇u 2 dx ≤4 δ 2 1 ∇u 2 ,
where we used the Cauchy-Schwarz inequality, the Hardy inequality, and the iden-

tity i |λ i | = ||δ|| 1 .
Similar estimates can be found for the direct part,

Lemma A.4. If δ ∈ F , then ϕ (δ) u ≤ 2 δ 1 ∇u , for all u ∈ H 1 (R 3 ) ⊗ C 4 .
Proof. We have successively

ϕ (δ) u 2 = |u(y)| 2 δ(x, x)δ(z, z) |x -y| |z -y| dxdydz = δ(x, x)δ(z, z) |u(y)| 2 |x -y| |z -y| dy dxdz ≤ 1 2 |δ(x, x)| |δ(z, z)| |u(y)| 2 |x -y| 2 dy + |u(y)| 2 |z -y| 2 dy dxdz ≤ 4 |δ(x, x)|dx |δ(z, z)|dz ∇u 2 ≤ 4 δ 2 1 ∇u 2 , (88)
where we used Hardy's inequality in (88).

A direct consequence of Lemma A.4 and the fact that square root is operator monotone is Proof. This is a direct consequence of Lemmata A.3 and A.4, since we have for all u in D(D (δ) )

Lemma A.5. If δ ∈ F , then |ϕ (δ) | ≤ 2 δ 1 |D 0 | .
D (δ) u = D Z + αϕ (δ) -αX (δ) u ≥ D Z u -α ϕ (δ) u -α X (δ) u ≥ (d -4α δ 1 ) D 0 u . (90)
Therefore

|D (δ) | 2 ≥ (d -4α δ 1 ) 2 |D 0 | 2 ,
which concludes the proof since the square root is operator monotone.

Lemma A.7. Assume δ ∈ F and ǫ > 0, then ϕ (δ) , ϕ (|δ|) ∈ L 3+ǫ (R 3 ) ∩ L ∞ (R 3 ).

Proof. As before we denote by (λ n ) n∈N the eigenvalues of δ and by (ξ n ) a corresponding orthonormal basis of eigenfunctions. Since |ϕ (δ) |(x)| ≤ ϕ (|δ|) (x), it is sufficient to prove the result for ϕ (|δ|) . We first prove that ϕ (|δ|) ∈ L 3+ǫ (R 3 ). We write χ R for the characteristic function of the ball with center 0 and radius R, and set χ R := 1 -χ R . We get

|ξ n | 2 * | • | -1 3+ǫ ≤ |ξ n | 2 3 2 χ R | • | -1 3/(2+ǫ) + χ R | • | -1 3+ǫ ≤ c[ ξ n , |p| ξ n + 1] ,
where we used the Hausdorff-Young inequality and the Sobolev inequality. Multiplication by |λ n |, summation over n, and the triangular inequality yields the result.

ϕ (|δ|) 3+ǫ ≤ c n |λ n |[ ξ n , |∇| ξ n + 1] ≤ c δ F .
Next we estimate ϕ (|δ|) ∞ . Using Kato's inequality, we get

|ϕ (|δ|) (x)| ≤ n |λ n | dy |ξ n (y)| 2 /|x -y| ≤ π 2 n |λ n | ξ n , |p| ξ n ≤ π 2 δ F . Lemma A.8. If δ ∈ F , then |X (δ) | ≤ ϕ (|δ|) .
Proof. This is a straightforward consequence of the spectral decomposition of δ and the Cauchy-Schwarz inequality.

Lemma A.9. If δ ∈ F , then W (δ) is relatively compact with respect to D 0 .

Straightforward calculations using scaling arguments and the specific form of ψ R give

D 0 - αZ |x| + αϕ (δ) -αX (δ) ψ R 2 = α • 1 i ∇ψ R 2 + mβψ R 2 + - αZ |x| + αϕ (δ) -αX (δ) ψ R 2 + 2ℜ mβψ R , α • 1 i ∇ψ R + 2ℜ - αZ |x| + αϕ (δ) -αX (δ) ψ R , α • 1 i ∇ψ R + 2ℜ mβψ R , - αZ |x| + αϕ (δ) -αX (δ) ψ R = ∇ψ R 2 + m 2 + α 2 - Z |x| + ϕ (δ) -X (δ) ψ R 2 + 2mα ψ R , - Z |x| + ϕ (δ) -X (δ) ψ R ≤ 1 R 2 ∇f 2 + m 2 + 2α 2 R 2 Z |x| f 2 + 8(q ′ ) 2 ∇f 2 + 2m ψ R , - αZ |x| + αϕ (δ) -αX (δ) ψ R ≤ m 2 + c 1 R 2 + 2m ψ R , - αZ |x| + αϕ (δ) -αX (δ) ψ R , (93)
where c 1 is a constant independent of R. Note that we have used Lemmata A.3 and A.4 for getting an upper bound of (φ (δ) -X (δ) )ψ R 2 . Now, since f is spherically symmetric, using Riesz's rearrangement inequality and Newton's Theorem, we obtain Note also that ψ R , X (δ) ψ R ≥ 0 and ψ R ,

1 |x| ψ R = 1 R f, 1 |x| f = c 2 R ,
with c 2 := f, 1 |x| f > 0. Thus, we have

ψ R , - Z |x| + ϕ (δ) -X (δ) ψ R ≤ - c 2 (Z -q ′ ) R . (94)
Since -(d -4αq ′ ) + 2αq < 0, we have .

Since from Theorem A.11 the operator D (γ) has infinitely many eigenvalues in (0, m), using (95), we finally get the expected result.

If we have a non-negative spherical symmetric density ρ with q := ρ(y)dy ≤ Z ,

then -Z/| • | + ρ * | • | -1 ≥ 0 .
This implies that the n-th eigenvalue of D Z + ϕ can be estimated from below by the n-th eigenvalue of D Z-q . For a non-spherical symmetric potential this situation for the positive eigenvalues is disturbed only slightly.

Lemma A.13. Assume 0 ≤ δ ∈ F . Then χ (0,m) (D (δ) )(D (δ) -m) is a Hilbert-Schmidt operator.

Proof. Since δ ∈ F , by writing W (δ) D -1 Z = W (δ) D -1 0 D 0 D -1 z , Lemmata A.1 and A.9 imply that W (δ) is relatively compact with respect to D Z . Moreover, since δ ≥ 0, we have W (δ) ≥ 0 (see lemma A.8).

Let λ 0 (0) ≤ λ 1 (0) ≤ . . . be the ordered positive eigenvalues of D Z , including multiplicity. We first prove that for all ǫ ∈ [0, 1], there exist N + (ǫ) and N - Using Kato's perturbation theory for isolated eigenvalues and numbering the eigenvalues that are crossing with respect to their ordering -namely the largest after crossing gets the highest index -, yields i) and ii). Continuity is also a consequence of perturbation theory for eigenvalues. The asserted monotonicity is a consequence of positivity of W (δ) .

We next show that the number M of eigenvalues λ k (ǫ) that crosses zero when ǫ increases from 0 to 1 is finite. By the Birman-Schwinger principle,

M = #{λ ∈ [1, ∞) | λ eigenvalue of R} ,

Finally

  , since |D (δ) | ≥ c α,Z,m,δ |D 0 | by definition of c α,Z,m,δ (Equation (87)) and since |D 0 | > 0, the operator D (δ) has a bounded inverse, as soon as c α,Z,m,δ > 0.

±

  leave the domain of A invariant).

3 . 3 .

 33 Reduction to Projection. Following an argument of Bach[START_REF] Bach | Error bound for the Hartree-Fock energy of atoms and molecules[END_REF], we getLemma 3.4. Assume q ∈ N, δ ∈ F , 0 ≤ γ ∈ S (δ)∂q with finite rank. Then there exists a projection P ∈ S (δ) ∂q such that (33) E(P ) ≤ E(γ) .

3. 5 .

 5 Reduction to Density Matrices of Maximal Charge.

∞

  . Then for all B ∈ S q1 (H) we have B ∈ S q2 (H) and tr

  Definition (23) and (19)) has a minimizer. Proof. Using lemma A.6 and the assumptions, we obtain (46) πα( 1 4 + max{tr δ, q}) < c α,Z,m,δ .

4 .

 4 ∂q . Therefore the functional E| S (δ) ∂q also has a minimizer. Properties of the Minimizers: No-Pair Dirac-Fock Equations 4.1. Infinitesimal Perturbations of Projections.

Lemma 4 . 2 .

 42 Assume 0 ≤ δ ∈ F and γ an orthogonal projection in S (δ) ∂q , q ∈ N. Then, for operators A such that Λ (δ) + AΛ (δ) + = A, |D 0 |A ∈ S 1 (H) and ǫ sufficiently close to zero, γ ǫ is again an orthogonal projection in S (δ) ∂q .

+

  = γ is a consequence of Lemma 3.1. The proof of γ 2 = γ follows exactly the lines of Lemma 3.4 except that the iteration of the process is superfluous here. 4.3. Minimizers Fulfill the No-Pair Dirac-Fock Equations. Eventually we derive the Euler equations for the minimizer of the energy.

Theorem 4 . 4 .

 44 Assume q ∈ N 0 and γ is an orthogonal projection minimizing E in S (δ) ∂q . Then γ commutes with the no-pair Dirac-Fock operator Λ

Lemma A. 6 . 3 / 2 ,

 632 If δ ∈ F and αZ ≤ √ the following operator inequality holds (89) c α,Z,m,δ ≥ (d -4α δ 1 ) .

ψ

  R , ϕ (δ) ψ R = |f R (x)| 2 ρ δ (y) |x -y| dx dy ≤ |f R (x)| 2 ρ * δ (y) |x -y| dx dy ≤ |f R (x)| 2 |x| dx ρ * δ (y) dy ≤ q ′ |f R (x)| 2 |x| dx ,where we used, since ρ δ (y) =4 s=1 δ(y, y) is positive, that ρ * δ (y) dy = ρ * δ 1 = ρ δ 1 = q ′ .

-

  ≤ 0 . Now, thanks to (97), one can apply[START_REF] Griesemer | A minimax principle for the eigenvalues in spectral gaps[END_REF] Theorem 3]. With their notations, sinceD(D (δ) ) = D(D (γ) ), we first define Q ± := D(D (γ) ) ∩ Λ (δ)± H. Then, denoting by µ n (A) the n th lowest eigenvalue of the operator A, we obtainm > µ n D (γ) | Λ+ (γ) n D (γ) | Λ (γ) + H ≥ µ n Λ

  (ǫ) in {-∞} ∪ Z ∪ {+∞}, with N -(ǫ) ≤ N + (ǫ), and (λ k (ǫ)) k∈{N-(ǫ),...,N+(ǫ)} such that i) σ D Z + ǫαW (δ) ∩ (-m, m) = {λ N-(ǫ) (ǫ), λ N-(ǫ)+1 (ǫ), . . . , λ N+(ǫ) (ǫ)} ii) λ N-(ǫ) (ǫ) ≤ λ N-(ǫ)+1 (ǫ) ≤ . . . ≤ λ N+(ǫ) (ǫ)iii) For all k ∈ {N -(ǫ), . . . , N + (ǫ)} the functions λ k (ǫ) are continuous, monotone increasing.

Proof. Since ϕ (δ) ∈ L 4 (R 3 ) by Lemma A.7, using an inequality of Seiler and Simon [START_REF] Simon | Trace Ideals and their Applications[END_REF]Theorem 4.1], we have (91)

0 ϕ (δ) ∈ S ∞ . We next prove that X (δ) is relatively compact with respect to D 0 . Let us denote by δ + (respectively δ -) the positive part (respectively the negative part) of δ: δ = δ + -δ -, δ ± ≥ 0.

We have, using δ ± (x, y) = δ ± (y, x), u, (X (δ±) ) 2 u = u(x) δ ± (x, y)δ ± (y, z) |x -y| |y -z| u(z)dydz dx

where G(x, z) := δ ± (x, y)δ ± (y, z) |x -y| |y -z| dy |u(x)| 2 and h is positive and measurable.

Applying the Cauchy-Schwarz inequality and

Now, similarly to (91), we have ϕ (δ±) (-∆ + m 2 ) -1/2 4 < ∞. Thus, since D -1 0 is bounded, we obtain D -1 0 ϕ (δ±) D -1 0 ∈ S 4 (H). Moreover, by Lemma A.7 we have ϕ (δ±) ∞ < ∞. Therefore, using (92), we get D -1 0 (X (δ±) ) 2 D -1 0 ∈ S 4 (H) and thus X (δ±) D -1 0 ∈ S 8 (H) ⊂ S ∞ (H), which implies X (δ) D -1 0 compact. Using this, and the fact that ϕ (δ) is relatively D 0 compact conclude the proof. Lemma A.9, the Kato-Rellich Theorem, and Weyl's theorem imply Lemma A.10. Let Z > 0 such that αZ ≤ 1 and let δ ∈ F . Then the operator

If δ is a positive density matrix of q electrons with q < Z + 1, then the potential of the nucleus prevails giving an attractive Coulomb tail at infinity. This leads us to expect that D (δ) has infinitely many bound states in the gap accumulating at m. The following theorem expresses this expectation formally.

Theorem A.11. Assume 0 ≤ δ ∈ F and tr δ < Z. Then the operator D (δ) has infinitely many eigenvalues in (0, m) accumulating at the point m.

), which is also spherically symmetric. We also define, for all R > 1, the functions

Now (93) and (94) imply

Therefore, if R is large enough, we get the inequality

which implies, by taking E = m/2, that D (δ) has at least one eigenvalue λ 1 in (0, m). Now by taking E = (m + λ 1 )/2, by the same argument as above, one gets a second eigenvalue λ 2 ∈ (λ 1 , m). The iteration of this procedure yields an infinite sequence of eigenvalues (λ n ) of D (δ) in (0, m) tending to m.

A similar result holds for the no-pair Dirac-Fock operator.

Lemma A.12. Assume that δ and γ are two positive definite finite rank density matrices. Assume, in addition, that γ is purely electronic having particle number not exceeding Z, i.e., we have

+ has infinitely many eigenvalues in (0, m).

Proof. We first prove that (95) σ ess (Λ

0 and W (γ) D -1 0 are compact. By assumption, we have (d -4α δ 1 ) > 0; thus (90) implies that D 0 (D (δ) ) -1 is bounded. This yields

Together with Lemma A.10, this proves (95).

Set q ′ := tr δ and q := tr γ. We denote by ϕ (γ) and X (γ) respectively the direct and exchange operators associated to γ, defined by replacing δ with γ in [START_REF] Bjorken | Relativistic Quantum Fields. International Series in Pure and Applied Physics[END_REF] and [START_REF] Brummelhuis | Stability of the relativistic electron-positron field of atoms in Hartree-Fock approximation: Heavy elements[END_REF]. For all u ∈ Λ (δ) -

where, in Inequality (96), we used Lemma A.8 and the fact that X (γ) ≥ 0. Now, from Lemmata A.5 and A.6 we obtain

is the Birman-Schwinger kernel with energy 0. Thus, since we count only the eigenvalues of R larger than 1, we have, using the notation W := W (δ) and ϕ := ϕ (δ) ,

Using (86) and [20, Theorem 4.1] as for Inequality (91) in the proof of Lemma A.9 yields

Collecting (98)-(101) proves M < ∞. Therefore, apart from the eigenvalues λ 0 (1), λ 1 (1), . . ., the operator D (δ) has only finitely many other eigenvalues in [0, m]. Thus, χ (0,m) (D (δ) )(D (δ) -m) ∈ S 2 (H) follows if the series k≥0 (λ k (1) -m) 2 ≤ k≥0 (λ k (0) -m) 2 is convergent. At this point we remind the reader that the relativistic hydrogen eigenvalues λ k (0) can be grouped into "multiplets" of 2n 2 eigenvalues corresponding to one non-relativistic eigenvalue m -Z 2 α 2 /n 2 . Each element of such a multiplet can be bounded from below by the previous non-relativistic eigenvalue m -Z 2 α 2 /(n -1) 2 . Thus, up to an unessential multiplicative constant, n≥2 n 2 /(n -1) 4 is a convergent majorant. This proves the claim.