Existence of Infinitely Many Distinct Solutions to the Quasirelativistic Hartree-Fock Equations

equations forN-electron Coulomb systems with quasirelativistic kinetic energy √ −α−2Δxn α−4 − α−2 for the nth electron. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge Ztot of K nuclei is greater than N − 1 and that Ztot is smaller than a critical charge Zc. The proofs are based on a new application of the Fang-Ghoussoub critical point approach to multiple solutions on a noncompact Riemannian manifold, in combination with density operator techniques.


Introduction
In the present paper we prove existence of infinitely many solutions to the quasirelativistic Hartree-Fock equations associated with N electrons interacting with K static nuclei with charges Z 1 , . . ., Z K , where Z k > 0. The nonlinear coupled equations arise as the Euler-Lagrange equations of the total energy functional E • defined as the quantum energy restricted to antisymmetric Slater determinants see Section 3 constructed from L 2 -orthonormal functions {φ n } N n 1 belonging to the Sobolev space H 1/2 R 3 .Above T 0 −α −2 Δ x n α −4 − α −2 is the quasirelativistic kinetic energy of the nth electron located at x n ∈ R 3 Δ x n being the Laplacian with respect to x n , α is Sommerfeld's fine structure constant, V en is the attractive interaction between an International Journal of Mathematics and Mathematical Sciences electron and the nuclei, ρ N n 1 |φ n | 2 is the density, and K xc is the exchange operator defined in 4.1 below.For the nonrelativistic setting, a review on classical results on existence of a ground state and its properties is found in Lions 1 .In the latter paper, Lions studied both minimal and nonminimal excited states solutions to the equations by using critical point theory in conjunction with Morse data.Lions' idea is to construct convenient min-max levels which yield the desired solutions through abstract critical point theory.For the nonrelativistic HF model, Lions verifies a Palais-Smale compactness condition which, roughly speaking, amounts to "being away from the continuous spectrum" or, equivalently when the so-called Morse information is taken into account , showing that certain Schr ödinger operators with Coulomb type potentials have enough negative eigenvalues.
The novelty of the present paper is Theorem 7.1, wherein we establish the following results for the quasirelativistic Hartree-Fock equations.1 A ground state exists provided the total charge Z tot of K nuclei is greater than N − 1 and Z tot is smaller than a critical charge Z c to be defined below; 2 under the same assumptions, infinitely many distinct solutions to the quasirelativistic Hartree-Fock equations exist; we refer to the theorem for the full statement.We proceed to sketch the proof of Theorem 7.1, starting with the existence of a ground state.We consider the C 2 -functional E on a Hilbert manifold C N defined in 3.7 .Since E is bounded from below, we may try to find a critical point at the level l inf C N E by determining whether the infimum is achieved.As we will see, it is easy to find an almost critical sequence at the level l, that is, a sequence {h j } in C N satisfying lim The hard part is to prove existence of a converging subsequence of {h j }.Unfortunately, roughly speaking due to ionization, the energy functional will not satisfy a Palais-Smale condition at level l.To make sure that we can extract a convergent subsequence, we use second-order information of E.
In the process of implementing these ideas we have to overcome additional technicalities for the quasirelativistic setting compared to the nonrelativistic, for instance, the Coulomb potential is not relatively compact in the operator sense with respect to the quasirelativistic energy operator.In particular, compact Sobolev imbeddings are not available for a recent survey of such problems, we refer to Bartsch et al. 2 .To overcome this problem, it is necessary to switch to a density operator formalism, as pioneered by Solovej 3 , and use that for an enlarged set of admissible density operators, one can, at least for the certain sequences, establish the inequality 6.20 below.A different proof for existence of a ground state was given by Dall'Acqua et al. 4 .Moreover, regularity of the ground state away from the nucleus and pointwise exponential decay of the orbitals were established therein.
In the opposite direction, Lieb 5 has proved that for N ≥ 2Z tot K there never exists a quasirelativistic Hartree-Fock ground state see Enstedt and Melgaard 6 for an analogous result .For the nonrelativistic setting, Solovej has improved Lieb's result by proving that there exists a universal constant Q > 0 such that N ≥ Z Q ensures that there are no minimizers 7 and Lewin 8 has applied Lions' approach to the nonrelativistic MCSCF equations.For further references, we refer to the survey by Le Bris and Lions 9, Section 3.1.6 .
We invoke a direct method developed by Fang and Ghoussoub 10,11 to address the existence of infinitely many nonminimal solutions.Since we are looking for nonminimal or unstable critical points, we consider a collection H of compact subsets of C N which is stable under a specific class of homotopies and then we show that E has a critical point at the level As we will see, the method by Fang and Ghoussoub gives us an almost critical sequence at the level l, that is, a sequence {h j } in C N satisfying 1.2 , with additional Morse information as mentioned above which is crucial for proving that the sequence is convergent.
Work related to our study of semilinear elliptic equations and critical point theory includes existence of solutions with finite Morse indices established by Dancer

Preliminaries
Throughout the paper we denote by c and C with or without indices various positive constants whose precise value is of no importance.Moreover, we will denote the complex conjugate of z ∈ C by z.

Function Spaces
For 1 ≤ p ≤ ∞, let L p R 3 be the space of equivalence classes of complex-valued functions φ which are measurable and satisfy The measure dx is the Lebesgue measure.For any p the L p R 3 space is a Banach space with norm In the case p 2, L 2 R 3 is a complex and separable Hilbert space with scalar product φ, ψ L 2 R 3 R 3 φψ dx and corresponding norm φ The space of infinitely differentiable complexvalued functions with compact support will be denoted C ∞ 0 R 3 .The Fourier transform is given by Define which, equipped with the scalar product International Journal of Mathematics and Mathematical Sciences becomes a Hilbert space; evidently, H 1 R 3 ⊂ H 1/2 R 3 .We have that C ∞ 0 R 3 is dense in H 1/2 R 3 and the continuous embedding H 1/2 R 3 → L 3 R 3 holds; more precisely, the Sobolev inequality is valid with c sob 2 −1/2 π −2/3 .Moreover, we will use any weakly convergent sequence that in H 1/2 R 3 has a pointwise convergent subsequence.

Operators
Let T be a self-adjoint operator on a Hilbert space H with domain D T .The spectrum and resolvent set are denoted by σ T and ρ T , respectively.We use standard terminology for the various parts of the spectrum; see, for example, 24, 25 .The resolvent is R ζ T − ζ −1 .The spectral family associated to T is denoted by E T λ , λ ∈ R. For a lower semibounded self-adjoint operator T , the counting function is defined by The space of trace operators, respectively, Hilbert-Schmidt operators, on h L 2 R 3 is denoted by S 1 h , respectively, S 2 h .
We need the following abstract operator result by Lions

The Quasirelativistic Hartree-Fock Model
Within the Born-Oppenheimer approximation, the quantum energy of N quasirelativistic electrons interacting with K static nuclei with charges Z Z 1 , . . ., Z K , Z k > 0, is, in Rydberg units, given by ∈ R 3 is the position of the nth electron, α is Sommerfeld's fine structure constant, and the potentials V ee and V en are given by with R k ∈ R 3 being the position of the kth nucleus.Here it is important that Z tot < Z c : 2/ απ .See Section 3.1 for details.In what follows, we ignore the spin variable but the entire contents can be trivially carried over to the spin-valued setting.Above F n Ψ e is the Fourier transform of Ψ e x 1 , . . ., x n−1 , •, x n 1 , . . ., x N , in the case when N 1 we will just write FΨ e , and dμ ξ : The interpretation of the quadratic form 3.1 is as follows see Section 3.1 for its well definedness .The first term corresponds to the quasirelativistic kinetic energy of the electrons, the second term is the one-particle attractive interaction between the electrons and the nuclei, and the third term is the standard two-particle repulsive interaction between the electrons.The wave function Ψ e : R 3N → C in 3.1 belongs to H e ∧ N H 1/2 R 3 , that is, the N-particle Hilbert space consist of antisymmetric functions expressing the Pauli exclusion principle where S N is the group of permutations of {1, . . ., N}, with the signature of a permutation σ being denoted by sign σ , and H 1/2 R 3 is the Sobolev space introduced in Section 2. The ground state energy is defined as

3.5
To determine E QM N, Z directly turns out to be too difficult, even for small N. One of the classical approximation methods for determining E QM N, Z is the Hartree-Fock theory, introduced by Hartree and improved by Fock and Slater in the late 1920s see, e.g., 26 , which consists of restricting attention to simple wedge products Ψ e ∈ S N , where This space is clearly a complete metric space and also an Hilbert manifold.A function Ψ e ∈ S N is sometimes called a Slater determinant, and the φ n are called orbitals 26 .

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In fact, if Ψ e ∈ S N then, by simple algebraic calculations, E QM Ψ e E Ψ e , where the quasirelativistic Hartree-Fock functional or simply the energy functional E • :

3.8
Here is the density matrix, and is the density associated to the state Ψ e ; when there is no risk of confusion we will suppress the dependence of D.

Atomic and Molecular Hamiltonians
By p we denote the momentum operator The following facts are well known for the perturbed one-particle operator H 1,1,α T 0 − S x 25, 28 .

Small Perturbations
If Z < π/2 Z c then S is T 0 -bounded with relative bound equal to two.If, on the other hand, 2α −1 < Z < Z c then S is T 0 -form bounded with relative bound less than one.
We prove the above-mentioned form boundedness.It follows from the following inequality first observed, it seems, by Kato 25, paragraph V-5.4 : Indeed, if, for any ψ, φ ∈ H 1/2 R 3 , we define the sesquilinear forms 3.13 then 3.12 shows that s is well defined and also, by invoking | − i∇| ≤ T 0 , we infer that, for all φ ∈ H 1/2 R 3 , s φ, φ < t 0 φ, φ provided Z < Z c .

3.14
This is the Coulomb uncertainty principle in the quasirelativistic setting.The KLMN theorem see, e.g., 25, paragraph VI-1.7 implies that there exists a unique self-adjoint operator, denoted H 1,1,α , generated by the closed sesquilinear form

3.16
In particular,

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The form construction of the atomic Hamiltonian H 1,1,α can be generalized to the molecular case, describing a molecule with N electrons and where V k is defined in 3.2 and by assuming that Z tot < Z c .Under the same hypothesis, we note that the discussion on the forms s, t 0 , and v en immediately gives us that the form 3.1 and thus E • is well defined closed and bounded from below.

Density Operator Formalism
We can re-express E • and the Hartree-Fock ground state energy via the one-to-one correpondence between elements of C N and projections onto finite-dimensional subspaces of L 2 R 3 .Indeed, given an element {φ n } N n 1 in C N we can associate a canonical projection operator, D N n 1 •, φ n φ n with trace equal to N. We may therefore write where v en φ n , φ n .

3.20
The direct Coulomb energy defined in terms of the Coulomb inner product and the exchange Coulomb energy defined by Furthermore, it is not hard to verify that given a projection operator with trace N defined on L 2 R 3 we can also find an element in C N corresponding to this operator.It is therefore clear that the Hartree-Fock ground state energy can be expressed as

3.23
International Journal of Mathematics and Mathematical Sciences 9 More generally, a density operator D is a trace class operator on h L 2 R 3 , in symbols D ∈ S 1 h , which satisfies the operator inequality 0 ≤ D ≤ I.This motivates the following definition:

3.24
Using standard arguments 3 in combination with 3.12 , and the Sobolev inequality 2.4 , it is easy to show that E • is well defined on the following enlarged set of density operators:

3.25
For later purpose we also introduce

The Quasirelativistic Fock Operator
Herein we introduce the quasirelativistic Fock operator.
be the integral kernel of the exchange operator K xc .Then the unique self-adjoint operator F associated with the differential expression is generated by the sesquilinear form x − y dy dx.

4.3
Proof.Bear in mind the definitions of t 0 , t 0 , and v en from Section 3.1.Define v ρ * 1/|x| as the third form on the right-hand side of 4.3 .Then 3.12 yields the estimate Under the hypothesis, we already know from Section 3.1 that the quadratic form t 0 v en is nonnegative on H 1/2 R 3 .Evidently, v ρ * 1/|x| is a nonnegative form and, consequently, International Journal of Mathematics and Mathematical Sciences Closedness of the nonnegative quadratic form f is equivalent to lower semicontinuity of f on H 1/2 R 3 .In fact, f is continuous.Indeed, 3.12 , respectively, 4.4 enables us to show continuity of the second, respectively, the third terms, in f.For instance, we consider v en and assume that φ j → φ in H 1/2 R 3 .Then an application of H ölder's inequality and 3.12 yields

4.5
We conclude that f is a closed quadratic form on H 1/2 R 3 .The first representation theorem 24, Theorem VI.2.4 informs us that the nonnegative closed form f is associated to a unique self-adjoint operator, say F. Furthermore, the exchange operator K xc is a Hilbert-Schmidt operator.Indeed, using, in this particular order, the weak Young inequality, the H ölder inequality and 3.12 we find that closed and, once again applying the first representation theorem, we obtain a unique self-adjoint operator F associated with the form in 4.3 .

Lower Spectral Bound
We will later need the following spectral result.

5.2
Proof.By a minor modification of 28, page 291 , which carries over the result 3.17 from the one-nucleus to the many-nuclei cases, we deduce that the essential spectrum of T 0 V en equals the semiaxis 0, ∞ .Next, a standard perturbation argument and yet an application of Weyl's essential spectrum theorem prove that σ ess T 0, ∞ .Let t μ denote the quadratic form defined by
Within the nonrelativistic context a similar result was first given by Lions 1, Lemma II.1 .

Relative Compactness of Palais-Smale Type Sequences
In this section we give the main auxiliary result that will be used in the proof of Theorem 7.1.We emphasize that the functional E • is not weakly lower semicontinuous on H 1/2 R 3 N and, in the proof below, it is thus necessary to switch to a density operator formalism.In particular, we use that for a specific sequence of density operators see the proof for details , one can establish the inequality 6.20 below replacing the notion of weak lower semicontinuity which is absent .Proposition 6.1.Assume that l ∈ R, that m ∈ N, and let N − 1 < Z tot < Z c .Then any sequence {φ j } ∞ j 1 ⊂ C N satisfying a Palais-Smale condition at level l and of order less than m is relatively compact in C N , that is, any sequence {φ j } ∞ j 1 in C N is relatively compact whenever the sequence satisfies the following conditions: ii lim j → ∞ E φ j 0; iii there exists a sequence of positive reals {δ j } ∞ j 1 with δ j → 0 such that for each j, E φ j has at most m eigenvalues below −δ j .Moreover, the components of the limit element φ φ 1 , . . ., φ N of {φ j } ∞ j 1 in C N satisfy the quasirelativistic Hartree-Fock equations where λ n ≥ 0 for Z tot > N − 1, respectively, λ n > 0 for Z tot > N, and F is the Fock operator defined in Lemma 4.1.

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Proof.First we treat the case N < Z tot .Henceforth we let {φ j n } N n 1 ∈ C N be the canonical sequence associated with an operator D j in M proj defined in Section 3.2.The hypotheses i and ii give us that sup where λ j n is a sequence of reals and f j is a sequence of quadratic forms associated with {φ j n }, defined as in 4.3 .
Let us now extract some subsequences that we will need.Let us start by proving existence of 0 < λ < λ < ∞ such that λ < λ j n < λ.To prove existence of a lower bound we note that from hypothesis iii we get in particular that with δ j n → 0 in the standard Euclidean metric for each fixed n and ψ in a closed subspace of H 1/2 R 3 with finite codimension N m.By invoking Lemma 2.1 we deduce that the quasirelativistic Schr ödinger operator has at most N m eigenvalues strictly less than − λ j n δ j n .Moreover, since Lemma 5.1 ensures that there exists δ > 0 independent of j such that T j has at least N m eigenvalues strictly below −δ.As a consequence, we infer that Since δ j n → 0 as j → ∞, we conclude that, for j large enough, λ j n ≥ λ > 0, ∀n.

6.7
We note that the hypothesis Z tot < Z c and the fact that {D j } ⊂ M proj satisfies International Journal of Mathematics and Mathematical Sciences 13 ensure the existence of a constant C, depending on N, such that Tr T 0 D j ≤ C 1 Tr D j .6.9 To prove existence of an upper bound we note that which follows from the Cauchy-Schwarz inequality and 6.8 .Now, perhaps after going to a subsequence using the Bolzano-Weierstrass theorem, we may assume that We know from 6.9 that Tr T 0 D j 6.12 is uniformly bounded in j.Then some straightforward calculations give us that is also uniformly bounded in j.Here the T 1/2 0 is defined using Kato's second representation theorem 24, Theorem VI.2.4 .Hence we may, using the Banach-Alaoglu theorem, extract a subsequence such that T 1/2 0 D j T 1/2 0 converges weakly in S 2 to an element D. Fix any ψ ∈ L 2 R 3 , then is a linear bounded functional.We get that lim Then a direct application of Fatou's lemma with respect to a counting measure gives us that Tr D ≤ lim inf j Tr D j ≤ N.

6.16
Mutatis mutandis it is clear that 6.17

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We note that D j converges weakly to D in S 2 and hence that the kernels of the operators D j will converge weakly in L 2 R 3 × R 3 to the kernel of D. In view of 6.17 and the fact that Tr D j ≤ C we infer that there exists a subsequence such that weakly in H 1/2 R 3 and, by invoking weak compactness see Section 2 , the convergence holds almost everywhere.Since weak limits are unique, we may assume that the kernel associated with D can be written as The inequality can be derived by arguments similar to the ones in 4, pages 722-724 , wherein it is proven for a minimizing sequence bearing in mind the spectral properties of the one-particle operator H 1,1,α in 3.15 which we summarized in Section 3.1 .More specically, using arguments by Barbaroux et al. 26 and Solovej 30 , the inequality 6.20 was proved by Dall'Acqua et al 16 for specfic sequences in the quasirelativistic setting and their proof carries over to our sequence.As a consequence, we have that lim sup

6.21
From this we conclude that Tr D N and therefore that φ n L 2 R 3 1.Repeating the argument above, we obtain the convergence in H 1/2 R 3 N .We recall the regularity property of E and that the quasirelativistic Hartree-Fock equations are the Euler-Lagrange equations corresponding to this functional.The last assertion then follows from hypothesis ii and the relative compactness that was just proved.
Finally, we consider the case N − 1 < Z tot .By going to the limit in 6.3 , the resulting inequality holds on a closed subspace of H 1/2 R 3 with finite codimension; this requires that φ j n → φ n weakly in H 1/2 R 3 inspection of the argument above justifies this .Hence we infer that T defined similar to T j with ρ j replaced by ρ has at most finitely many eigenvalues less than or equal to −λ n ≤ 0. If Tr D N, then we are done.If, on the other hand, Tr D < N then we apply Lemma 5.1 and repeat the reasoning above.This completes the proof.
The density operator argument in the proof of Proposition 6.1 is inspired by Solovej 3 .Remark 6.2.It is worth to mention that from the perspective of Physics, there is no difference between the requirements Z tot > N − 1 and Z tot ≥ N because Z tot is integer valued.

Existence of a Ground State and Excited States
The main result is the following theorem.δ mn for all 1 ≤ m, n ≤ N and, furthermore, the Lagrange multipliers λ k n are positive, respectively, nonnegative, when Z tot > N, respectively, Z tot > N−1.Moreover, the following properties are valid as k → ∞: 3 any solution to 7.1 belongs to C ∞ R 3 \ {R 1 , . . ., R N } N and ϕ n decays exponentially sufficiently far away from the locations of the nuclei.
Before proving assertion 1 of Theorem 7.1, let us give a few explanations.To ensure that a Palais-Smale sequence converges, one needs to somehow "improve" it.Since E is a C 2functional, one may try to obtain an almost critical sequence with some information on the second derivative.This enables us to built an almost critical sequence which satisfies 6.3 .

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Due to lack of compactness, one cannot find critical points of E and therefore one perturbs the functional while, simultaneous, ensuring that the new functional has critical points of the kind, one expects for the original one.The way one will obtain such sequences consist in applying a "perturbed variational principle" by Borwein and Preiss 30 .
Proof of Theorem 7.1 (assertion 1).First of all we note that using 3.12 and the Cauchy-Schwarz inequality that E is bounded from below uniformly on C N and we may therefore conclude existence of a minimizing sequence, { Φ j } ∞ j 1 to 3.11 .To prove relative compactness we will now prove that the hypotheses ii and iii in Proposition 6.1 are satisfied.An application of the Borwein and Preiss variational principle 30, Theorem 2.6 provides us with a new minimization sequence {Φ j } ∞ j 1 , such that We will also have that for some Ψ j ∈ H 1/2 R 3 N and γ j > 0, where γ j → 0. From this we can conclude that hypothesis ii is satisfied.If we then follow the idea to prove a lower bound on the reals in the proof of Proposition 6.1 it is not difficult to show that hypothesis iii is satisfied for N. Existence of a minimum follows from Proposition 6.1.To show that λ n > 0 one argues by contradiction as in 27, page 2139 .The last assertion on the Lagrange multipliers and its relation to the Fock operator has been proven in 4 .
Proceeding towards the second assertion of Theorem 7.1 which addresses the existence of infinitely many nonminimal solutions, one would expect from the previous proof that a more involved perturbed variational principle is needed.For our specific setting, however, it suffices to apply the direct method by Fang and Ghoussoub 10 see also 11, 31 .Again, due to the lack of weakly lower semicontinuity, it is necessary to switch to density operators in the proof below.
Proof of Theorem 7.1 (assertions 2 and 3).We will prove that there exists a critical point at infinitely many distinct levels.We will use abstract critical point theory by Fang and Ghoussoub 10 .Consider the C 2 -functional E on the C 2 -Riemannian manifold C N .We consider Z 2 {0, 1} to be the compact 0-dimensional Lie group, with groups actions 0, φ → φ and 1, φ → − φ φ ∈ C N .We note that the functional is even, in fact it is invariant under unitary transformation, this can be seen by repeating the proof from the nonrelativistic Hartree-Fock case see, e.g., 27, Lemma 2.3 .Next we make preparations for the min-max principle: For each k ∈ N, we consider the following homotopic classes of order k where S k−1 is the unit sphere in the Euclidean space R k .Let We claim that −∞ < l k ≤ l k 1 < 0 for each k ∈ N and that lim k → ∞ l k 0, the proof of this fact will be given last in this proof.We may of course, after perhaps going to a subsequence, assume that l k < l k 1 for each k.Now, we will use the abstract results by Fang and Ghoussoub 10 in particular 31, Theorem 11.1 and Remark 11.13 , to extract a sequence satisfying the assumptions of Palais-Smale condition at level l k and of order less than k, but such a sequence is according to Proposition 6.1 relatively compact in C N .
Let us now prove the properties of the sequence {ϕ k } k≥1 of distinct solutions.We have already seen that we may assume that −∞ < l k < l k 1 < 0 so we may find a sequence such that −∞ < l k−1 < l k E ϕ k < l k 1 < 0. tends to zero.This together with 7.7 allows us to conclude that we can find perhaps after going to a subsequence a weak limit, ϕ ∈ H 1/2 R 3 N , for our sequence.Due to the assumption Z tot < Z c we may find a constant C > 0 such that International Journal of Mathematics and Mathematical Sciences and we may therefore conclude that ϕ 0. This finishes the part on the properties of the sequence.It remains to prove the claim stated above.The monotonicity of {l k } ∞ k 1 is a direct consequence of how we have defined H k and since E is uniformly bounded below on C N , we immediately get that l k > −∞.An application of Lemma 5.1 ensures that there exists a kdimensional subspace H k of H 1/2 R 3 such that for all φ ∈ H k with φ L 2 R 3  1 we denote the unit sphere in this subspace by S k−1 , one has t 0 φ, φ v e φ, φ for some k > 0. It is not hard to find a continuous isomorphism g : H k → R k such that g S k−1 S k−1 , now denote by e the natural embedding of S k−1 into C N due to the monotonicity we may assume k to be sufficiently large and therefore g • e will be an odd and continuous mapping from S k−1 into C N where E negative and therefore we can conclude that l k < 0. Hence we can find M k ∈ H k such that

7.13
To prove that lim k → ∞ l k 0, we use the separability of H 1/2 R 3 by considering a nested sequence of finite-dimensional subspaces W k of H 1/2 R 3 such that dim W k k and ∪ k W k is dense in H 1/2 R 3 .Define V k as the orthogonal complement of W k−1 .Now, assume that M k ∩ V k ∅, let π k−1 be the orthogonal projection from 7.14 and by following Rabinowitz 32 and using the Borsuk-Ulam theorem we will now arrive at a contradiction.Using that zero is an upper bound for the functional we can extract a sequence h k ∈ M k ∩ V k such that h k tends weakly to some element that must be equal to 0. By repeating the arguments in Proposition 6.1 we may find a subsequence which is of course sufficient in our case such that D k , the density operator corresponding to h k , tends weakly to 0 in S 2 .We get by the same type of argument as for 6.20 that The latter together with 7.13 implies that lim k → ∞ , l k 0. The regularity and decay properties of our sequence were proved in 4 for an atom and it carries over to our setting mutatis mutandis.

Theorem 7 . 1 . 1 whereF
Assume that the total nuclear charge Z tot K k 1 Z k satisfies Z tot < Z c and let N ∈ N satisfy N − 1 < Z tot .Then 1 every minimizing sequence of the quasirelativistic Hartree-Fock functional E • is relatively compact in C N .In particular, there exists a minimizer ϕ of E • on the admissible set C N and (up to unitary transformations) the components of ϕ ϕ 1 , . . ., ϕ N satisfy the quasirelativistic Hartree-Fock equations Fϕ n λ n ϕ n 0, ϕ m , ϕ n L 2 R 3 δ mn , 7.is the quasirelativistic Fock operator defined in Lemma 4.1, and the numbers −λ n are the N lowest negative eigenvalues of F, 2 there exists a sequence {ϕ k } k≥1 , with entries ϕ k ϕ k 1 , . . ., ϕ k N , of distinct solutions of the quasirelativistic Hartree-Fock equations 7.1 in H 1/2 R 3 N which satisfy the constraints ϕ

6 ρ k x ρ k x − D k x, x 2 |x − x | dx dx 7. 10
in k for each n.We note that the right-hand side of 2 12 , de Figueiredo et al. 13 , Flores et al. 14 , and Tanaka 15 , existence of multiple solutions established by Cingolani and Lazzo 16 and Ghoussoub and Yuan 17 , "relaxed" Palais-Smale sequences as in Lazer and Solimini 18 and Jeanjean 19 , and problems on noncompact Riemannian manifolds found in Fieseler and Tintarev 20, 21 , Mazepa 22 , and Tanaka 23 .
1, Lemma II.2 .Hilbert space H, and let H 1 , H 2 be two subspaces of H such that H H 1 ⊕ H 2 , dim H 1 h 1 < ∞ and P 2 TP 2 ≥ 0, where P 2 is the orthogonal projection onto H 2 .Then T has at most h 1 negative eigenvalues.