Page 1

SOLUTIONS OF THE DIRAC-FOCK EQUATIONS AND THE

ENERGY OF THE ELECTRON-POSITRON FIELD

MATTHIAS HUBER AND HEINZ SIEDENTOP

Abstract. We consider atoms with closed shells, i.e., the electron number

N is 2, 8, 10,..., and weak electron-electron interaction. Then there exists

a unique solution γ of the Dirac-Fock equations [D(γ)

ditional property that γ is the orthogonal projector onto the first N positive

eigenvalues of the Dirac-Fock operator D(γ)

ergy of the relativistic electron-positron field in Hartree-Fock approximation,

if the splitting of H := L2(R3) ⊗ C4into electron and positron subspace, is

chosen self-consistently, i.e., the projection onto the electron-subspace is given

by the positive spectral projection of D(γ)

g,α. For fixed electron-nucleus coupling

constant g := αZ we give quantitative estimates on the maximal value of the

fine structure constant α for which the existence can be guaranteed.

g,α,γ] = 0 with the ad-

g,α. Moreover, γ minimizes the en-

1. Introduction

Heavy atoms should be described by relativistic quantum electrodynamics. Fol-

lowing this idea, Bach et al. [1] showed that the energy of the relativistic electron-

positron field in Hartree-Fock approximation interacting with the second quantized

Coulomb field of a nucleus is non-negative (if the quantization is chosen with re-

spect to external field) and that the vacuum is a minimizer. Moreover, they showed

that the quantization with respect to the external field is optimal in the sense that

any other quantization yields a lower ground state energy.

Barbaroux et al. [3] addressed the existence of atoms in the above model, i.e.,

they prescribed the charge of the electron-positron field and showed that the corre-

sponding functional has a minimizer which fulfills the no-pair Dirac-Fock equations.

The existence of solutions of the Dirac-Fock equations was shown by Esteban

and S´ er´ e [6] and Paturel [11]. Moreover, Esteban and S´ er´ e [5] considered the non-

relativistic limit of the Dirac-Fock equations. They showed that certain solutions of

the Dirac-Fock equations converge to the energy minimizing solutions of the non-

relativistic Hartree-Fock equations when the speed of light tends to infinity. This

allows them to define the notion of ground state solutions and ground state energy

of the Dirac-Fock equations.

In the spirit of Mittleman [9] the physical energy should be obtained by maxi-

mizing the ground state energy (as defined, e.g., in [3]) over all allowed one-particle

electron subspaces. One might conjecture that a corresponding ground state is

a solution of the Dirac-Fock equations. Moreover, such a solution of the Dirac-

Fock equations should minimize the energy among all solutions of the Dirac-Fock

equations. We call this for brevity the “Mittleman conjecture”.

The validity of Mittleman’s conjecture was already addressed by Barbaroux et al.

[2]. They confirmed it when the atomic shells are closed and the electron-electron

interaction is weak (large velocity of light). In the open shell case it was only proven

by Barbaroux et al. [4] in the case of hydrogen. All other cases are unknown.

Date: December 5, 2005.

1

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2 M. HUBER AND H. SIEDENTOP

A stronger conjecture – for brevity called in the paper BES conjecture – would

be: the maximin pair ( maximizing Λ and minimizing γ) is a projector onto the

first N eigenfunctions of the self-consistent Dirac-Fock operator and that Λ is the

spectral projector onto the negative spectral subspace of this operator. Barbaroux

et al. [2] showed that this conjecture is incorrect in the open shell case in the

non-relativistic limit. (For N = 1 this result can be extended beyond the limiting

case (Barbaroux et al. [4]).) However, they confirm their conjecture for closed shell

atoms in the non-relativistic limit.

In this paper – following Barbaroux et al. [2] – we also consider the limit of

weak electron-electron interaction. Similarly to Barbaroux et al.[2, Proposition 8]

and Esteban and S´ er´ e [5, Theorem 5] we prove the existence of a unique solution

of the Dirac-Fock equations with the property that the eigenvalues of the solutions

are the lowest eigenvalues of the corresponding self-consistent Dirac-Fock operator

and that the next eigenvalue is strictly bigger. Again similarly to Barbaroux et al.

and Esteban and S´ er´ e [5, Theorem 6] this allows us to prove that this solution is

the minimizer of the Dirac-Fock energy on the set of all solutions of the Dirac-Fock

equations with non-negative eigenvalues. However, we can prove that this solution

minimizes the Dirac-Fock energy even on the set of all charge density matrices (see

the corresponding result of Barbaroux et al. [2, Proposition 8, Equation (15)]) if

the quantization is chosen with respect to this solution. We emphasize that we

do not only admit positrons in the charge density matrices γ; in fact we can drop

the assumption that off-diagonal elements of γ vanish, a requirement inherent in

Barbaroux et al. (Corollary 6). Eventually, we show that the minimizer is uniquely

determined and spherically symmetric in a certain sense. It has eigenfunctions

(orbitals) that respect the Aufbau principle.

The essential novelty of our result is twofold: First, our proof is sufficiently

direct and simple allowing for explicit estimates. This enables us to show not only

existence results (Esteban and S´ er´ e [6] and Paturel [11]) but also to prove important

properties of the solutions. In addition we obtain these properties not only in the

non-relativistic limit (Barbaroux et al. [2]) but we get explicit estimates on the

allowed coupling constants for which these results hold. Second, we can show the

minimization property among all density matrices of the electron-positron field in

the self-consistent quantization.

2. Definition of the Model

The notation and estimates used are mainly those of Barbaroux, Farkas, Helffer,

and Siedentop [3]. For the convenience of the reader we give here nevertheless their

main definitions and results. The technical tools from [3] are listed in an appendix.

For any further details we refer the reader to [3].

The Coulomb-Dirac operator is written as

Dg:= −iα · ∇ + β − g| · |−1.

Physically g = Zα where α is the Sommerfeld fine structure constant and Z is the

atomic number of the considered element. The operator is essentially self-adjoint

on S(R3) ⊗ C4if g ∈ [0,√3/2).

It is convenient to introduce the set G := R3× {1,2,3,4} and the measure

dx := dx ⊗ dµ, where dx is the Lebesgue measure on R3and dµ the counting

measure of the set {1,2,3,4}. We denote the Banach space of trace class operators

on H by S1(H). Furthermore,

F := {γ ∈ S1(H)|γ = γ∗, D0γ ∈ S1(H)}.

Note that Barbaroux et al. [3] use a slightly different definition of the space F.

Moreover, F is a Banach space when equipped with the norm ?γ?F := ?D0γ?1=

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DIRAC-FOCK EQUATIONS OF THE ELECTRON-POSITRON FIELD3

?|D0||γ|?1.

0 ≤ g <√3/2 because of Lemma 12.

We write the integral kernel of any given γ ∈ F using its eigenvalues λn and

eigenspinors ξnas

∞

?

The one-particle density associated to γ is

Finally, we note that ?γ?F,g := ?Dgγ?1 is an equivalent norm for

γ(x,y) =

n=1

λnξn(x)ξn(y).

ργ(x) :=

4

?

s=1

∞

?

n=1

λn|ξn(x)|2.

Its electric potential operator is φ(γ):= ργ∗ | · |−1. The exchange operator X(γ)

associated to γ is given by its integral kernel

X(γ)(x,y) := γ(x,y)/|x − y|.

The total interaction operator is defined as

W(γ)= φ(γ)− X(γ).

The Coulomb scalar product is defined as

?

and the exchange scalar product as

?

The total interaction energy is defined as

Q(γ,γ?) := D(ργ,ργ?) − E(γ,γ?)

For α ≥ 0 and γ ∈ F the Dirac-Fock operator is defined as

D(γ)

D(ρ,σ) :=1

2

R3dx

?

R3dyρ(x)σ(y)

|x − y|

E(γ,γ?) :=1

2

G

dx

?

G

dyγ(x,y)γ?(x,y)

|x − y|

.

g,α:= Dg+ αW(γ).

Some useful properties of the operators defined above are listed in Appendix B.

For N ∈ N and δ ∈ F we define

?S(δ)

SN:={γ ∈ F |0 ≤ γ, trγ ≤ N},

and

Eg,α(γ) := trDgγ + αQ(γ,γ)

where Λ(δ)

and Λ(δ)

+ is the projector onto the negative spectral subspace.

Moreover, we will frequently use the abbreviations

cg,α,N:= (bg− 4αN)−1,

where bg:=

?(γ)

j

(j = 1,...) the eigenvalues of Dg and D(γ)

counting multiplicities).

We will be interested in solutions of the Dirac-Fock equations.

∂N:={γ ∈ F |−Λ(δ)

?S(δ):={γ ∈ F |−Λ(δ)

−≤ γ ≤ Λ(δ)

−≤ γ ≤ Λ(δ)

+,trγ = N},

+},

+= χ[0,∞)(D(δ)

−= 1 − Λ(δ)

g,α) is the projector on the positive spectral subspace of D(δ)

g,α

˜ cg,α,N:= (π/4)αNcg,α,N

?1 − g2(?4g2+ 9−4g)/3 (see also Lemma 12). We denote by ?0

jand

g,α respectively (ordered by size and

Definition 1. We denote the set of solutions to the Dirac-Fock equations by DF,

i.e.,

DF := {γ ∈ F|γ = γ2, [D(γ)

g,α,γ] = 0, γΛ(γ)

+ = γ}.

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4M. HUBER AND H. SIEDENTOP

For fixed g and small α we get: for closed shell atoms there exists a solution δ ∈

DF such δ is the projection onto the first N positive eigenvalue of D(δ)

1). We will prove this result using the Banach fixed point theorem yielding even

uniqueness of the solution. We also show that the fixed point (Corollary 3) and the

energy functional Eg,α (Theorem 2) are spherically symmetric in a certain sense,

and that the fixed point minimizes Eg,α on DF (Corollary 2). To this end the

uniqueness of the fixed point is crucial.

Moreover, we show (Theorem 3) that this solution minimizes Eg,αeven on the

set?S(δ)

[3].

Note, that the notion of closed shells – as used in this article – refers to the

Coulomb-Dirac operator, i.e., for N ∈ N we are in the closed shell case, if ?0

?0

N. It does not matter, if the gap is the gap between shells with different principal

quantum numbers. This means that N = 2,8,10,.... For brevity, we denote the set

of all such N by CS.

g,α (Theorem

∂qWe emphasize that we do not need to require Λ(δ)

had to be left open in the context of the no-pair Hartree-Fock theory discussed in

+γΛ(δ)

− = 0, a fact that

N+1>

3. Controlling the Spectrum of Dirac-Fock Operators

The aim of this section is to derive some estimates which control the eigenvalues

of Dirac-Fock operators by the corresponding eigenvalues of Coulomb-Dirac oper-

ators. The main tool of this section is the minimax principle of Griesemer and

Siedentop [8], which is formulated in Appendix A (Theorem 4). We are going to

use the Coulomb-Dirac operator as unperturbed operator and the Dirac-Fock oper-

ator as perturbed operator. First, we check the hypotheses of the minimax theorem

(Theorem 4).

Lemma 1. Let A = D(γ)

χ(0,∞)(Dg), Λ−:= χ(−∞,0)(Dg), and h±:= Λ±h and assume 0 / ∈ σ(D(γ)

the hypotheses of Theorem 4 are fulfilled, if (π/2)α?γ?1≤ bg.

Proof. Let f ∈ Q−. Then

(f,D(γ)

≤ (f,Dgf) +π

g,α with 0 ≤ γ ∈ F, h = L2(R3)4, Q = D(A). Let Λ+:=

g,α). Then

g,αf) = (f,Dgf) + α(f,W(γ)f) ≤ (f,Dgf) + α(f,φ(γ)f)

2α?γ?1(f,|∇|f) ≤ (f,Dgf) +π

2α?γ?1

1

bg(f,|Dg|f) ≤ 0

where we used Lemmata 8 and 12, and (17),. The condition

(f,D(γ)

g,αf) > 0

for all f ∈ Q(A)∩H+is trivially fulfilled since W(γ)≥ 0 (Lemma 8). It remains to

check the boundedness of (|D(γ)

As in [7, Lemma 1],

?∞

i.e., we have to estimate the expression

?∞

Now, ?(Dg− iη)−1? ≤ [(?0

[λ0,∞) → R,fη(λ) :=

g,α| + 1)

1

2P−Λ+. To this end we proceed as follows:

P−Λ+= −α

2π

−∞

(D(γ)

g,α− iη)−1W(γ)(Dg− iη)−1dηΛ+,

−∞

(|D(γ)

g,α| + 1)

1

2(D(γ)

g,α− iη)−1W(γ)(Dg− iη)−1dη.

1)2+ η2]−1/2. Moreover, we look at the function

?

(λ + 1)/(λ2+ η2)

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DIRAC-FOCK EQUATIONS OF THE ELECTRON-POSITRON FIELD5

where λ0:= inf σ(|D(γ)

the point max{λ0,−1 +

g,α|) > 0 by assumption. This function has its maximum at

?1 + η2}, i.e.,

fη(λ) =

We conclude that

?∞

is finite, which implies the boundedness of (|D(γ)

Lemma 12 shows that the condition 0 ∈ ρ(D(γ)

bg> 4α?γ?1. One can relax this condition adapting an argument of Barbaroux et

al. ([4]); but since it is not the most restrictive condition, we refrain from doing so.

Lemma 1 enables us to control the eigenvalues of the Dirac-Fock operator by the

eigenvalues of the Coulomb-Dirac operator. Since the minimax principle yields the

eigenvalues ordered by size and counting multiplicities, we do not only get some

information on the localization of the eigenvalues but also on the dimension of

the projector onto a given part of the discrete spectrum. Note that the estimate

depends only on ?γ?1but not on γ itself.

Lemma 2. Let 0 ≤ γ ∈ F and let the hypotheses of Lemma 1 be fulfilled. Then,

for all n ∈ N

?0

n

≤ (1 + (π/2)α?γ?1b−1

Proof. Since 0 ≤ X(γ), 0 ≤ φ(γ)and 0 ≤ W(γ)(Lemma 8), we have using (17) and

Lemma 12

sup

λ

?(λ0+ 1)/(λ2

(−1+√

0+ η2)|η| ≤?(λ0+ 1)2− 1

?

(−1+√

1+η2)+1

1+η2)2+η2

|η| >?(λ0+ 1)2− 1.

−∞

?(|D(γ)

g,α| + 1)1/2(D(γ)

g,α− iη)−1??W(γ)??(Dg− iη)−1?dη

g,α| + 1)

g,α) in Lemma 1 is fulfilled, if

1

2P−Λ+.

??

n≤ ?(γ)

g)?0

n.

Dg≤ Dg+ αW(γ)≤ Dg+ αφ(γ)≤ Dg+π

≤ Dg+π

where (Dg)+ and (Dg)− denote the positive and negative part of the Coulomb-

Dirac operator respectively. We choose now h±:= Λ±h. Then the above operator

inequality yields immediately for all n ∈ N the inequality λn(Dg) ≤ λn(D(γ)

(π/2)α?γ?1b−1

4. Now, by Lemma 1 the hypotheses of Theorem 4 are fulfilled for D(γ)

the hypotheses are trivially fulfilled by the choice of h±:= Λ±h. For the operator

(1 + (π/2)α?γ?1/dg)(Dg)++ (1 − (π/2)α?γ?1/dg)(Dg)− the hypotheses are also

fulfilled, since it has got the same positive and negative spectral subspaces as the

operator Dg. Thus Theorem 4 immediately yields the claimed inequality.

2α?γ?1|∇|

g)(Dg)++ (1 −π

2α?γ?1b−1

g|Dg| ≤ (1 +π

2α?γ?1b−1

2α?γ?1b−1

g)(Dg)−

g,α) ≤ (1+

g)λn(Dg), where the λnare the minimax values defined in Theorem

g,α. For Dg

??

Lemma 3. Let 0 ≤ γ ∈ F with ργspherically symmetric and let the hypotheses of

Lemma 1 be fulfilled. Then, for all n ∈ N

?0

n

≤ ?(g − α?γ?1)0

where ?(g − α?γ?1)0

replaced by g − α?γ?1.

Proof. The first inequality is the same as in Lemma 2. For the second inequality

note that φ(γ)≤ ?γ?1| · |−1by Newton’s inequality. Thus, by Lemma 8

n≤ ?(γ)

n,

ndenote the eigenvalues of the Coulomb-Dirac operator with g

D(γ)

g,α= Dg+ αφ(γ)− αX(γ)≤ Dg+ α?γ?1| · |−1.

Since

(f,(D0− g| · |−1+ α?γ?1| · |−1)f) = (f,Dg−α?γ?1f) ≤ 0

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6M. HUBER AND H. SIEDENTOP

by Kato’s inequality and Lemma 12 for all f ∈ Q−, we get by the Minimax Theorem

4

λn(Dg−α?γ?1) ≤ ?(g − α?γ?1)0

Using Lemma 1, this implies the claim.

n.

??

We define now c := (1 + (π/2)αN/bg)?0

(π/2)αN/bg)?0

Nand η := ?0

N+1− c = ?0

N+1− (1 +

N. Let

α0:= sup{α ∈ R|?0

To simplify the notation, the dependence of these quantities on g and α is sup-

pressed. Eventually, we set

N+1− (1 +π

2αN/bg)?0

N> 0} = 2(?0

N+1− ?0

N)bg/(πN?0

N).

gg,N(α) := (?0

+ [(π − 12)N?0

N)2π2N3α3+ [−6πN2?0

N?0

N?0

N+1+?4π − 1/4π2?N2(?0

N)2]bgα2

N)2]bg2α

N+ ?0

N+1− 4N?0

1?0

N+ 12n(?0

N+1)2+ (4 − π)N(?0

− bg3(?0

N+1)2

and define α?

Z ≤ α−1

is fulfilled for these values of Z.

0be the smallest root of the cubic equation gg,N(α) = 0. Furthermore,

1 − (1 +√33)/16)2≈ 124.23 implies ?0

phys

?

1≥ ?0

3− ?0

1holds, so that ?0

1≥ η

Theorem 1. Assume N ∈ CS and α < min{α0,α?

as

Then the mapping

0,bg/(4N)}. We pick a path C

C(t) :=

?0

c +η

c +η

?0

1−η

2+ t(c +η

2− iη

2+ (t − 2)(?0

1−η

2− (?0

1−η

2)) − iη

2

0 ≤ t ≤ 1

1 ≤ t ≤ 2

2 ≤ t ≤ 3

3 ≤ t ≤ 4.

2+ (t − 1)η

1−η

2− (c +η

2)) + iη

2

2+ iη

2− i(t − 3)η

T : SN→ SN, γ ?→ −(2πi)−1

?

C

(D(γ)

g,α− z)−1dz

has a unique fixed point.

We remark that our bound on the range of allowed fine structure constants α

in the hypothesis tends to zero as 1/N. This has one main technical reason: the

control on eigenvalues of the Dirac-Fock operator in terms of the Coulomb-Dirac

operator becomes worse as the particle number grows, since the gap between the

eigenvalues becomes smaller for larger eigenvalues (see Lemma 2). This is also the

reason why our estimates on the contraction properties of the map T becomes worse

as N growths.

Proof. Step 1: Note that SN is a closed subset of F. Pick γ ∈ SN. Because of the

inequalities (Lemma 2)

?0

1≤ ?0

k≤ ?(γ)

k

≤ (1 +π

2αNb−1

g)?0

k≤ (1 +π

2αNb−1

g)?0

N< ?0

N+1

for k = 1,...,N and because T(γ) is the projector onto the spectral subspace

of D(γ)

Theorem XII.6]) we have γ ∈ F, trT(γ) = N and γ ≥ 0. Thus, T is well defined.

Step 2: We show that the mapping T is a contraction: pick γ,γ?∈ SN. Let P be

the projector on range(|Dg|(T(γ) − T(γ?)). Since dimrange((T(γ) − T(γ?)) ≤ 2N,

g,α corresponding to the N lowest eigenvalues of D(γ)

g,α (Reed and Simon [12,

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DIRAC-FOCK EQUATIONS OF THE ELECTRON-POSITRON FIELD7

we have dimrange(|Dg|(T(γ) − T(γ?)) ≤ 2N, implying ?P?1≤ 2N. Thus,

?T(γ) − T(γ?)?F,g= ?|Dg|(T(γ) − T(γ?))?1= ?P|Dg|(T(γ) − T(γ?))?1

≤ ?P?1?|Dg|(T(γ) − T(γ?))? ≤

≤ 2Ncg,α,Nbg?|D(γ)

≤ π−1cg,α,NbgN?|D(γ)

C

≤αcg,α,NNbg

π

C

≤ αcg,α,NN(2η + (c − ?0

max

2N

1 − 4αN/bg?|D(γ)

g,α|(T(γ) − T(γ?))?

(D(γ)

g,α|(T(γ) − T(γ?))?

g,α|

?

g,α− z)−1− (D(γ?)

g,α− z)−1dz?

?

?

|D(γ)

g,α|(D(γ)

g,α− z)−1W(γ−γ?)(D(γ?)

g,α− z)−1dz?

1))×

z∈C([0,4)](?|D(γ)

g,α|(D(γ)

g,α− z)−1?)max

z∈C([0,4)]?(D(γ?)

g,α− z)−1??γ − γ??F,g

where we used Lemma 12, the resolvent identity, Lemma 7 and (16).

Pick an arbitrary z = x+iy ∈ C (x,y ∈ R). We derive estimates for ?|D(γ)

z)−1?. Let A := R \ [(0,?0

cause E[(0,?0

N+1)] = 0, it follows that

?|D(γ)

g,α|(D(γ)

g,α. Be-

g,α−

1) ∪ (c,?0

N+1)] and E the spectral resolution of D(γ)

1) ∪ (c,?0

g,α|(D(γ)

g,α− z)−1? ≤ sup

λ∈A|λ|/|λ − z| = sup

λ∈Afx,y(λ)

where

fx,y(λ) := |λ|/|λ − z| = |λ|((λ − x)2+ y2)−1/2.

First assume x = c + η/2 and y arbitrary. Since fx,y(λ) ≤ |λ|/|λ − x|, we get

sup

λ∈Afx,y(λ) ≤ |?0

N+1|/|?0

N+1− x| = 2?0

N+1/η.

Similarly for x = ?0

1− η/2 and arbitrary y, we get

?|D(γ)

1−η/2,c+η/2]. Obviously with B := [?0

sup

λ∈Afx,y(λ) = sup

A little calculation shows that fx,yattains its maximum on [0,∞) at λ0= (x2+

y2)/x. Moreover, fx,y(λ) ≤ |λ|/|y| for all λ ∈ R, implying

?|D(γ)

Since the function hy(b) := (b2+y2)/(b|y|) attains its minimum on (0,∞) at b = |y|

and is monotonously increasing for b > |y|, we get

?|D(γ)

because x > y by the remark before the theorem. Now a little calculation shows

that

(c + η/2)2+ (η/2)2

(c + η/2)η/2

implying ?|D(γ)

?(D(γ?)

for z ∈ ρ(D(γ?)

?T(γ) − T(γ?)?F,g≤αcg,α,NN · (2η + (c − ?0

=α[4Ncg,α,N?0

g,α|(D(γ)

g,α− z)−1? ≤ 2?0

1/η.

Now let y = ±η/2 and x ∈ [?0

1,c]∪[?0

n+1,∞),

λ∈Bfx,y(λ).

g,α|(D(γ)

g,α− z)−1? ≤ fx,y(λ0) ≤ (x2+ y2)/(x|y|).

g,α|(D(γ)

g,α− z)−1? ≤ [(c + η/2)2+ η2/4]/[(c + η/2)η/2],

≤2?0

N+1

η

g,α|(D(γ)

g,α− z)−1? ≤ 2?0

g,α− z)−1? ≤ 1/dist(z,σ(D(γ?)

g,α) yielding altogether

N+1/η for all z ∈ C([0,4]). We also use

g,α)) = 2/η

1))2?0

N+1η−12η−1?γ − γ??F,g

1))]η−2?γ − γ??F,g.

n+1(2η + (c − ?0

Page 8

8M. HUBER AND H. SIEDENTOP

Now the condition 4αNcg,α,N?0

gg,N(α) < 0 which proves the claim.

N+1(2η + (c − ?0

1))η−2< 1 leads to the inequality

??

Note that the set of density matrices γ ∈ F with spherical density ργ is closed

in the F-norm: taking a F-convergent sequence γn of such density matrices we

merely have to show that the limiting density matrix has spherical density, too.

However, convergence in F implies convergence of the corresponding densities ρn

in L1. Suppose that R is a rotation, then ρn,R(x) := ρn(Rx)

L1−limn→∞ρn= L1−limn→∞ρn,R. Thus we may apply the fixed point theorem to

this smaller set. This improves the estimates in the proof Theorem 1 slightly using

Lemma 3. We display the result of a numerical evaluation of the corresponding –

more complicated condition – in Figure 1.

a.e.

= ρn(x). Thus, also

100 806040

0.0015

0.001

200

0.0005

0.002

0

with sph. symmetry

without sph. symmetry

Figure 1. The maximal value of α, for which the contraction

property of T can be guaranteed, in dependence on the nuclear

charge Z = 137g for N = 2 with and without assuming spherical

symmetry. (Note: in our proves we use – because of theoretical

reasons – two independent parameters, namely g and α. Physically,

one would choose: (i) g = αphysZ where Z is the atomic number

of the considered element. (ii) α = αphys where αphys ≈ 1/137.

To make contact with the physics, we make this first choice and

plot the maximal value of α fulfilling our hypotheses. The plot

shows that we do not reach αphys; however, our result is on the

right order of magnitude for highly ionized medium sized atoms.)

Corollary 1. If N ∈ CS and α fulfills the hypotheses of Theorem 1, then there

exists a unique δ ∈ F which is the projector onto the eigenspace of the N lowest

eigenvalues of D(δ)

g,α.

Proof. Theorem 1 ensures the existence of such a δ.

projector γ onto the N lowest positive eigenvalues of D(γ)

T(γ) = γ which has the unique solution δ.

On the other hand, any

g,α, fulfills the equation

??

We set

EN:= ?0

1+ ... + ?0

4)?0

N−1

a := 2πN2(EN+ ?0

c := (−?0

N)

b := [−3

4πEN− (4 +π

ag,N:= (−b −√b2− 4ac)/(2a).

N+1+ (4 −π

2)?0

N]bgN

N+ ?0

N+1)b2

g

Page 9

DIRAC-FOCK EQUATIONS OF THE ELECTRON-POSITRON FIELD9

Corollary 2. If N ∈ CS and there is a unique solution δ of the equation T(γ) = γ,

then it minimizes the energy among all Dirac-Fock solutions, i.e., this solution

fulfills

Eg,α(δ) = min{Eg,α(γ)|γ ∈ DF, trγ = N},

if α ≤ min{ag,N,bg/(4N),α0}.

Proof. Because of Lemma 9 we have for all γ ≥ 0

(1)Eg,α(γ) = trD(γ)

We pick an arbitrary solution γ of (DF) with trγ = N. With Lemma 9 and 12 we

get

g,αγ − αQ(γ,γ) ≤ trD(γ)

g,αγ.

(2)Eg,α(γ) = trD(γ)

≥ trD(γ)

g,αγ − αQ(γ,γ) ≥ trD(γ)

g,αγ −π

g,αγ − αD(ργ,ργ)

g,αγ −π

4αN tr|∇||γ| ≥ trD(γ)

4αNcg,α,Ntr|D(γ)

= (1 − ˜ cg,α,N)trD(γ)

g,α||γ|

g,αγ.

We denote by ?k, k = 1,...,N, the eigenvalues (ordered by size and counting

multiplicities) of D(γ)

g,α whose eigenvectors are in the range of γ. If the ?k, k =

1,...,N, fulfill the inequality

1≤ ?k≤?1 +π

for k = 1,...,N, the γ is a projector onto the eigenspace of the N lowest eigenvalues

of D(γ)

g,α and hence equal to the unique fixed point δ of T. By equation (1) and

Lemma 2, we get for the energy of the fixed point

?0

2αNb−1

g

??0

N

Eg,α(δ) ≤

N

?

m=1

?(δ)

m≤ (1 +π

2αNb−1

g)

N

?

m=1

?0

m.

If, on the other hand, there is a l ∈ {1,...,N} such that ?j≥ ?(0)

and ?j≤ (1 +π

N+1for all j ≥ l

2αNb−1

g)?(0)

n

for all j ≤ l − 1, then, by (2), we get

Eg,α(γ) ≥ (1 − ˜ cg,α,N)trD(γ)

g,αγ = (1 − ˜ cg,α,N)

N

?

m=1

?≥ (1 − ˜ cg,α,N)?N−1

?m

≥ (1 − ˜ cg,α,N)?l−1

N+1> ?(0)

?

Nfor m ≥ l, it follows that

m=1

?0

m+

N

?

m=l

?m

?

m=1

?0

m+ ?0

N+1

?.

Now, because ?m≥ ?(0)

Eg,α(γ) − Eg,α(δ)

= (1 − ˜ cg,α,N)

N−1

?

m=1

?0

m+ (1 − ˜ cg,α,N)?0

N+1− (1 +π

2αN

bg)

N

?

m=1

?0

m

= −α(π

4Ncg,α,N+πN

2bg)

N−1

?

m=1

?0

m+ (1 −π

4αNcg,α,N)?0

N+1− (1 +π

2αN

bg)?0

N.

The condition Eg,α(γ) − Eg,α(δ) ≥ 0 yields the quadratic equation in α

2πN2(EN+ ?0

N)α2+?−3π

4EN−?4 +π

4

??0

n+1+ (4 −π

2)?0

+?−?0

?

N

?bgNα

N+ ?0

N+1

?b2

g= 0,

whose smallest root ag,N is relevant.

?

Page 10

10M. HUBER AND H. SIEDENTOP

Z

10080604020

0.004

0.002

0.003

0

0

0.001

Figure 2. The constant min{ag,n,bg/(4N),α0} in dependence on

the nuclear charge Z = 137g for N = 2.

4. Spherical Symmetry

As a next step – following [2] – we show that the fixed point of T is spherically

symmetric in a certain sense: For any R ∈ SO(3) there is a UR∈ SU(2) such that

(Rx)·? σ = UR(x·? σ)U−1

choices differ only by −1. Since we are only interested in eigenvectors, we do not

care about this ambiguity. Pick

?f(u)

We define

?URf(u)(R−1x)

Obviously, fR∈ L2(R3;C4). For γ =?∞

γR:=

n=1

We first show the following

Rfor all x ∈ R3. Note that URis not unique; the two possible

f =

f(l)

?

∈ L2(R3;C4).

fR(x) =

URf(l)(R−1x)

?

.

n=1λn|ξn??ξn| ∈ F we define

∞

?

λn|(ξn)R??(ξn)R|.

Lemma 4. If f ∈ L2(R3;C4) is an eigenfunction of D(γ)

fR∈ L2(R3;C4) is eigenfunction of D(γR)

Proof. We treat the Dirac-Fock operator term by term.

Step 1: Let Q = R−1. We have

g,α with eigenvalue ?, then

g,α with eigenvalue ?.

(3)(−iα · ∇fR)(x) = VR(−iα · ∇f)(Qx)

?UR

where

VR=

0

0UR

?

.

Proof of Step 1: We denote by ∂ the total derivative and by ∂j the respective

partial derivatives.

∂jfR=

?

UR∂jf(u)

UR∂jf(d)

R

R

?

=

?UR[(∂f(u))◦Q]Qej

UR[(∂f(l))◦Q]Qej

?

=

UR

3 ?

k=1

3 ?

[(∂kf(u))◦Q]Qkj

UR

k=1

[(∂kf(l))◦Q]Qkj

Page 11

DIRAC-FOCK EQUATIONS OF THE ELECTRON-POSITRON FIELD11

It follows that

α · ∇fR=

3

?

?URU−1

j=1

3

?

k=1

?σjUR[(∂kf(l))◦Q]Qkj

σjUR[(∂kf(u))◦Q]Qkj

?

=

3

?

k=1

R(? σ · Rek)UR(∂kf(l))◦Q

URU−1

R(? σ · Rek)UR(∂kf(u))◦Q

?

=

3

?

k=1

?URσk(∂kf(l)) ◦ Q

= VR(α · ∇f) ◦ Q

URσk(∂kf(u)) ◦ Q

?

Second term: We have

(4) βfR= β

?UR

0

0UR

??f(u)◦ Q

f(l)◦ Q

?

= VR(βf) ◦ Q

Third term:

(5) | · |−1fR= VR|Q · |−1f ◦ Q = VR| · |−1f ◦ Q

Fourth term:

(6)φ(γR)fR= VR

??

??

n

λn

?

?

UR[ξ(u)

UR[ξ(l)

?

ξ(l)

n

◦ Q]

n ◦ Q]

ξ(u)

n (y)

?

?

,

?

UR[ξ(u)

UR[ξ(l)

?

n

◦ Q]

n ◦ Q]

?

?

?

C4∗

1

| · |

?(f ◦ Q)

= VR

n

λn

?

n (y)

?

= VR[(φ(γ)◦ Q)(f ◦ Q)] = VR[(φ(γ)f) ◦ Q]

,

ξ(u)

n (y)

ξ(l)

n (y)

C4

dy

| · −Ry|f ◦ Q

Fifth term:

(7)(X(γR)fR)(x)

??

=

n

λn

?

URξ(u)

URξ(l)

n (Qx)

n (Qx)

?

?

?

URξ(u)

URξ(l)

n (Qy)

n (Qy)

?

,

?URf(u)(Qy)

URf(l)(Qy)

??

C4

dy

|x − y|

= VR

?

dy

?

n

λn

?

ξ(u)

n (Qx)

ξ(l)

n (Qx)

?

?

?

ξ(u)

n (y)

ξ(l)

n (y)

|x − Ry|

?

,

?f(u)(y)

f(l)(y)

??

C4

= VR(X(γ)f)(Qx)

Thus,

(D(γR)fR)(x) = VR(D(γ)

g,αf)(Qx) = ?VRf(Qx) = ?fR(x)

?

which proves the claim.

?

Corollary 3. If α < α0and if T has a unique fixed point δ, then

δR= δ

for all R ∈ SO(3) i.e., δ is spherically symmetric.

Proof. Because of the preceding Lemma the claim follows from the uniqueness of

the fixed point of T.

??

The following is indicated in [2, p. 4].

Theorem 2. The energy functional Eg,αis invariant under rotations of the density

matrices, i.e., for all γ ∈ F and all R ∈ SO(3) we have

Eg,α(γ) = Eg,α(γR).

Page 12

12M. HUBER AND H. SIEDENTOP

Proof. We remark that 2E(γ,γ) = trX(γ)γ and 2D(ργ,ργ) = trφ(γ)γ. But for any

f ∈ H1(R3)4, any γ ∈ F and any R ∈ SO(3) we get, using (3), (4), (5)(6), and (7),

?

(fR,X(γR)fR) =

G

?

This proves the claim.

(fR,DgfR) =

G

VRf(Qx)

tVR(Dgf)(Qx)dx = (f,Dgf) (8)

?

VRf(Qx)

tVR(X(γ)f)(Qx)dx = (f,X(γ)f)(9)

(fR,φ(γR)fR) =

G

VRf(Qx)

tVR(φ(γ)f)(Qx)dx = (f,φ(γ)f)(10)

??

5. Solutions of the Dirac-Fock Equations Minimize the Energy of

Electron-Positron Field

In this section we show that the solution of the Dirac-Fock equations, which we

constructed above, yields a minimizer of the Dirac-Fock functional on the set of

all density matrices, if the quantization is chosen with respect to this solution. To

prove this, we need a technical remark:

Definition 2. Let γ ∈ F and P−an orthogonal projector. γ is called density matrix

with respect to P−, if and only if the operator inequality

0 ≤ γ + P−≤ 1

is fulfilled.

For density matrices with respect to P−the following lemma is valid:

Lemma 5. Let γ be a density matrix with respect to P− and let P+:= 1 − P−.

Then the following operator inequalities hold:

P−γP−P−γP−+ P−γP+P+γP−≤ −P−γP−

P+γP+P+γP++ P+γP−P−γP+≤ P+γP+.

Proof. The proof is a consequence of the fact that from 0 ≤ γ + P−≤ 1 it follows

that (γ + P−)2≤ γ + P−(see [1], Equations (18) and (19)).

With these preparations the main result of this section is a corollary of the

following theorem:

??

Theorem 3. Assume δ = χ[0,?(δ)

Moreover, assume 0 <π

N](D(δ)

g,α) and N = trδ. Let γ ∈?S(δ), and γ?:= γ−δ

Eg,α(γ) ≥ Eg,α(δ),

4cg,α,Nα < 1. Then

(11)

if one of the following conditions is fulfilled:

(1) The density matrix γ?is an orthogonal perturbation of δ, i.e., δγ?δ = 0.

(2) γ ∈?S(δ)

(12)

∂N, and the difference ?(δ)

N+1− ?(δ)

N+1− (1 +π

Nis so big that

(1 −π

4cg,α,Nα)?(δ)

4cg,α,Nα)?(δ)

N≥ 0.

N+1> ?(δ)

Proof. First of all we note that the hypothesis implies ?(δ)

N](D(δ)

the Dirac sea in such a way that the eigenfunctions of the occupied orbitals belong

to the Dirac sea.

We now choose the density matrix γ ∈?S(δ)arbitrarily. We have

N. We set P−:=

χ(−∞,?(δ)

g,α) and P+:= 1 − P−. This choice of the projectors means shifting

0 ≤ γ + Λ−= γ?+ δ + Λ−= γ?+ P−≤ 1,

Page 13

DIRAC-FOCK EQUATIONS OF THE ELECTRON-POSITRON FIELD 13

i.e., γ?is a density matrix with respect to P−, where Λ+:= Λ(δ)

We now plug γ into the functional and get

+ and Λ−:= Λ(δ)

−.

Eg,α(γ) = tr(Dg(γ?+ δ)) + αQ(γ?+ δ,γ?+ δ)

= tr(Dgδ) + αQ(δ,δ) + tr(Dgγ?) + 2αQ(δ,γ?) + αQ(γ?,γ?)

= Eg,α(δ) + trD(δ)

Moreover,

g,αγ?+ αQ(γ?,γ?) ≥ Eg,α(δ) + trD(δ)

g,αγ?− αE(γ?,γ?).

γ?γ?= P+γ?P+γ?P++ P+γ?P+γ?P−+ P+γ?P−γ?P++ P+γ?P−γ?P−

+ P−γ?P−γ?P−+ P−γ?P−γ?P++ P−γ?P+γ?P−+ P−γ?P+γ?P+.

Using Lemma 10 and Lemma 12 we calculate E(γ?,γ?) because of the inequalities

of Lemma 5. Also all terms of the form

tr(|D(δ)

g,α|

1

2P−γ?P+P+γ?P+|D(δ)

g,α|

g,α.

1

2)

vanish, since the spectral projectors commute with D(δ)

E(γ?,γ?) ≤π

4tr(γ?|∇|γ?) ≤π

≤π

4cg,α,Ntr(γ????D(δ)

=π

4cg,α,Ntr

g,α

???γ?)

4cg,α,Ntr

?

|D(δ)

g,α|

1

2(P+γ?P+− P−γ?P−)?D(δ)

?

g,α|

1

2

?

D(δ)

g,α(P+γ?P++ Λ−γ?Λ−− δγ?δ)

?

Moreover,

trD(δ)

g,αγ?= tr

?

D(δ)

g,α(P+γ?P++ Λ−γ?Λ−+ δγ?δ

?

,

i.e., we get altogether

Eg,α(γ) = Eg,α(δ) + trD(δ)

g,αγ?+ αQ(γ?,γ?)

≥ Eg,α(δ) +?1 −π

4cg,α,Nα?trD(δ)

g,αP+γ?P+

4cg,α,Nα?trD(δ)

+?1 +π

4cg,α,Nα?trD(δ)

g,αδγ?δ +?1 −π

g,αΛ−γ?Λ−.

We see that the energy grows, if γ?is an orthogonal perturbation of δ, because in

this case δγ?δ = δ(γ − δ)δ = 0. This shows the first part of the claim.

We prove now the second claim: from now assume γ ∈?S(δ)

trγ?= trΛ−γ?Λ−+ trδγ?δ + trP+γ?P+,

∂N. Since trγ = trδ,

we have trγ?= 0. Moreover,

such that

0 ≤ −trΛ−γ?Λ−= trδγ?δ + trP+γ?P+,

because Λ−γ?Λ−= Λ−γΛ−is a negative operator. It follows

trδγ?δ ≥ −trP+γ?P+.

Note that trδγ?δ ≤ 0 holds. We even have −1 ≤ δγ?δ ≤ 0, since

(f,δγ?δf) = (fDF,δγ?δfDF)

= (fDF,δγδfDF) − (fDF,δfDF) ≤ (fDF,Λ+fDF) − (fDF,fDF) = 0

and

(f,δγ?δf) = (fDF,δγ?δfDF) = (fDF,δγδfDF) − (fDF,δfDF)

≥ −(fDF,Λ−fDF) − (fDF,fDF) = −(fDF,fDF) ≥ −(f,f),

where we set fDF:= δf.

Page 14

14M. HUBER AND H. SIEDENTOP

Let now λi, i = 1,...,N, and fi, i = 1,...,N, eigenvalues and the corresponding

eigenvectors of δγ?δ, and λi,i > N, and fi,i > N, eigenvalues and corresponding

eigenvectors of P+γ?P+. Then

trD(δ)

g,αδγ?δ =

N

?

i=1

λi

?

fi,D(δ)

g,αfi

?

≥ ?(δ)

Ntrδγ?δ ≥ −?(δ)

NtrP+γ?P+

and

trD(δ)

g,αP+γ?P+=

∞

?

i=N+1

λi

?

fi,D(δ)

g,αfi

?

≥ ?(δ)

N+1trP+γ?P+

hold. It follows that

(13)Eg,α(γ) − Eg,α(δ) ≥ (1 −π

4cg,α,Nα)trD(δ)

+ (1 +π

g,αP+γ?P+

4cg,α,Nα)trD(δ)

g,αδγ?δ

≥ (1 −π

4cg,α,Nα)?(δ)

N+1trP+γ?P+− (1 +π

= [(1 −π

4cg,α,Nα)?(δ)

4cg,α,Nα)?(δ)

NtrP+γ?P+

4cg,α,Nα)?(δ)

N+1− (1 +π

N]trP+γ?P+

Since trP+γ?P+= trP+γP+≥ 0, this shows the claim.

Setting a := (2πN2− 1/8π2N)?0

1/4π)?0

??

N, b := ((4N − 1/4π − 1/2πN)?0

N)(bg)2we define

?

N− (4N +

N+1)bgand c := (?0

N+1− ?0

kg,N:= (−b −

b2− 4ac)(2a)−1.

Corollary 4. Assume N ∈ CS, δ as in Theorem 3, and α ≤ min{kg,N,

(14)Eg,α(δ) = EDF

Proof. It suffices to verify (12) of Theorem 3. By Lemma 2 it suffices to show

?1 −π

in order to fulfill inequality (12) of Theorem 3. This condition leads to the quadratic

equation

?2πN2−1

whose relevant smaller solution is given by kg,N.

bg

4N}. Then

N

:= inf{Eg,α(γ)|γ ∈?S(δ)

N+1−?1 +π

∂N}.

4cg,α,Nα??0

4cg,α,Nα?· (1 +π

2αNb−1

g)?0

N≥ 0

8π2N??0

Nα2+??4N−1

4π−1

2πN??0

N−?4N+1

4π??0

N+1

N+1− ?0

?

?bgα

+??0

N

?(bg)2= 0

?

We close with some remarks:

(1) In the spirit of Mittleman the ground state energy EM

electrons in the field of a nucleus with coupling constant g in Hartree-Fock

approximation is defined as

Nof N relativistic

Ge”andert

(15)EM

N= sup{inf{Eg,α(γ)|γ ∈?S(δ)

there – that EM

tion to closed shells – for α small enough (Barbaroux et al. [2, Formula

(13)]), thus confirming that these different definitions of the ground state

energies for closed shell atoms agree in the non-relativistic limit. In fact

this shows even the BES conjecture in this case. (See Barbaroux et al. [2,

Theorem 5]).

(2) Since the BES conjecture is false in the open shell case [2, 4], the previous

remark shows also that the restriction to the closed shell case is not of mere

technical nature.

∂N}|δ ∈ F, δ = δ2, trδ = N}.

Corollary 4 together with Equation (11) shows – under the hypotheses made

N≥ EDF

N. The reverse inequality is valid – without restric-

Eingef”ugt

Page 15

DIRAC-FOCK EQUATIONS OF THE ELECTRON-POSITRON FIELD 15

(3) Under the assumption that δ has a spherically symmetric density ρδ, one

can show the assertion of Corollary 4 even for bigger α, using Lemma 3

instead of Lemma 2. The results of this (numerical) computation are shown

additionally in Figure 3.

0.01

0.008

0.006

60

0

100 20

0.012

0.004

0.002

80400

with sph. symmetry

without sph. symmetry

fine structure const.

Figure 3. The maximal value of α for which we can guarantee

that a projection δ ∈ DF onto the lowest eigenvalues of D(δ)min-

imizes the energy in dependence on the nuclear charge Z = 137g

and N = 2.

Appendix A. The Minimax Principle of Griesemer and Siedentop

In [8] the following minimax principle for eigenvalues of self-adjoint operators in

spectral gaps was proven:

Theorem 4. Suppose that A is a self-adjoint operator in a Hilbert space h =

h+⊕ h− where h+ ⊥ h−. Let Λ± be the orthogonal projectors onto h± and let

Q be a subspace with D(A) ⊂ Q ⊂ Q(A) and Λ±h ⊂ Q, where D(A) and Q(A)

denote operator domain and form domain of A respectively. Let P+:= χ(0,∞)(A),

P−:= χ(−∞,0)(A), Q±:= Q ∩ h±, and

λn(A) := inf

M+⊂Q+

dim(M+)=n

sup

φ∈M+⊕Q−

?φ?=1

(φ,Aφ).

(1) If (φ,Aφ) ≤ 0 for all φ ∈ Q−, then

λn(A) ≤ µn(A|P+h)

(2) If (φ,Aφ) > 0 for all non-vanishing φ ∈ Q(A)∩h+and (|A|+1)

bounded, then

1

2P−Λ+is

λn(A) ≥ µn(A|P+h).

Here the µndenote the standard (Courant) minimax values of an operator bounded

from below.

Page 16

16M. HUBER AND H. SIEDENTOP

Appendix B. Properties of Dirac-Fock Operators

We list some useful inequalities from [1] and [3] and some slight improvements

of these.

Lemma 6. For any γ ∈ F we have

(16)

φ(γ)≤π

φ(γ)≤π

2?|∇|γ?1≤π

2?γ?1|∇| ≤π

2?|D0|γ?1=π

2?γ?1|D0|.

2?γ?F,

(17)

Lemma 7. If γ ∈ F, then

X(γ)≤ φ(|γ|), ?X(γ)? ≤ ?φ(|γ|)?, and ?W(γ)? ≤ ?φ(|γ|)?.

Proof. We prove only the third statement. Let γ = γ+− γ−, i.e., γ+and γ−are

the positive and negative parts of γ respectively. Then

W(γ)= φ(γ+)− φ(γ−)− X(γ+)+ X(γ−)≤ φ(γ+)+ X(γ−)≤ φ(γ+)+ φ(γ−)= φ(|γ|),

where we used Lemmata 8 and 7. In the same way we get W(γ)≥ −φ(γ−)−

X(γ+)≥ −φ(γ−)− φ(γ+)= −φ(|γ|), so |(f,W(γ)f) ≤ (f,φ(|γ|)f) for all f ∈ H. This

immediately implies the claim.

??

Lemma 8. Let 0 ≤ γ ∈ F, then 0 ≤ X(γ)≤ φ(γ); in particular 0 ≤ W(γ).

An immediate consequence of the preceding lemmata is

Lemma 9. If γ = γ∗∈ S1(H) and γ?∈ F, then

|D(ργ,ργ?)| ≤π

E(γ,γ?) ≤D(ρ|γ|,ρ|γ?|).

4?γ?1tr(|∇||γ|),

We also need

Lemma 10 (Bach et al. [1]). For all γ ∈ F we have E(γ,γ) ≤π

Lemma 11. Pick γ ∈ F, g ∈ (−√3/2,√3/2), α ∈ R. W(γ)is relatively compact

with respect to D0. The operator D(γ)

H1(R3)4and

σess(D(γ)

(1) Set Cg:= (?4g2+ 9 − 4g)/3 and, for 0 ≤ g <√3/2,

dg:= (1 + C2

(1 − C2

Then, we for g ∈ [0,√3/2] according to Morozov ([10])

|Dg|2≥ d2

If we assume in addition γ ∈ F and dg− 4|α|?γ?1> 0, then

|D(γ)

(2) Setting bg:=

|Dg|2≥ b2

Assuming in addition bg− 4|α|?γ?1> 0 implies the inequalities

|D(γ)

|D(γ)

Acknowledgement: We thank Jean-Marie Barbaroux for explaining the proof

of Equation (13) in [2]. We acknowledge partial support through the European

Union’s IHP network Analysis & Quantum, HPRN-CT-2002-0277.

4tr(γ|∇|γ).

g,α is self-adjoint with D(D(γ)

g,α) = D(Dg) =

g,α) = σess(Dg) = (−∞,−1] ∪ [1,∞).

Lemma 12.

g−

?

g)2+ 4g2C2

g)/2.

g|D0|2.

g,α|2≥ (dg− 4|α|?γ?1)2|D0|2

?1 − g2(?4g2+ 9 − 4g)/3 and g ∈ (0,√3/2) we have

g|∇|2.

g,α|2

g,α|2

≥

≥

(bg− 4|α|?γ?1)2|∇|2,

(1 − 4|α|?γ?1b−1

g)2|Dg|2.

Page 17

DIRAC-FOCK EQUATIONS OF THE ELECTRON-POSITRON FIELD17

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Mathematik, Theresienstraße 39, 80333 M¨ unchen, Germany

E-mail address: mhuber@math.lmu.de and h.s@lmu.de