The Energy of Heavy Atoms According to Brown

and Ravenhall: The Scott Correction

Rupert L. Frank, Heinz Siedentop, and Simone Warzel

Received: March 9, 2009

Communicated by Alfred K. Louis

Abstract. We consider relativistic many-particle operators which – according to Brown and Ravenhall – describe the electronic states of heavy atoms. Their ground state energy is investigated in the limit of large nuclear charge and velocity of light. We show that the leading quasi-classical behavior given by the Thomas-Fermi theory is raised by a subleading correction, the Scott correction. Our result is valid for the maximal range of coupling constants, including the critical one. As a technical tool, a Sobolev-Gagliardo-Nirenberg-type inequality is established for the critical atomic Brown-Ravenhall operator. Moreover, we prove sharp upper and lower bounds on the eigenvalues of the hydrogenic Brown-Ravenhall operator up to and including the critical coupling constant.

2000 Mathematics Subject Classification: Primary 81V45; Secondary 81V55, 35Q40, 46N50, 47N50. Keywords and Phrases: Heavy atoms, ground state energy, relativistic Coulomb system, Scott correction, Brown-Ravenhall operator.

Contents

1 Introduction and main result 464 1.1 Relativistic energy form . . . . . . . . . . . . . . . . . . . . . . 466 1.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 1.3 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . 469 2 The hydrogenic Brown-Ravenhall operator 470

2.1 Estimates on eigenvalues of the hydrogen atom . . . . . . . . . 470 2.1.1 Upper bound on hydrogen eigenvalues . . . . . . . . . . 472 2.1.2 Lower bounds on hydrogen eigenvalues. Subcritical case 473 2.1.3 Lower bounds on hydrogen eigenvalues. Critical case . . 474

2.2 Sobolev inequality for the critical Brown-Ravenhall operator . . 476 2.2.1 Comparison of critical operators . . . . . . . . . . . . . 477 2.2.2 Proof of the Sobolev inequality . . . . . . . . . . . . . . 479

2.3 Estimates on the electric potential . . . . . . . . . . . . . . . . 480

3 Spectral shift from Schrödinger to Brown-Ravenhall operators 483 3.1 Estimate on the spectral shift . . . . . . . . . . . . . . . . . . . 483 3.2 Properties of the spectral shift . . . . . . . . . . . . . . . . . . 486

4 Proof of the Scott correction 486 4.1 Upper bound on the energy difference . . . . . . . . . . . . . . 487 4.2 Lower bound on the energy difference . . . . . . . . . . . . . . 492

4.2.1 Controlling the electron projection for high angular mo-

menta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 4.2.2 Contribution from low angular momenta to the Coulomb

energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 4.2.3 Finishing the proof . . . . . . . . . . . . . . . . . . . . . 497

A Partial wave analysis 498

B Properties of the twisting operators 499

C Basics of relativistic hydrogenic operators 501 C.1 Massless case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 C.2 Massive case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

D Critical Chandrasekhar operator on a finite domain 503

E The trial density matrix 506 E.1 Low angular momenta . . . . . . . . . . . . . . . . . . . . . . . 506 E.2 High angular momenta . . . . . . . . . . . . . . . . . . . . . . . 507 E.3 Energy estimates for high angular momenta . . . . . . . . . . . 508

1 Introduction and main result

The description of atoms and molecules, in particular of their energies, has been a primer for the development of quantum mechanics. However, it became soon clear that atoms with more than one electron are not accessible to explicit solutions. This motivated the development of approximate models for large Coulomb systems. One of the most simple and – simultaneously – the most fundamental models was introduced by Thomas [73], Fermi [27, 28], and Lenz [41] who proposed the energy functional which we will also use here. It predicts that the ground state energy of atoms would decrease with the atomic number Z to leading order as Z^{7/3}. In order to get a refined description, Scott [54] conjectured that the electrons close to the nucleus should raise the energy by Z^{2}/2. Considerably later Schwinger [52] argued also for Scott’s prediction; Schwinger [53] and Englert and Schwinger [13, 14, 15] even refined these considerations by adding more lower order terms [53] (see also Englert [12]). In fact, a contribution to the Z^{5/3}-term can be traced back to Dirac [10]. — The challenge to address the question whether the predicted formulae would yield asymptotically correct results when compared with the N-particle Schrödinger theory was for a long time unsuccessful. It was Lieb and Simon who proved in their seminal paper [44] that the prediction of Thomas, Fermi, and Lenz is indeed asymptotically correct. However, establishing the Scott correction resisted the mathematical efforts and became Problem 10B of Simon’s 15 Problems in Mathematical Physics [62]. Eventually, the Scott correction was established mathematically by Hughes [37, 38] (lower bound), and Siedentop and Weikard [55, 56, 57, 58, 59] (lower and upper bound). In fact, even the existence of the Z^{5/3}-correction conjectured by Schwinger was proved by Fefferman and Seco [23, 24, 25, 18, 26, 21, 19, 20, 22]. Later these results were extended in various ways, e.g., to ions and molecules [1, 39, 66, 4]. Despite of the mathematical success in establishing the large Z asymptotics of the Schrödinger theory, these considerations remain questionable from a physical point of view, since large atoms force electrons into orbits that are close to the nucleus where the electrons move with high speed which should require a relativistic treatment. The atom is shrinking with increasing Z: already in nonrelativistic quantum mechanics the bulk of the electrons has a distance Z^{−1/3} from the nucleus; the electrons contributing to the Scott correction even live on the scale Z^{−1}. Schwinger [53] has estimated these effects concluding that a correction to the Scott correction occurs whereas the leading term should be unaffected by the change of model. Sørensen [51] was the first who proved that the latter is indeed the case for a simplified ad hoc naive relativistic model, the Chandrasekhar multi-particle operator, in the limit of large Z and large velocity of light c. The value of the Scott correction is again of order Z^{2}, a result which was announced [64] and proven [65] by Solovej, Sørensen, and Spitzer (see also Sørensen [50] for the non-interacting case). In a previous paper [31] we gave a short alternative proof, roling the problem back to the non-relativistic Scott correction. Nevertheless, a question from the physical point of view remains: Although the Chandrasekhar model is believed to represent some qualitative features of relativistic systems, there is no reason to assume that it should give quantitatively correct results. Therefore, to obtain not only qualitatively correct results it is interesting, in fact mandatory, to consider a Hamiltonian which – as the one by Brown and Ravenhall [6] – is derived from QED such that it yields the leading relativistic effects in a quantitative correct manner. (See also Sucher [69, 70, 71].) The first step in this direction was taken by Cassanas and Siedentop [7] who showed that, similarly to the Chandrasekhar case, the leading energy is not affected. To show in which way the Scott correction is changed for this model is our concern in this paper.

1.1 Relativistic energy form

According to Brown and Ravenhall [6] the energy of an atom with N electrons in a state ψ ∈ 𝔔_{N}^{B} is given by

E_{N}^{B}(ψ)

:= ⟨ψ, [∑_{ν=1}^{N} (c α_{ν} · p_{ν} + c^{2}β_{ν} − c^{2} − \frac{Z}{|x_{ν}|}) + ∑_{1≤μ<ν≤N} \frac{1}{|x_{μ} − x_{ν}|}] ψ⟩ . (1.1)

This involves the free Dirac operator reduced by the rest mass, acting in L^{2}(ℝ^{3}, ℂ^{4}), with the four Dirac matrices in standard representation,

α = (\begin{matrix}0 & σ \\ σ & 0 \end{matrix}) , β = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) ,

where σ are the three Pauli matrices in standard representation, i.e.,

σ_{1} = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , σ_{2} = (\begin{matrix}0 & −i \\ i & 0 \end{matrix}) , σ_{3} = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) .

We use atomic units in which m = e^{2} = ℏ = 1. The parameter Z is the atomic number and c the velocity of light. The Hilbert space of an electron is chosen as the positive spectral subspace of the Dirac operator,

ℌ^{B} := χ_{[c^{2},∞)}(c α · p + c^{2}β) (L^{2}(ℝ^{3}, ℂ^{4})) ,

and, correspondingly, the Hilbert space of N electrons ℌ_{N}^{B} is the antisymmetric tensor product of the one-particle space, i.e., ℌ_{N}^{B} := ⋀_{ν=1}^{N} ℌ^{B}. Finally, the form domain of (1.1) is 𝔔_{N}^{B} := ℌ_{N}^{B} ∩ 𝔖(ℝ^{3N}, ℂ^{4^{N}}) with 𝔖 the Schwartz space of rapidly decreasing functions. As is shown in [17], the Brown-Ravenhall form E_{N}^{B} is closable and bounded from below if and only if

κ := \frac{Z}{c} ≤ κ^{B} := \frac{2}{2/π + π/2}.  (1.2)

(See also Tix [74, 76], who improved the bound given in [17] to an explicit positive bound.) For the physical value, about 1/137, of the Sommerfeld fine structure, which equals 1/c in atomic units used here, the critical atomic number Z exceeds 124 slightly. This includes all known elements.

In the following we will assume that the atom described by (1.1) is neutral, i.e., Z = N, an assumption that we make mainly for the sake of brevity and clarity of presentation, since the Scott correction is independent of the ionization degree N/Z ≥ const > 0. Similarly, it might seem that our treatment is restricted to spherically symmetric systems (atoms). However, on the energy scale considered here, molecular Hamiltonians essentially separate – in nature the distances between nuclei with charges ZZ_{1}, ..., ZZ_{K} remain on a scale much larger than Z^{−1/3} – into spherically symmetric one-center problems (atoms). Therefore, the molecular case follows from the atomic case by additional localization. However, for the sake of brevity and clarity, we will spare the reader the corresponding tedious technicalities, restrict to the atomic case, and freely use the resulting symmetry. Thus, according to Friedrichs, the one-particle form E_{1}^{B} defines for κ ≤ κ^{B} a distinguished self-adjoint operator in ℌ^{B}. Through a unitary transformation it may be represented as a self-adjoint operator in the Hilbert space ℌ := L^{2}(ℝ^{3}, ℂ^{2}) of two-spinors. More precisely, using the notation p := |p|, ω_{p} := p/p we set

E(p) :=\sqrt{p^{2} + 1},  ϕ_{ν}(p) := \sqrt{\frac{E(p) + (−1)^{ν}}{2E(p)}} , ν = 0, 1,  (1.3)

and introduce the following bounded operators on ℌ,

Φ_{0}(p) := ϕ_{0}(p), Φ_{1}(p) := ϕ_{1}(p) σ · ω_{p}.  (1.4)

The operator Φ_{c} : ℌ → ℌ^{B}, ψ ↦ (Φ_{0}(p/c) ψ, Φ_{1}(p/c) ψ), maps ℌ unitarily onto ℌ^{B} [7]. Therefore, the form E_{1}^{B} defines the (two-spinor) Brown-Ravenhall operator in ℌ,

B_{c}[Z/|x|] :=Φ_{c}^{−1} (cα · p + c^{2}β − c^{2} − Z/|x|) Φ_{c}
=c^{2}E(p/c) − c^{2} − U_{c}(Z/|x|),  (1.5)

where U_{c}(A) := Φ_{0}(p/c) A Φ_{0}(p/c) + Φ_{1}(p/c) A Φ_{1}(p/c). In the case c = 1 we denote this operator by B_{Z}. Further properties of B_{Z} and its relation to the corresponding Chandrasekhar operator and Schrödinger operator

C_{Z} := (p^{2} + 1)^{1/2} − 1 − Z/|x|,  S_{Z} := \frac{1}{2}p^{2} − Z/|x|  (1.6)

all realized in ℌ, can be found in Sections 2 and 3 below and in Appendix C.

1.2 Main result

We are interested in the ground state energy

E_{c}^{B}(Z) := inf{E_{Z}^{B}(ψ) | ψ ∈ 𝔔_{Z}^{B}, ∥ψ∥ = 1}

of the energy form (1.1) for large atomic number Z and large velocity of light c satisfying (1.2). Note that we picked N = Z. It was shown in [7], that similarly to the Chandrasekhar case [51], the leading behavior of E_{c}^{B}(Z) is not affected by relativistic effects and is, as in the Schrödinger case [44], given by the minimal Thomas-Fermi energy

E_{TF}(Z) := inf{E_{TF}(ρ) | ρ ∈ L^{5/3}(ℝ^{3}), ρ ≥ 0, D(ρ, ρ) < ∞}.  (1.7)

The latter is defined in terms of the Thomas-Fermi energy functional

E_{TF}(ρ) := ∫_{ℝ^{3}} [\frac{3}{5}γ_{TF} ρ(x)^{5/3} − \frac{Z}{|x|}ρ(x)] dx + D(ρ, ρ)

where, in our units, γ_{TF} = (3π^{2})^{2/3}/2 and

D(ρ, σ) := \frac{1}{2} ∫_{ℝ^{3}} ∫_{ℝ^{3}} \frac{\overline{ρ(x)}σ(y)}{|x − y|} dxdy

is the Coulomb scalar product. By scaling, one finds E_{TF}(Z) = E_{TF}(1) Z^{7/3}. This paper concerns the correction to the leading behavior. For the formulation of the main result, we abbreviate the negative part of an operator by A_{−} := −Aχ_{(−∞,0)}(A) and introduce for 0 < κ ≤ κ^{B} the spectral shift

s(κ) := κ^{−2} tr_{ℌ} [(B_{κ})_{−} − (S_{κ})_{−}] .  (1.8)

(We use the term “spectral shift” for s for convenience although it is used in slightly different meaning otherwise.) It describes the shift of the Brown-Ravenhall bound state energies compared to those of the Schrödinger operator. In Section 3 we show that s is well-defined and discuss some of its properties. In particular, we prove that the function s is non-negative on the interval (0, κ^{B}] and can be continuously extended to zero where it satisfies

s(κ) = O(κ^{2})  as κ → 0.  (1.9)

We are now ready to state our main result.

Theorem 1.1 (Scott correction). There exists a constant C > 0 such that for all Z ≥ 1 and all c ≥ Z/κ^{B} one has

|E_{c}^{B}(Z) − E_{TF}(Z) − (\frac{1}{2} − s(Z/c)) Z^{2}| ≤ CZ^{47/24}.  (1.10)

Put differently, Theorem 1.1 asserts that in the limit Z → ∞ we have uniformly in the quotient κ = Z/c ∈ (0, κ^{B}]

E_{c}^{B}(Z) = E_{TF}(Z) + (\frac{1}{2} − s(κ)) Z^{2} + o(Z^{2}).  (1.11)

(We do not claim that the error Z^{47/24} in (1.10) is sharp, so we only write o(Z^{2}) here.) The second term (\frac{1}{2} − s(κ)) Z^{2} in (1.11) is the so-called Scott correction in the Brown-Ravenhall model. It does not exceed the Scott correction Z^{2}/2 in the non-relativistic model [55]. Indeed, if κ = Z/c stays away from zero then there is a relativistic lowering of the ground state energy at order Z^{2}. On the other hand, in the non-relativistic limit c → ∞ with κ = Z/c → 0, one recovers – non-surprisingly – the value of the Schrödinger case. In this case (1.9) implies

E_{c}^{B}(Z) = E_{TF}(Z) + \frac{1}{2}Z^{2} + O(c^{−2}Z^{4} + Z^{47/24}).  (1.12)

The Scott correction in the Brown-Ravenhall model, however, exceeds the Scott correction predicted by the naive Chandrasekhar model treated in [65] and [31]. This follows from the fact that sums of bound state energies of the atomic Chandrasekhar operator are dominated by those of the Brown-Ravenhall operator, cf. the proof of Theorem 3.1 below.

1.3 Outline of the paper

The central strategy of our paper is to compare the ground state energy of the Brown-Ravenhall operator with that of the Schrödinger operator. The latter is known up to the required accuracy o(Z^{2}) and the leading contribution agrees with the Brown-Ravenhall energy. The subtraction of the corresponding ground state energies results in a renormalized effective model which accurately describes the energy differences and is amenable to analysis. The germ of this idea has been presented in the simpler context of the Chandrasekhar model [31]. The full blown renormalization required is developed in this paper. A virtue of our approach is that it leads to an explicit formula for the spectral shift which can be evaluated numerically. We believe it would be interesting to compare this formula with experimental data. We show that the difference between the Brown-Ravenhall and Schrödinger ground state energies on the multi-particle level coincides, up to the required accuracy, with a spectral shift on the one-particle level. A crucial step in our analysis is therefore a bound on the corresponding spectral shift for rather general spherically symmetric potentials. This is presented in Section 3, where we show that sums of differences of Brown-Ravenhall and Schrödinger eigenvalues decay rather rapidly as the angular momentum increases. In Section 2 we address various aspects of hydrogenic Brown-Ravenhall operators. An essential feature and source of difficulties, which does not occur in the naive Chandrasekhar model, is the non-locality of the potential energy. In particular, instead of the usual Coulomb potential |x|^{−1} we face the ‘twisted’ non-local operator U_{c}(|x|^{−1}). Estimating the difference between the corresponding potential energies is the topic in Subsection 2.3. Since, in contrast to the Schrödinger case, the eigenvalues of the hydrogenic Brown-Ravenhall operator are not known explicitly, we prove upper and lower bounds in Subsection 2.1. Our bounds are sharp with respect to their dependence on the quantum numbers n and l. An upper bound is given by the Dirac eigenvalues, a consequence of the mini-max principle for eigenvalues in the gap. For the lower bound we overcome the non-locality of the potential by a non-trivial comparison argument with a super-critical Chandrasekhar operator. In Subsection 2.2 we prove a new Sobolev-type inequality, from which we derive estimates on the eigenfunctions of the hydrogenic Brown-Ravenhall operator. The technical challenge here is to prove such a result up to and including the critical coupling constant. Finally, we present the proof of our main result, Theorem 1.1, in Section 4. For the readers’ convenience we collect various facts in the appendices. Appendix A recalls the partial wave decomposition of the Hilbert space of twospinors, Appendix B establishes some useful properties of the twisting operators, and Appendix C collects basic facts on hydrogenic Brown-Ravenhall and Chandrasekhar operators. Appendix D fills in some details in the proof of Theorem 2.2 and, eventually, Appendix E defines the one-particle density matrix giving the main contribution of the energy.

2 The hydrogenic Brown-Ravenhall operator

In this section we set c = 1 and investigate the Brown-Ravenhall operator with Coulomb potential

B_{κ} = \sqrt{p^{2} + 1} − 1 − κU(|x|^{−1})  (2.1)

in the Hilbert space ℌ = L^{2}(ℝ^{3}, ℂ^{2}) of two-spinors, where we recall that

U(|x|^{−1}) = Φ_{0}(p)|x|^{−1}Φ_{0}(p) + Φ_{1}(p)|x|^{−1}Φ_{1}(p)  (2.2)

with Φ_{ν} defined in (1.4). In Subsection 2.1 we prove sharp upper and lower bounds on the eigenvalues of B_{κ}. In Subsection 2.2 we prove L^{p} estimates on the eigenfunctions of this operator. Technically, this is expressed as a Sobolevtype inequality for the massless version of B_{κ}, which is a non-negative operator. Finally, in Subsection 2.3 we compare the potential energy of the operator B_{κ}, namely ⟨ψ, U(|x|^{−1})ψ⟩, with the corresponding local potential energy ⟨ψ, |x|^{−1}ψ⟩. For comparison purpose also the corresponding Chandrasekhar and Schrödinger operator C_{κ} and S_{κ} occur (see (1.6)). According to [17] and [40] the operators B_{κ} and C_{κ} are well-defined for all κ ≤ κ^{#} with # = B, C and

κ^{B} = \frac{2}{2/π + π/2},  κ^{C} := 2/π;  (2.3)

see also Appendix C. Of course, for the Schrödinger operator no upper bound on κ is needed.

2.1 Estimates on eigenvalues of the hydrogen atom

In contrast to the Schrödinger or Dirac models, the eigenvalues of B_{κ} and C_{κ} are not known explicitly. In order to obtain upper and lower bounds on these eigenvalues, we use that the spectra of B_{κ}, C_{κ} and S_{κ} may be classified in terms of angular momenta. As usual write L := x × p for the operators of orbital angular momentum and J := L + \frac{1}{2}σ for the operators of total angular momentum. The four operators B_{κ}, J^{2}, J_{3}, L^{2} commute pairwise, and this also holds, if C_{κ} or S_{κ} replace B_{κ}. This allows us to decompose the Hilbert space ℌ into orthogonal subspaces which reduce such a quadruple of operators, i.e.,

ℌ = ⨁_{j∈ℕ_{0}+ \frac{1}{2}} ⨁_{l=j±1/2} ℌ_{j,l},  ℌ_{j,l} := ⨁_{m=−j}^{j} ℌ_{j,l,m}.  (2.4)

Here ℌ_{j,l,m} is the maximal joint eigenspace of J^{2} with eigenvalues j(j + 1), of L^{2} with eigenvalue l(l + 1), and J_{3} with eigenvalue m. More details concerning the partial wave decomposition (2.4) can be found in Appendix A. We denote by b_{j,l}(κ), c_{l}(κ), and s_{l}(κ) the reduced operators corresponding to fixed angular momenta j and l, where, strictly speaking, we consider b_{j,l}(κ) and c_{l}(κ) in momentum space whereas s_{l}(κ) in position space. We refer to Appendix C for precise definitions of b_{j,l}(κ) and c_{l}(κ) and for further discussion. Of course, s_{l}(κ) = −\frac{1}{2} \frac{d^{2}}{dr^{2}} + \frac{l(l+1)}{2r^{2}} − \frac{κ}{r}. The main result of this subsection is that for large quantum numbers n, j, and l, the eigenvalues of b_{j,l}(κ) and c_{l}(κ) behave similarly to the explicitly known ones of the Schrödinger operator s_{l}(κ).

Theorem 2.1 (Energies of Brown-Ravenhall hydrogen). There is a constant C < ∞ such that for all j ∈ ℕ_{0} + \frac{1}{2}, and l = j ± \frac{1}{2}, n ∈ ℕ and κ ∈ (0, κ^{B}] one has

−C \frac{κ^{2}}{(n + l)^{2}} ≤ λ_{n}(b_{j,l}(κ)) ≤ −\frac{κ^{2}}{2(n + l)^{2}} .  (2.5)

Here and below, we denote by λ_{1}(A) ≤ λ_{2}(A) ≤ … the eigenvalues, repeated according to multiplicities, below the bottom of the essential spectrum of the self-adjoint, lower semi-bounded operator A. Note that −κ^{2}(2(n + l)^{2})^{−1} = λ_{n}(s_{l}(κ)) on the right hand side of (2.5) is the n-th eigenvalue of the Schrödinger operator corresponding to angular momentum l. In particular, we conclude from (2.5) that the partial traces in ℌ_{j,l} (cf. (3.1)) satisfy

0 ≤ tr_{j,l} ([B_{κ} + μ]_{−} − [S_{κ} + μ]_{−}) < ∞  (2.6)

for all μ ≥ 0. In the proof of Theorem 2.1 we use heavily the corresponding result for the Chandrasekhar case, which we state next.

Theorem 2.2 (Energies of Chandrasekhar hydrogen). There is a constant C < ∞ such that for all l ∈ ℕ_{0}, n ∈ ℕ and κ ∈ (0, κ^{C}] one has

−C \frac{κ^{2}}{(n + l)^{2}} ≤ λ_{n}(c_{l}(κ)) ≤ −\frac{κ^{2}}{2(n + l)^{2}} .  (2.7)

We break the proofs of Theorems 2.1 and 2.2 into three parts, corresponding to the upper bound and the lower bound for subcritical and, respectively, critical values of the coupling constant.

2.1.1 Upper bound on hydrogen eigenvalues

We begin with the Chandrasekhar case.

Proof of Theorem 2.2. Upper bound. The second inequality in (2.7) is an immediate consequence of the inequality \sqrt{p^{2} + 1}−1 ≤ p^{2}/2 and the known form of the Schrödinger eigenvalues in the subspace corresponding to fixed angular momentum l.

Next, we turn to the Brown-Ravenhall case.

Proof of Theorem 2.1. Upper bound. We first recall some facts about the eigenvalues of the hydrogenic Dirac operator D_{κ} := α · p + β − κ|x|^{−1}; see Darwin [8], Gordon [32] and also Bethe and Salpeter [5] for a textbook presentation. The following subspaces of L^{2}(ℝ^{3}, ℂ^{4}),

ℌ̃_{j,l,m} = \{x ↦ (\begin{matrix}ir^{−1}f(r)Ω_{j,l,m}(ω_{x}) \\ −r^{−1}g(r)Ω_{j,2j−l,m}(ω_{x}) \end{matrix}) : f, g ∈ L^{2}(ℝ_{+})\} ,

reduce the Dirac operator D_{κ} with κ ∈ (0, 1). Under the natural identification of ℌ̃_{j,l,m} with L^{2}(ℝ_{+}, ℂ^{2}) the part of D_{κ} in ℌ̃_{j,l,m} is unitarily equivalent to

d_{j,l}(κ) = (\begin{matrix}1 − \frac{κ}{r} & −\frac{d}{dr} − \frac{(j−l)(2j+1)}{r} \\ \frac{d}{dr} − \frac{(j−l)(2j+1)}{r} & −1 − \frac{κ}{r} \end{matrix}) .

The non-decreasing sequence λ_{n}(d_{j,l}(κ)) of eigenvalues of d_{j,l}(κ) in the gap (−1, 1) is independent of l and given explicitly by

λ_{n}(d_{j,l}(κ)) = (1 − \frac{κ^{2}}{(n − 1 + \sqrt{(j + 1/2)^{2} − κ^{2}})^{2} + κ^{2}} )^{1/2} , n ∈ ℕ. (2.8)

The Dirac eigenvalues reduced by the rest energy are bounded from above by the Schrödinger eigenvalues: for all n, l, j, and κ ∈ (0, 1)

1 − λ_{n}(d_{j,l}(κ)) ≥ \frac{κ^{2}}{2(n+l)^{2}} = −λ_{n}(s_{l}(κ)).  (2.9)

To show (2.9), we use

\sqrt{(j + 1/2)^{2} − κ^{2}} ≤ \sqrt{(l + 1)^{2} − κ^{2}} ≤ \sqrt{(n + l)^{2} − κ^{2}} + 1 − n

and expand the outer square root in (2.8) up to first order which gives an upper bound. Hence the upper bound in Theorem 2.1 will follow if we can show that

λ_{n}(b_{j,l}(κ)) ≤ −1 + λ_{n}(d_{j,l}(κ)).  (2.10)

To prove this, we fix (j, l) and abbreviate Λ_{+} := χ_{[1,∞)}(d_{j,l}(0)) and Λ_{−} := 1 − Λ_{+}. It follows from the definition of the Brown-Ravenhall operator that b_{j,l}(κ) is unitarily equivalent to the operator Λ_{+}(d_{j,l} − 1)Λ_{+} in the Hilbert space Λ_{+}(L^{2}(ℝ_{+}, ℂ^{2})). The variational principle for eigenvalues in gaps by Griesemer et al. [34, 35] under the weakened hypotheses of Dolbeault et al.

[11] states that

λ_{n}(d_{j,l}(κ))

= inf_{{V ⊂Λ_{+}(L^{2}(ℝ_{+},ℂ^{2})), \atop dim V =n}} sup \{\frac{(f, d_{j,l}(κ)f)}{∥f∥^{2}} : 0 ≠ f ∈ V ⊕ Λ_{−}(L^{2}(ℝ_{+}, ℂ^{2}))\} .

Since the supremum decreases when restricted to 0 ≠ f ∈ V, one obtains (2.10).

2.1.2 Lower bounds on hydrogen eigenvalues. Subcritical case

Proof of Theorem 2.2. Subcritical case. Since we will reduce the Brown-Ravenhall case in Theorem 2.1 to the Chandrasekhar case, we actually prove a slightly stronger statement. As explained in (C.8), the operators c_{l}(κ) are lower bounded for all l ≥ 1 up to κ_{l}^{C} > κ^{B}. We assume that either l ≥ 1 and 0 < κ ≤ κ^{B} or else that l = 0 and 0 < κ ≤ κ^{B}κ^{C}/κ_{1}^{C}. For any 0 < δ < 1 there exist M_{δ} > 0 and c_{δ} > 0 such that

\sqrt{p^{2} + 1} − 1 ≥ \begin{cases}(1 − δ)p & if p ≥ M_{δ} \\ c_{δ} p^{2}/2 & if p ≤ M_{δ}. \end{cases} .

Denoting by χ_{i} the characteristic function of the centered ball in ℝ^{3} with radius M_{δ}, and putting χ_{o} := 1 − χ_{i}, the Schwarz inequality implies the operator inequality

|x|^{−1} ≤ (1 + δ^{−1})χ_{i}(p)|x|^{−1}χ_{i}(p) + (1 + δ)χ_{o}(p)|x|^{−1}χ_{o}(p),

and hence

\sqrt{p^{2} + 1} − 1 − κ|x|^{−1} ≥χ_{i}(p) (c_{δ}p^{2}/2 − (1 + δ^{−1})κ|x|^{−1}) χ_{i}(p) (2.11)
+ χ_{o}(p) ((1 − δ)|p| − (1 + δ)κ|x|^{−1}) χ_{o}(p) .

Now choose δ as the the unique solution of the equation (1+δ)/(1−δ) = κ_{1}^{C}/κ^{B} in the interval (0, 1). Then the restrictions on κ imply that (1 + δ)κ ≤ (1 − δ)κ_{1}^{C} ≤ (1 − δ)κ_{l}^{C} for l ≥ 1 and (1 + δ)κ ≤ (1 − δ)κ^{C} for l = 0. In any case, the second operator in the above sum is non-negative. The variational principle hence implies that the n-th eigenvalue of c_{l}(κ) is greater or equal to the n-th eigenvalue of χ_{i}(p) (c_{δ}p^{2}/2 − (1 + δ^{−1})κ|x|^{−1}) χ_{i}(p). Again by the variational principle, the latter is greater or equal to the n-th eigenvalue of c_{δ}p^{2}/2 − (1 + δ^{−1})κ|x|^{−1}, which is −const κ^{2}(n + l)^{−2}.

Proof of Theorem 2.1. Subcritical case. We assume that either j ≥ 3/2 and 0 < κ ≤ κ^{B} or else that j = 1/2 and 0 < κ ≤ κ^{B}κ^{C}/κ_{1}^{C}. We claim that

λ_{n}(c_{l}(κ)) = λ_{2n−1}(c_{l}(κ) ⊗ 1_{ℂ^{2}}) ≤ λ_{2n−1}(b_{j,l}(κ)) .  (2.12)

Once we have proved this, the assertion follows easily from what we have shown in the proof of Theorem 2.2 above. To establish (2.12) we use the same notation as in the proof of the upper bound in Theorem 2.1. By the variational principle,

λ_{n}(b_{j,l}(κ))

= sup_{\begin{matrix}f_{1},...,f_{n−1}∈ \\ Λ_{+}(L^{2}(ℝ_{+},ℂ^{2})) \end{matrix}}inf\{⟨f, (d_{j,l}(κ) − 1)f⟩ : ∥f∥ = 1, f ∈ Λ_{+}(L^{2}(ℝ_{+}, ℂ^{2})), f ⊥ f_{ν}\}

= sup_{\begin{matrix}f_{1},...,f_{n−1}∈ \\ L^{2}(ℝ_{+},ℂ^{2}) \end{matrix}}inf\{⟨F_{l}f, c_{l}(κ)F_{l}f⟩ : ∥f∥ = 1, f ∈ Λ_{+}(L^{2}(ℝ_{+}, ℂ^{2})), f ⊥ f_{ν}\}

with F_{l} the Fourier-Bessel transform, see (A.5). The infimum does not increase if the condition f ∈ Λ_{+}(L^{2}(ℝ_{+}, ℂ^{2})) is relaxed to f ∈ L^{2}(ℝ_{+}, ℂ^{2}). This gives the eigenvalues of the operator c_{l}(κ) ⊗ 1_{ℂ^{2}}, proving (2.12).

2.1.3 Lower bounds on hydrogen eigenvalues. Critical case

Proof of Theorem 2.2. Critical case. It remains to prove that

λ_{n}(c_{0}(κ)) ≥ −const κ^{2}n^{−2}

for κ^{B}κ^{C}/κ_{1}^{C} ≤ κ ≤ κ^{C}. We may assume that κ = κ^{C} and will prove that for all τ > 0

N(−τ, c_{0}(κ^{C})) := tr χ_{(−∞,−τ)}(c_{0}(κ^{C})) ≤ const τ^{−1/2}.  (2.13)

Let χ_{i}^{2} + χ_{o}^{2} = 1 be a smooth radial quadratic partition of unity with χ_{i} supported in the unit ball and χ_{o} supported outside the ball of radius 1/2 about the origin. It was shown in [31, Eq. (19)] that the localization error can be estimated by a bounded exponentially decaying potential v(r) ≤ const e^{−r}, i.e.,

\sqrt{p^{2} + 1} − 1 − κ^{C}|x|^{−1} ≥ χ_{i} (\sqrt{p^{2} + 1} − 1 − κ^{C}|x|^{−1} − v(|x|)) χ_{i}

+ χ_{o} (\sqrt{p^{2} + 1} − 1 − κ^{C}|x|^{−1} − v(|x|)) χ_{o}.

By the variational principle it suffices to consider the eigenvalue counting function corresponding to the interior and exterior term separately. The interior term is further estimated according to

χ_{i} (\sqrt{p^{2} + 1} − 1 − κ^{C}|x|^{−1} − v(|x|)) χ_{i}

≥ χ_{i} (|p| − κ^{C}|x|^{−1} − const ) χ_{i}.

As shown by Lieb and Yau [46] and explained in Corollary D.1, the number of negative eigenvalues of the latter operator acting in the subspace corresponding to l = 0 is finite, i.e., for all τ > 0

N_{l=0} (−τ, χ_{i} (|p| − κ^{C}|x|^{−1} − const ) χ_{i}) ≤ const .  (2.14)

For the exterior problem, we note that by the variational principle

N_{l=0} (−τ, χ_{o} (\sqrt{p^{2} + 1} − 1 − κ^{C}|x|^{−1} − v(|x|)) χ_{o})

≤ N_{l=0} (−τ, \sqrt{p^{2} + 1} − 1 − χ(x)(κ^{C}|x|^{−1} + v(|x|))) (2.15)

where χ denotes the characteristic function of the support of χ_{o}. With the singularity gone, the result follows as in the subcritical case. Namely, similarly as in (2.11) we cut in momentum space according to small and large momenta. Again, by the variational principle, the right-hand side of (2.15) is then bounded from above by

N_{l=0}(−const τ, |p| − w(|x|)) + N_{l=0}(−const τ, p^{2} − w(|x|)),

where w(r) = const χ(r)(κ^{C}r^{−1} + v(r)). The first term is estimated with the help of Daubechies’ inequality [9]

N_{l=0}(−τ, |p| − w(|x|)) ≤ τ^{−1/2} tr_{l=0}(|p| − w(|x|))_{−}^{1/2}

≤ const τ^{−1/2} ∫_{0}^{∞} w(r)^{3/2} dr

with the latter integral being finite. For the second term we estimate w(r) ≤ const r^{−1} and use that

N_{l=0}(−τ, p^{2} − const |x|^{−1}) ≤ const τ^{−1/2}.

This concludes the proof of Theorem 2.2.

Our proof of Theorem 2.1 in the critical Brown-Ravenhall case is based on a reduction to the Chandrasekhar case. The next lemma compares the number of eigenvalues of the critical operators b_{1/2,l}(κ^{B}) with those of the two operators c_{l′}(κ_{l′}^{C}) with l′ = 0, 1 and critical coupling constants κ_{0}^{C} = 2/π and κ_{1}^{C} = π/2, cf. (C.8).

Lemma 2.3. There is a constant such that for l = 0, 1 and all τ > 0 one has

N (−τ, b_{1/2,l}(κ^{B})) ≤ const [N (−\frac{τ}{const} , c_{0}(κ_{0}^{C})) + N (−\frac{τ}{const} , c_{1}(κ_{1}^{C}))] .

Proof. We start with the observation that (κ_{0}^{C})^{−1} + (κ_{1}^{C})^{−1} = 2(κ^{B})^{−1}. Using the explicit form of the reduced operators (cf. Appendix C), this implies the identities

\begin{aligned}b_{1/2,0}(κ^{B}) & = κ^{B} ((κ_{0}^{C})^{−1}ϕ_{0}b̃_{0,0}ϕ_{0} + (κ_{1}^{C})^{−1}ϕ_{1}b̃_{1,1}ϕ_{1}) , \\ b_{1/2,1}(κ^{B}) & = κ^{B} ((κ_{0}^{C})^{−1}ϕ_{1}b̃_{0,1}ϕ_{1} + (κ_{1}^{C})^{−1}ϕ_{0}b̃_{1,0}ϕ_{0}) , \end{aligned}  (2.16)

where the operators b̃_{l,ν} are defined in L^{2}(ℝ_{+}) through quadratic forms

⟨f, b̃_{l,ν}f⟩ := ∫_{0}^{∞} \frac{E(p) − 1}{2ϕ_{ν}(p)^{2}}|f(p)|^{2} dp − κ_{l}^{C} ∫_{0}^{∞} ∫_{0}^{∞} \overline{f(p)}k_{l}^{C}(p, q)f(q) dp dq.

In case ν = 1 it hence follows from 2ϕ_{1}(p)^{2} ≤ 1 that ⟨f, b̃_{l,1}f⟩ ≥ ⟨f, c_{l}(κ_{l}^{C})f⟩. In case ν = 0 we use the inequality

(E(p) − 1)ϕ_{0}(p)^{−2} ≥ \sqrt{p^{2} + 4} − 2 = 2(E(p/2) − 1)  (2.17)

which is most easily seen by writing both sides in terms of E(p). It implies

⟨f, b̃_{l,0}f⟩ ≥ 2⟨uf, c_{l}(κ_{l}^{C}) uf⟩

where the unitary scaling transformation u is defined through (uf)(p) := \sqrt{2}f(2p). The proof is completed by the variational principle.

We are now ready to give a

Proof of Theorem 2.1. Critical case. The previous lemma implies that it suffices to show that for l = 0, 1

N (−τ, c_{l}(κ_{l}^{C})) ≤ const τ^{−1/2}.

In case l = 0 this was established in (2.13), and the case l = 1 follows similarly with the analogue of (2.14) given in Corollary D.1.

2.2 Sobolev inequality for the critical Brown-Ravenhall operator

Having studied the eigenvalues of B_{κ} in the previous subsection, we now turn to integrability properties of its eigenfunctions. The L^{q}-norm of two-spinors ψ is given by

∥ψ∥_{q} := (∫_{ℝ^{3}} |ψ(x)|^{q}dx)^{1/q} ,

where the modulus, | · |, refers to the Euclidean norm in ℂ^{2}. For q = 2 we drop the subscript. We aim at proving the following

Theorem 2.4 (L^{q}-properties of eigenfunctions). Let 2 ≤ q < 3. There exists a constant C_{q} < ∞ such that for any κ ∈ (0, κ^{B}] and all ψ ∈ 𝔔(B_{κ}) with ⟨ψ, B_{κ}ψ⟩ ≤ 0 one has ψ ∈ L^{q} with

∥ψ∥_{q} ≤ C_{q} ∥ψ∥ .  (2.18)

Note that (2.18) applies, in particular, to eigenfunctions of B_{κ} corresponding to negative eigenvalues. The proof of Theorem 2.4, which is spelled out below, relies on a Sobolev inequality for the massless atomic Brown-Ravenhall operator in ℌ given by

B_{κ}^{(0)} := |p| − \frac{κ}{2} (|x|^{−1} + ω_{p} · σ |x|^{−1} ω_{p} · σ) .

This operator is bounded below (in fact, non-negative) if and only if κ ≤ κ^{B}.

Theorem 2.5 (Sobolev inequality). For any 2 ≤ q < 3 there exists a constant C_{q} > 0 such that for all ψ ∈ 𝔔(B_{κ^{B}}^{(0)}),

∥ψ∥_{q}^{2} ≤ C_{q} ⟨ψ, B_{κ^{B}}^{(0)}ψ⟩^{θ} ∥ψ∥^{2(1−θ)},  θ = 6(\frac{1}{2} − \frac{1}{q}).  (2.19)

It is illustrative to compare (2.19) with the ‘standard’ Sobolev-Gagliardo-Nirenberg inequalities,

∥ψ∥_{q}^{2} ≤ C′_{q} ⟨ψ, |p|ψ⟩^{θ} ∥ψ∥^{2(1−θ)},  θ = 6(\frac{1}{2} − \frac{1}{q}),  2 ≤ q ≤ 3, (2.20)

see, e.g., [43, Thm. 8.4]. Hence Theorem 2.5 says that, if the endpoint q = 3 is avoided, an inequality of the same form remains true after subtracting the maximal possible multiple of \frac{1}{2} (|x|^{−1} + ω_{p} · σ |x|^{−1} ω_{p} · σ) from |p|. Moreover, one can show that (2.19) does not hold with q = 3, not even if the L^{3}-norm is replaced by the weak L^{3}-norm. Note that if κ < κ^{B} then (2.19) with B_{κ}^{(0)} instead of B_{κ^{B}}^{(0)} follows from (2.20)

– but with a constant that deteriorates as κ → κ^{B}. The main point is to derive an inequality which holds uniformly in κ up to and including the critical constant. Our proof is based on the somewhat surprising fact that the Brown-Ravenhall operator with coupling constant κ^{B} can be bounded from below by the Chandrasekhar operator with smaller coupling constant κ^{C}. Before we start the proof of (2.19), we provide the

Proof of Theorem 2.4. The Sobolev inequality (2.19) implies

∥ψ∥_{q}^{2} ≤ C_{q} ⟨ψ, B_{κ}^{(0)}ψ⟩^{θ} ∥ψ∥^{2(1−θ)} ≤ C_{q}⟨ψ, [B_{κ}^{(0)} − B_{κ}]ψ⟩^{θ}∥ψ∥^{2(1−θ)}
≤ C_{q}∥B_{κ}^{(0)} − B_{κ}∥^{θ}∥ψ∥^{2}.

Tix showed [75, Thm. 1] (see also Balinsky and Evans [3]) that the difference B_{κ}^{(0)} − B_{κ} extends to a bounded operator with norm uniformly bounded for any κ ∈ (0, κ^{B}].

2.2.1 Comparison of critical operators

The first step in the proof of the Sobolev inequality (2.19) is a comparison of B_{κ}^{(0)} with the massless atomic Chandrasekhar operator in ℌ, which is given by

C_{κ}^{(0)} := |p| − κ|x|^{−1}.

It is bounded below if and only if κ ≤ κ^{C}. As discussed in Appendix C the parts of B_{κ}^{(0)} and C_{κ}^{(0)} in the subspace ℌ_{j,l,m} are unitarily equivalent to operators b_{j}^{(0)}(κ) and c_{l}^{(0)}(κ) in L^{2}(ℝ_{+}), which depend only on j in the Brown-Ravenhall case and only on l in the Chandrasekhar case. For the comparison argument it is important to note that the reduced operators b_{j}^{(0)}(κ) and c_{l}^{(0)}(κ) are lower bounded for κ up to and including the critical coupling constants κ_{j}^{B} and κ_{l}^{C} respectively. They are defined in (C.7) and, as is explained there, exceed κ^{B} and κ^{C}, if j ≥ 3/2 or l ≥ 1. We begin by observing that all the critical operators b_{j}^{(0)}(κ_{j}^{B}) and c_{l}^{(0)}(κ_{l}^{C}) have the same ‘generalized ground state’, namely p. The corresponding ground state representation formula (in momentum space) is given in

Lemma 2.6 (Ground state representation). If f ∈ 𝔔(b_{j}^{(0)}(κ_{j}^{B})) and g(p) = pf(p), then

⟨f, b_{j}^{(0)}(κ_{j}^{B})f⟩ = \frac{κ_{j}^{B}}{2} ∫_{0}^{∞} ∫_{0}^{∞} |g(p) − g(q)|^{2}k_{j}^{B}(p, q) \frac{dp}{p} \frac{dq}{q} .  (2.21)

Similarly, if f ∈ 𝔔(c_{l}^{(0)}(κ_{l}^{C})) and g(p) = pf(p), then

⟨f, c_{l}^{(0)}(κ_{l}^{C})f⟩ = \frac{κ_{l}^{C}}{2} ∫_{0}^{∞} ∫_{0}^{∞} |g(p) − g(q)|^{2}k_{l}^{C}(p, q) \frac{dp}{p} \frac{dq}{q} .  (2.22)

where k_{j}^{B} and k_{l}^{C} are defined in (C.4).

Proof. We write k for one of the functions k_{j}^{B} or k_{l}^{C} and κ for the corresponding constant κ_{j}^{B} or κ_{l}^{C}. Expanding the square and using k(p, q) = k(q, p), we find

\frac{1}{2} ∫_{0}^{∞} ∫_{0}^{∞} |g(p) − g(q)|^{2}k(p, q) \frac{dp}{p} \frac{dq}{q}

= ∫_{0}^{∞} |g(p)|^{2} (∫ k(p, q) \frac{dq}{q} ) \frac{dp}{p} − ∫_{0}^{∞} ∫_{0}^{∞} \overline{g(p)}k(p, q)g(q)\frac{dp}{p} \frac{dq}{q}
= ∫_{0}^{∞} p|f(p)|^{2} (∫ k(p, q) \frac{dq}{q} ) dp −∫_{0}^{∞} ∫_{0}^{∞} \overline{f(p)}k(p, q)f(q) dp dq.

By definitions (C.7) and (C.8) of κ we have

∫_{0}^{∞} k(p, q) \frac{dq}{q} = κ^{−1},

which implies the assertion.

Next, we bound B_{κ^{B}}^{(0)} from below by C_{κ^{C}}^{(0)}. Lemma 2.7 (Comparison of critical operators). There is a positive constant such that for any ψ ∈ 𝔔(B_{κ^{B}}^{(0)}) ∩ ℌ_{1/2,1}^{⊥}

⟨ψ, B_{κ^{B}}^{(0)}ψ⟩ ≥ const ⟨ψ, C_{κ^{C}}^{(0)}ψ⟩ .  (2.23)

An inequality of the form (2.23) cannot hold for ψ ∈ ℌ_{1/2,1}, since the right hand side is bounded from below by a constant times ⟨ψ, |p|ψ⟩ while the left hand side is not.

Proof. By orthogonality it suffices to prove the inequality on each subspace ℌ_{j,l}. First let (j, l) = (1/2, 0). We may also fix m = ±1/2 and choose ψ ∈ ℌ_{1/2,0,m}. Its Fourier transform is of the form ψ̂(p) = p^{−1}f(p)Ω_{\frac{1}{2} ,0,m}(ω_{p}), see

Appendix A. Setting f(p) =: pg(p) one finds using (C.2) and Lemma 2.6

⟨ψ, B_{κ^{B}}^{(0)}ψ⟩ = ⟨f, b_{1/2}^{(0)}(κ^{B})f⟩ = \frac{κ^{B}}{2} ∫_{0}^{∞} ∫_{0}^{∞} |g(p) − g(q)|^{2} k_{1/2}^{B}(p, q) \frac{dp}{p} \frac{dq}{q} .

Similarly, using (C.3)

⟨ψ, C_{κ^{C}}^{(0)}ψ⟩ = ⟨f, c_{0}^{(0)}(κ^{C})f⟩ = \frac{κ^{C}}{2} ∫_{0}^{∞}∫_{0}^{∞} |g(p) − g(q)|^{2} k_{0}^{C}(p, q) \frac{dp}{p} \frac{dq}{q} .

Recalling the explicit expressions (C.4) and (C.5) for k_{1/2}^{B} and k_{0}^{C} and estimating Q_{1} ≥ 0 we conclude that

⟨ψ, B_{κ^{B}}^{(0)}ψ⟩ ≥ (1 + (2/π)^{2})^{−1}⟨ψ, C_{κ^{C}}^{(0)}ψ⟩,

which proves the assertion on the subspace ℌ_{1/2,0}. Now assume that ψ ∈ (ℌ_{1/2,0} ⊕ ℌ_{1/2,1})^{⊥}. Since κ_{j}^{B} is monotone increasing in j, see Appendix C, we have on that subspace

|p| ≥ \frac{κ_{3/2}^{B}}{2} (|x|^{−1} + ω_{p} · σ |x|^{−1} ω_{p} · σ) .

We conclude that

⟨ψ, B_{κ^{B}}^{(0)}ψ⟩ ≥ \frac{κ_{3/2}^{B} − κ_{1/2}^{B}}{κ_{3/2}^{B}} ⟨ψ, |p|ψ⟩ ≥ \frac{κ_{3/2}^{B} − κ_{1/2}^{B}}{κ_{3/2}^{B}} ⟨ψ, C_{κ^{C}}^{(0)}ψ⟩,

proving the assertion.

2.2.2 Proof of the Sobolev inequality

We are now ready to give a

Proof of Theorem 2.5. By scaling, (2.19) is equivalent to the inequality

∥ψ∥_{q}^{2} ≤ C′_{q} (⟨ψ, B_{κ^{B}}^{(0)}ψ⟩ + ∥ψ∥^{2}) .

This, together with the triangle inequality, shows that it is enough to prove the inequality separately on the subspaces ℌ_{1/2,1} and ℌ_{1/2,1}^{⊥}. On the latter subspace, the claim follows immediately from Lemma 2.7 above and the Sobolev inequality for the critical Chandrasekhar operator [30, Corollary 2.5]. We now reduce the claim for the subspace ℌ_{1/2,1} to that for ℌ_{1/2,0}. For this purpose, we note that the helicity operator H = ω_{p} · σ, cf. (B.1), commutes with B_{κ}^{(0)} and, by (B.2), maps ℌ_{j,l} into ℌ_{j,2j−l}. Hence if ψ ∈ ℌ_{1/2,1} then by the Sobolev inequality on ℌ_{1/2,0}

⟨ψ, B_{κ^{B}}^{(0)}ψ⟩ + ∥ψ∥^{2} = ⟨Hψ, B_{κ^{B}}^{(0)}Hψ⟩ + ∥Hψ∥^{2} ≥ const ∥Hψ∥_{q}^{2}.

By Lemma B.1 the helicity H = H^{−1} is bounded on L^{q}(ℝ^{3}, ℂ^{2}).

2.3 Estimates on the electric potential

The goal of this subsection is to compare twisted and untwisted electric potentials. We begin with an estimates for point charges and then turn to smeared out charges.

Lemma 2.8. Let l ≥ 1 and ψ ∈ ℌ_{j,l}. Then

|⟨ψ, (|x|^{−1} − U(|x|^{−1})) ψ⟩| ≤ \frac{const}{l^{2}} ⟨ψ, p^{2}ψ⟩.  (2.24)

Proof. By orthogonality it suffices to prove the assertion for ψ ∈ ℌ_{j,l,m}. Its Fourier transform is of the form ψ̂(p) = f(p)p^{−1}Ω_{j,l,m}(ω_{p}), cf. Appendix A, and we compute similarly as in (C.11)

⟨ψ, (|x|^{−1} − U(|x|^{−1})) ψ⟩

= \frac{1}{π} ∫_{0}^{∞} dp \overline{f(p)} ∫_{0}^{∞} dq f(q) \{[1 − ϕ_{0}(p)ϕ_{0}(q)] Q_{l} (\frac{1}{2} (\frac{q}{p} + \frac{p}{q}))

−ϕ_{1}(p)ϕ_{1}(q)Q_{2j−l} (\frac{1}{2} (\frac{q}{p} + \frac{p}{q}))\}

= \frac{1}{2π} (A_{1} + A_{2})

with

A_{1} := ∫_{0}^{∞} dp \overline{f(p)} ∫_{0}^{∞} dq f(q) ∑_{ν=0}^{1} (ϕ_{ν}(p) − ϕ_{ν}(q))^{2} Q_{l} (\frac{1}{2} (\frac{q}{p} + \frac{p}{q})) ,

A_{2} :=2 ∫_{0}^{∞} dp \overline{f(p)} ∫_{0}^{∞} dq f(q)ϕ_{1}(p)ϕ_{1}(q)
× [Q_{l} (\frac{1}{2} (\frac{q}{p} + \frac{p}{q})) − Q_{2j−l} (\frac{1}{2} (\frac{q}{p} + \frac{p}{q}))] .

We estimate these terms separately. For the first term we use (B.5) and (B.6) together with Abel’s argument to turn Hermitian integral operators into multiplication operators by means of the Schwarz inequality (see also [46, Ineq.

(6.9)]). Since the Q_{l} are positive, we obtain

A_{1} ≤ ∫_{0}^{∞} dp\frac{|f(p)|^{2}}{E(p)^{4}} ∫_{0}^{∞} dq (\frac{p}{q})^{2} E(p)^{2}

× ∑_{ν=0}^{1} (ϕ_{ν}(p) − ϕ_{ν}(q))^{2} E(q)^{2}Q_{l} (\frac{1}{2} (\frac{q}{p} + \frac{p}{q}))

≤\frac{5}{8} ∫_{0}^{∞} dp\frac{|f(p)|^{2}}{E(p)^{4}} ∫_{0}^{∞} dq (\frac{p}{q})^{2} (p − q)^{2}Q_{l} (\frac{1}{2} (\frac{q}{p} + \frac{p}{q}))
=\frac{5}{8} ∫_{0}^{∞} dp\frac{|f(p)|^{2}}{E(p)^{4}} p^{3} ∫_{0}^{∞} \frac{dq}{q^{2}} (1 − q)^{2}Q_{l} (\frac{1}{2} (q + q^{−1})) .

We now use the bounds p^{3}/E(p)^{4} ≤ p^{2} and, for q ≥ 1, (1 − q)^{2} ≤ q^{2} − 1 which yield

∫_{0}^{∞} \frac{dq}{q^{2}} (1 − q)^{2}Q_{l} (\frac{1}{2} (q + q^{−1})) = 2 ∫_{1}^{∞} \frac{dq}{q^{2}} (1 − q)^{2}Q_{l} (\frac{1}{2} (q + q^{−1}))

≤ 4 ∫_{1}^{∞} dx Q_{l}(x) = \frac{4}{l(l + 1)},

where the last step involved [16, 324(18)]. Thus,

A_{1} ≤ \frac{5}{2l(l + 1)} ∫_{0}^{∞} dp p^{2}|f(p)|^{2} = \frac{5}{2l(l + 1)}⟨ψ, p^{2}ψ⟩.

We estimate the term A_{2} similarly by the Schwarz inequality,

A_{2} ≤ 2 ∫_{0}^{∞} dp|f(p)|^{2} |ϕ_{1}(p)|^{2}

× ∫_{0}^{∞} dq \frac{p}{q} |Q_{l} (\frac{1}{2} (\frac{q}{p} + \frac{p}{q})) − Q_{2j−l} (\frac{1}{2} (\frac{q}{p} + \frac{p}{q}))|

≤ 4 ∫_{0}^{∞} dp|f(p)|^{2}p^{2}

× ∫_{1}^{∞} \frac{dq}{q} |Q_{l} (\frac{1}{2} (q + q^{−1})) − Q_{2j−l} (\frac{1}{2} (q + q^{−1}))| .

Due to the pointwise monotonicity (C.10) the difference inside the modulus is of definite sign. Without loss of generality, we may therefore assume 2j = 2l+1. Using the integral representation (C.1) we can bound

∫_{1}^{∞} \frac{dq}{q} [Q_{l} (\frac{1}{2} (q + q^{−1})) − Q_{l+1} (\frac{1}{2} (q + q^{−1}))]

= ∫_{1}^{∞} dzz^{−l−2} (z − 1) ∫_{1}^{\frac{1}{2}(z+z^{−1})} \frac{dx}{\sqrt{x^{2} − 1}} \frac{1}{\sqrt{1 − 2xz + z^{2}}}
≤\frac{π}{\sqrt{2}} ∫_{1}^{∞} dzz^{−l−5/2} (z − 1) = \frac{π}{\sqrt{2}(l + \frac{1}{2})(l + \frac{3}{2})}.

Documenta Mathematica 14 (2009) 463–516

Adding the estimates for A_{1} and A_{2} we arrive at (2.24).

Note that our proof shows that one can choose different powers of |p| on the right hand side of (2.24).

Lemma 2.9. There exists a constant such that for any electric potential v of a spherically symmetric non-negative charge density

|⟨ψ, (v − U(v)) ψ⟩| ≤ const v(0) ⟨ψ, p^{2}ψ⟩.

Proof. We denote by τ : ℝ^{3} → [0, ∞) the spherically symmetric, non-negative charge density corresponding to v, i.e., v(x) = ∫ τ(x − y) |y|^{−1}dy. The Fourier transform of τ obeys the estimates

|τ̂(p)| = \sqrt{\frac{2}{π p^{2}}} |∫_{0}^{∞} r sin(|p|r) τ(r)dr| ≤ \frac{v(0)}{(2π)^{3/2} |p|}.

By Fourier transform the scalar product on the left side of the assertion becomes

⟨ψ, (v − U(v)) ψ⟩

= \sqrt{\frac{2}{π}} ∫∫ ψ̂(p)^{∗} \frac{τ̂(p − q)}{|p − q|^{2}} (1 − Φ_{0}(p)Φ_{0}(q) − Φ_{1}(p)Φ_{1}(q)) ψ̂(q)dpdq.

Using Lemma B.2 we estimate the absolute value of the preceding expression from above by the sum of two terms, B_{1} and B_{2}. The first term can be further bounded as follows,

B_{1} = const ∫∫ |τ̂(p − q)||ψ̂(p)||ψ̂(q)| dpdq

≤ const v(0) ∫ dp |ψ̂(p)|^{2} ∫ (\frac{|p|}{|q|})^{5/2} \frac{1}{|p − q|} dq

≤ const v(0) ∫ |ψ̂(p)|^{2}p^{2} dp,

where we use the Schwarz inequality in the second step. The second term is estimated similarly

B_{2} = const ∫∫ |τ̂(p − q)|\frac{\sqrt{|p| |q|}}{|p − q|}|ψ̂(p)||ψ̂(q)| dpdq

≤ const v(0) ∫ dp |ψ̂(p)|^{2} ∫ (\frac{|p|}{|q|})^{2} \frac{\sqrt{|p| |q|}}{|p − q|^{2}} dq

≤ const v(0) ∫ |ψ̂(p)|^{2}|p|^{2} dp.

This proves the assertion.

3 Spectral shift from Schrödinger to Brown-Ravenhall operators

The main theme of this section is the (integrated) spectral shift, i.e., the difference of sums of eigenvalues of the Brown-Ravenhall or Chandrasekhar operator

B[v] := \sqrt{p^{2} + 1} − 1 − U(v),  C[v] := \sqrt{p^{2} + 1} − 1 − v,

(cf. (2.2)) and the Schrödinger operator S[v] := \frac{1}{2}p^{2} − v, all acting in the Hilbert space ℌ = L^{2}(ℝ^{3}, ℂ^{2}) of two-spinors. We have set c = 1. Concerning the potential v : ℝ^{3} → ℝ we will always assume that the above operators can be defined through the Friedrichs extension starting from 𝔖(ℝ^{3}, ℂ^{2}). For example, the condition 0 ≤ v(x) ≤ κ^{#} |x|^{−1} with # = B, C (cf. (2.3)) ensures that the Brown-Ravenhall, respectively the Chandrasekhar operator are well-defined and bounded from below (see [17] and [40]). We assume throughout that the potential v is radially symmetric which allows us to investigate the spectral shift on each subspace ℌ_{j,l} in the decomposition (2.4) separately. We write Λ_{j,l} for the orthogonal projection onto ℌ_{j,l}. For the reduced traces we use the notations

tr_{j,l}(A) := tr(Λ_{j,l}A),  tr_{j}(A) := tr_{j,j+1/2}(A) + tr_{j,j−1/2}(A).  (3.1)

3.1 Estimate on the spectral shift

One of the key observations in our proof of the Scott correction is that the spectral shift between the one-particle Brown-Ravenhall and the Schrödinger operator decreases sufficiently fast for high angular momenta.

Theorem 3.1 (Spectral shift: Brown-Ravenhall case). There exists a constant C < ∞ such that for any κ ≤ κ^{B}, any v : [0, ∞) → [0, ∞) satisfying

v(r) ≤ κ r^{−1},  (3.2)

any μ > 0 and any j ∈ ℕ_{0} + 1/2 one has

tr_{j} ([B[v] + μ]_{−} − [S[v] + μ]_{−}) ≤ C κ^{4} j^{−2}.  (3.3)

We derive this result from a corresponding theorem for the Chandrasekhar operator. For a proof of the latter we need to strengthen [31, Thm. 2]. In particular, we need to consider C[v] for potentials v satisfying (3.2) also in case κ^{C} < κ ≤ κ^{B}. Those operators are not densely defined in the Hilbert space ℌ. However, according to (C.8) below, they are densely defined in the subspaces ℌ_{j,l} with j ≥ 3/2. Another new aspect is that we trace the dependence on the coupling constant.

Theorem 3.2 (Spectral shift: Chandrasekhar case). There exists a constant C < ∞ such that for all l ∈ ℕ_{0}, j = l ± \frac{1}{2}, for all κ satisfying

κ ≤ \{ \begin{matrix}κ^{C} & if l = 0, \\ κ^{B} & if l ≥ 1, \end{matrix}

for all μ ≥ 0 and for all v : [0, ∞) → [0, ∞) satisfying (3.2), one has

0 ≤ tr_{j,l} ([C[v] + μ]_{−} − [S[v] + μ]_{−}) ≤ C \frac{κ^{4}}{(l + \frac{1}{2})^{2}} .  (3.4)

One of the key points to be appreciated in the above theorems is an effective cancellation in the differences in (3.4) and (3.3). This can already be seen for Coulomb potentials v(r) = κr^{−1}, where

tr_{j,l} [S_{κ}]_{−} = (2j + 1)\frac{κ^{2}}{2} ∑_{n=1}^{∞} \frac{1}{(n + l)^{2}} ,

which does not decay at all as j → ∞. Moreover, for fixed j and l the above trace vanishes only like κ^{2} as κ → 0. It is rather remarkable that such cancellations occur uniformly for all attractive potential v satisfying (3.2). The following proof of Theorem 3.2 follows the ideas of [31, Thm. 2]. It is not only included to render the paper self-contained, but also to establish the above mentioned improvement, which is important for the present paper.

Proof of Theorem 3.2. We note that both traces tr_{j,l} [C[v] + μ]_{−} and tr_{j,l} [S[v] + μ]_{−} are finite. This follows by the variational principle from the case v(r) = κr^{−1}, cf. Theorem 2.2 in the Chandrasekhar case. Thus, for l < 3 say, it is enough to show the claim for κ in a neighborhood of 0. More precisely, we can assume κ ≤ \frac{1}{\sqrt{8}}(l + \frac{1}{2}) which covers all κ ≤ κ^{B} for l ≥ 3. Moreover, by an approximation argument it is sufficient to consider μ > 0 and bounded potentials v, cf. [31]. We denote by γ_{j,l} the orthogonal projection onto the eigenspace of C[v] corresponding to angular momenta j, l and eigenvalues less or equal than −μ. The identity

\frac{1}{2}p^{2} = C_{0} + \frac{1}{2}C_{0}^{2}  (3.5)

and the variational principle (cf. [43, Thm. 12.1]) imply

0 ≤ 2 tr_{j,l} ([C[v] + μ]_{−} − [S[v] + μ]_{−}) ≤ tr [C_{0}^{2}γ_{j,l}] .  (3.6)

Using the eigenvalue equation and the bound (3.2) on the potential we estimate this term further as follows.

tr [C_{0}^{2}γ_{j,l}] ≤ tr_{j,l} [C[v]]_{−}^{2} + tr [v^{2}γ_{j,l}] ≤ tr_{j,l} [C_{κ}]_{−}^{2} + κ^{2} tr [|x|^{−2}γ_{j,l}] .

Using Hardy’s inequality and (3.5)

tr [|x|^{−2}γ_{j,l}] ≤ (l + \frac{1}{2})^{−2} tr [p^{2}γ_{j,l}]
= (l + \frac{1}{2})^{−2} (tr [C_{0}^{2}γ_{j,l}] + 2 tr [C_{0}γ_{j,l}]) .

Since κ < l + \frac{1}{2}, the last two estimates may be summarized as

tr [C_{0}^{2}γ_{j,l}] ≤ (1 − \frac{κ^{2}}{(l + \frac{1}{2})^{2}} )^{−1} (tr_{j,l} [C_{κ}]_{−}^{2} + \frac{2κ^{2}}{(l + \frac{1}{2})^{2}} tr [C_{0}γ_{j,l}]) . (3.7)

We shall estimate the two terms on the right hand side separately. From [31, Lemma 3] we recall the following angular momentum barrier inequality on ℌ_{j,l},

C_{0} ≥ 2κr^{−1}χ_{\{r≤R_{l}(κ)\}},  R_{l}(κ) = \frac{1}{8κ}(l + \frac{1}{2})^{2}.  (3.8)

(Here we use that κ ≤ \frac{1}{\sqrt{8}}(l + \frac{1}{2}).) This implies

tr [C_{0}γ_{j,l}] ≤ κ tr [|x|^{−1}γ_{j,l}] ≤ \frac{1}{2} tr [C_{0}γ_{j,l}] + \frac{1}{4} tr [w_{l}γ_{j,l}]

= \frac{3}{4} tr [C_{0}γ_{j,l}] − \frac{1}{4} tr [C[w_{l}]γ_{j,l}]

where w_{l}(r) := 4κr^{−1}χ_{{r≥R_{l}(κ)}}. Hence, using the variational principle followed by Daubechies’ inequality [9] (cf. also [31, Prop. 1])

tr [C_{0}γ_{j,l}] ≤ tr_{j,l} [C[w_{l}]]_{−}

≤ const (2l + 1) (∫_{0}^{∞} w_{l}(r)^{3/2} dr + ∫_{0}^{∞} w_{l}(r)^{2} dr) ≤ const κ^{2}. (3.9)

In order to estimate the first term on the right hand side of (3.7) we use (3.8) to obtain on ℌ_{j,l}

C_{κ} ≥ \frac{1}{2}C_{0} − κr^{−1}χ_{\{r≥R_{l}(κ)\}} ≥ \frac{1}{2}C[w_{l}].

with w_{l} as above. Hence again by Daubechies’ inequality

tr_{j,l} [C_{κ}]_{−}^{2} ≤const (2l + 1) (∫_{0}^{∞}w_{l}(r)^{5/2} dr + ∫_{0}^{∞}w_{l}(r)^{3} dr)
≤const κ^{4}(l + \frac{1}{2})^{−2}.

Combing this with (3.9), (3.7), and (3.6) completes the proof.

Having finished the proof of Theorem 3.2 it is easy to give the

Proof of Theorem 3.1. Since the trace tr_{j} [B[v] + μ]_{−} is finite according to Theorem 2.1 we may assume that either κ ≤ κ^{C} and j = 1/2, or else that j ≥ 3/2. In this case, the claim essential boils down to Theorem 3.2. To see this, we note the identity

B[v] = U(C[v]) = \frac{1}{2} (U(p)^{∗}C[v]U(p) + U(p)C[v]U(p)^{∗})  (3.10)

involving the unitary operator U(p) := Φ_{0}(p)+iΦ_{1}(p) (see also (2.1)). Equality (3.10) as well as the unitarity of U(p) are easily derived from the fact that Φ_{0}^{2}(p) + Φ_{1}^{2}(p) = 1. Even if v satisfies (3.2) only with a κ^{C} < κ ≤ κ^{B}, identity (3.10) remains valid on all subspaces ℌ_{j} with j ≥ 3/2. Hence by the concavity of the sum of negative eigenvalues [72] of B[v] + μ one has for any μ ≥ 0

tr_{j} [B[v] + μ]_{−}
≤\frac{1}{2} tr_{j} [U^{∗}(p)C[v]U(p) + μ]_{−} + \frac{1}{2} tr_{j} [U(p)C[v]U^{∗}(p) + μ]_{−}
= tr_{j} [C[v] + μ]_{−} .  (3.11)

By (3.4) the trace in (3.3) is thus bounded from above by

tr_{j} ([C[v] + μ]_{−} − [S[v] + μ]_{−}) ≤ const κ^{4}j^{−2},

as claimed.

3.2 Properties of the spectral shift

In this subsection we discuss some properties of the spectral shift s(κ) defined in (1.8).

Lemma 3.3 (Properties of the spectral shift). The spectral shift s is a continuous, non-negative function on (0, κ^{B}] satisfying s(κ) = O(κ^{2}) as κ ↓ 0.

Proof. According to (2.6) and Theorem 3.1 one has

0 ≤ s_{j}(κ) := κ^{−2} tr_{j} ([B_{κ}]_{−} − [S_{κ}]_{−}) ≤ const κ^{2}j^{−2}.

Therefore the sum s(κ) = ∑_{j} s_{j}(κ) converges, is non-negative and satisfies the claimed asymptotic estimate as κ ↓ 0. By the min-max principle each eigenvalue depends continuously on κ. Thus the continuity of their sum follows from the estimates in Theorem 2.1 and the Weierstraß criterion for uniform convergence.

4 Proof of the Scott correction

The strategy of the proof of the main results is similar to the one used for the Chandrasekhar operator [31]. We employ the Schrödinger operator as a regularization for the relativistic problem, i.e., we will use it to eliminate the main contribution to the energy (the Thomas-Fermi energy) and focus only on the energy shift of the low lying states. For these the electron-electron interaction plays no role and the unscreened problem remains. We define

E^{S}(Z) := inf{E_{Z}^{S}(ψ) | ψ ∈ 𝔔_{Z}^{S}, ∥ψ∥ = 1}

to be the ground state energy in the Schrödinger case,

E_{N}^{S}(ψ) := ⟨ψ, [∑_{ν=1}^{N} (\frac{1}{2}p_{ν}^{2} − Z|x_{ν}|^{−1}) + ∑_{1≤μ<ν≤N} |x_{μ} − x_{ν}|^{−1}] ψ⟩ .

It is defined on 𝔔_{N}^{S} := ℌ_{N}^{S} ∩ 𝔖(ℝ^{3N}, ℂ^{2^{N}}), where ℌ_{N}^{S} := ⋀_{ν=1}^{N} ℌ is the Hilbert space of anti-symmetric two-spinors. We recall that we suppose neutrality, i.e., N = Z. The asymptotics of the Schrödinger ground-state energy up to Scott correction reads [55]

E^{S}(Z) = E_{TF}(Z) + \frac{1}{2} Z^{2} + O(Z^{47/24}).  (4.1)

For our purpose this remainder estimate is sufficient. However, even the coefficient of the Z^{5/3}-term in the asymptotic expansion is known [23, 24, 25, 18, 26, 21, 19, 20, 22]. Our main result, Theorem 1.1, will follow from (4.1) if we can show that in the limit Z → ∞ the difference of the Schrödinger and Brown-Ravenhall groundstate energy satisfies

E^{S}(Z) − E_{c}^{B}(Z) = s(Z/c) Z^{2} + O(Z^{47/24})  (4.2)

uniformly in κ = Z/c ∈ (0, κ^{B}]. We break the proof of this assertion into an upper and lower bound.

4.1 Upper bound on the energy difference

The Thomas-Fermi functional (1.7) has a unique minimizer ϱ_{Z}, the Thomas-Fermi density (Lieb and Simon [44]). It scales as ϱ_{Z}(x) := Z^{2}ϱ_{1}(Z^{1/3}x). We set

ϕ_{TF}(x) := Z|x|^{−1} − ∫_{ℝ^{3}} \frac{ϱ_{Z}(y)}{|x − y|} dy,  (4.3)

the Thomas-Fermi potential, and

L_{TF}(x) := ∫_{|x−y|<R_{Z}(x)} \frac{ϱ_{Z}(y)}{|x − y|} dy,

the exchange hole potential. Here R_{Z}(x) is defined as the (unique) minimal radius for which ∫_{|x−y|≤R_{Z}(x)} ϱ_{Z}(y)dy = \frac{1}{2}. The corresponding one-particle operators – self-adjointly realized in ℌ – are

S_{TF} = S[ϕ_{TF} + L_{TF}], B_{TF} = B_{c}[ϕ_{TF} + L_{TF}].

Here we use a notation analogous to that in (1.5). We shall express the many-particle ground-state energies E^{S}(Z) and E_{c}^{B}(Z) in terms of quantities involving the above one-particle operators. In the Schrödinger case, this was achieved in [55, 58] in terms of the Thomas-Fermi potential ϕ_{TF}. Our point in the proof of the following proposition is to replace ϕ_{TF} by the exchange hole reduced potential ϕ_{TF} + L_{TF}.

Proposition 4.1. Let J := [Z^{1/9}] + \frac{1}{2}. Then, as Z → ∞,

E^{S}(Z)

= − ∑_{j=1/2}^{J−1} tr_{j} [S[Z|x|^{−1}]]_{−} − ∑_{j=J}^{Z+1/2} tr_{j} [S_{TF}]_{−} − D(ϱ_{Z}, ϱ_{Z}) + O(Z^{47/24}). (4.4)

Since ϕ_{TF} + L_{TF} has a Coulomb tail, the trace tr_{j} [S_{TF}]_{−} is finite for each j, but not summable with respect to j. It is therefore essential to restrict the second sum to a finite number of angular momenta. However, the value of the cut-off, j ≤ Z + 1/2, is not chosen optimally here, since for our argument it is largely arbitrary.

Proof of Proposition 4.1. According to the correlation inequality [47]

E^{S}(Z) ≥ − ∑_{j=1/2}^{Z+1/2} tr_{j} [S_{TF}]_{−} − D(ϱ_{Z}, ϱ_{Z}).

Note that the Z electrons can certainly be accommodated in the first Z angular momentum channels (which is a very crude bound). Estimating ϕ_{TF}+L_{TF} from above by the Coulomb potential for small angular momenta, we obtain

E^{S}(Z) ≥ − ∑_{j=1/2}^{J−1} tr_{j} [S[Z|x|^{−1}]]_{−} − ∑_{j=J}^{Z+1/2} tr_{j} [S_{TF}]_{−} − D(ϱ_{Z}, ϱ_{Z}). (4.5)

Moreover, see [55, 58],

E^{S}(Z)

≤ − ∑_{j=1/2}^{J−1} tr_{j} [S[Z|x|^{−1}]]_{−} − ∑_{j=J}^{∞} tr_{j} [S[ϕ_{TF}]]_{−} − D(ϱ_{Z}, ϱ_{Z}) + const Z^{47/24}.

Hence it suffices to prove that

− ∑_{j=J}^{Z+1/2} tr_{j} [S_{TF}]_{−} ≥ − ∑_{j=J}^{∞} tr_{j} [S[ϕ_{TF}]]_{−} − const Z^{5/3}  (4.6)

(Note that the lower bound in [31] contains an error by estimating [31, Equation (43)] too generously. Really, only the first Z lowest negative eigenvalues need to occur on the right hand side instead of all. In particular, there will be never more than Z total angular momentum channels occupied. This fact is taken into account here yielding a suitable lower bound. The problem in [31] can be circumvented in exactly the same way.) We decompose L_{TF} = L_{<} + L_{>} where

L_{<} = χ_{{|x|<R}}L_{TF}, L_{>} = χ_{{|x|≥R}}L_{TF}, with a constant R (independent of Z) to be chosen below. For ε > 0 to be specified later we estimate using the variational principle for sums of eigenvalues

tr_{j} [S_{TF}]_{−}

≤ tr_{j}(\frac{1}{2}(1 − 2ε^{2})p^{2} − ϕ_{TF})_{−} + ε^{2} tr_{j}(\frac{1}{2}p^{2} − ε^{−2}L_{<})_{−} + ε^{2} tr_{j}(\frac{1}{2}p^{2} − ε^{−2}L_{>})_{−}. (4.7)

By the subsequent lemma the first (and main) term is bounded according to

∑_{j=J}^{Z+1/2} tr_{j}(\frac{1}{2}(1 − 2ε^{2})p^{2} − ϕ_{TF})_{−} − ∑_{j=J}^{∞} tr_{j}(\frac{1}{2}p^{2} − ϕ_{TF})_{−}

≤ tr(\frac{1}{2}(1 − 2ε^{2})p^{2} − ϕ_{TF})_{−} − tr(\frac{1}{2}p^{2} − ϕ_{TF})_{−} ≤ const ε^{2}Z^{7/3}.

For the second term on the right side of (4.7) we use the Lieb-Thirring inequality [45] to obtain

ε^{2} ∑_{j=J}^{Z+1/2} tr_{j}(\frac{1}{2}p^{2} − ε^{−2}L_{<})_{−} ≤ ε^{2} tr(\frac{1}{2}p^{2} − ε^{−2}L_{<})_{−}

≤ const ε^{−3} ∫ L_{<}(x)^{5/2} dx ≤ const ε^{−3}Z^{2/3}.

In the last inequality we used a bound of Siedentop and Weikard [58, Proof of Lemma 2]. It is at this point that R is chosen. The penultimate inequality in [58, Proof of Lemma 2] asserts after scaling that L_{>}(x) ≤ const |x|^{−1}. Hence by comparison with the exact hydrogen solution

ε^{2} ∑_{j=J}^{Z+1/2} tr_{j}(\frac{1}{2}p^{2} − ε^{−2}L_{>})_{−} ≤ ε^{2} ∑_{j=1/2}^{Z+1/2} tr_{j}(\frac{1}{2}p^{2} − ε^{−2}const |x|^{−1})_{−}

=const ε^{−2} ∑_{j=1/2}^{Z+1/2} ∑_{n=1}^{∞} \frac{2j + 1}{(n + j − 1/2)^{2}} ≤ const ε^{−2}Z.

Choosing ε = Z^{−1/3} all the error terms are O(Z^{5/3}), proving (4.6).

In the previous proof we used

Lemma 4.2. For all 0 < ε ≤ 1/2, as Z → ∞,

tr(\frac{1}{2}(1 − ε^{2})p^{2} − ϕ_{TF})_{−} ≤ tr(\frac{1}{2}p^{2} − ϕ_{TF})_{−} + const ε^{2}Z^{7/3}.  (4.8)

Note that there are only a finite number of eigenvalues, since ϕ_{TF} decays like |x|^{−4}.

Proof. Let d_{TF}^{ε} be the projection onto the negative eigenvalues of \frac{1}{2}(1−ϵ^{2})p^{2} − ϕ_{TF}. Then, by the variational principle

tr(\frac{1}{2}(1 − ε^{2})p^{2} − ϕ_{TF})_{−} − tr(\frac{1}{2}p^{2} − ϕ_{TF})_{−}

≤ − tr d_{TF}^{ε}(\frac{1}{2}(1 − ε^{2})p^{2} − ϕ_{TF}) + tr d_{TF}^{ε}(\frac{1}{2}p^{2} − ϕ_{TF}) = \frac{ε^{2}}{2} tr d_{TF}^{ε}p^{2}. (4.9)

Hence the claim will follow, if we show that tr d_{TF}^{ε}p^{2} ≤ const Z^{7/3}. Note that d_{TF}^{ε} depends on both ε and Z, and by rescaling one may get rid of the ε dependence at the expense of changing Z. We may therefore assume that ε = 0 and write d_{TF} = d_{TF}^{0}. Thus, it remains to prove

tr d_{TF}p^{2} ≤ const Z^{7/3}.  (4.10)

Note that this says that the kinetic energy is bounded by the order of the total energy tr d_{TF}(\frac{1}{2}p^{2} − ϕ_{TF}), which is well-known to be of order Z^{7/3}. Using that ϕ_{TF} is bounded by a constant times min{Z|x|^{−1}, |x|^{−4}} (see [44]) we get for any R > 0

\frac{1}{2} tr d_{TF}p^{2} ≤ tr d_{TF}ϕ_{TF}

≤ const ((∫_{{|x|<R}}(Z|x|^{−1})^{5/2} dx)^{2/5} (∫ d_{TF}(x, x)^{5/3} dx)^{3/5}

+R^{−4} ∫ d_{TF}(x, x) dx) .

The Cwikel-Lieb-Rozenblum inequality (for a textbook presentation, see, e.g., [63]) guarantees that

∫ d_{TF}(x, x) dx ≤ const ∫ ϕ_{TF}(x)^{3/2}dx = const Z.

Moreover, by the Lieb-Thirring inequality [45]

∫ d_{TF}(x, x)^{5/3} dx ≤ const tr d_{TF}p^{2}.

We can estimate for any δ > 0

(∫_{{|x|<R}}(Z|x|^{−1})^{5/2} dx)^{2/5} (∫ d_{TF}(x, x)^{5/3} dx)^{3/5}

≤ const ZR^{1/5} (tr d_{TF}p^{2})^{3/5}

≤ δ tr d_{TF}p^{2} + const δ^{−3/2}Z^{5/2}R^{1/2}.

In summary, we have shown that

(\frac{1}{2} − const δ) tr d_{TF}p^{2} ≤ const (δ^{−3/2}Z^{5/2}R^{1/2} + R^{−4}Z) .

Choosing δ small (of order one) and R = Z^{−1/3} we obtain (4.10).

Next, we bound the many-particle ground state energy of the Brown-Ravenhall operator from below by one-body quantities which match the corresponding quantities in the Schrödinger case (4.4).

Lemma 4.3. For all J ∈ ℕ_{0} + 1/2 and Z ∈ ℕ

E_{c}^{B}(Z) ≥ − ∑_{j=1/2}^{J−1} tr_{j} [B_{c}[Z|x|^{−1}]]_{−} − ∑_{j=J}^{Z+1/2} tr_{j} [B_{TF}]_{−} − D(ϱ_{Z}, ϱ_{Z}).

Proof. This follows by the same argument leading to (4.5).

We are now ready to give a

Proof of Theorem 1.1 – first part. Choosing J = [Z^{1/9}] + \frac{1}{2} and combining Proposition 4.1 and Lemma 4.3 we obtain

E^{S}(Z) − E_{c}^{B}(Z) ≤ ∑_{j=1/2}^{J−1} tr_{j} ([B_{c}[Z|x|^{−1}]]_{−} − [S[Z|x|^{−1}]]_{−})  (4.11)

+ ∑_{j=J}^{Z+1/2} tr_{j} ([B_{TF}]_{−} − [S_{TF}]_{−}) + O(Z^{47/24}).

We note that by scaling x ↦ x/c, the operators S[Z|x|^{−1}] and B_{c}[Z|x|^{−1}] are unitarily equivalent to the operators Z^{2}κ^{−2}S_{κ} and Z^{2}κ^{−2}B_{κ} where κ = Z/c. Similarly, S_{TF} and B_{TF} are unitarily equivalent to the operators Z^{2}κ^{−2}S[κ|x|^{−1} − χ_{c}] and Z^{2}κ^{−2}B[κ|x|^{−1} − χ_{c}] acting in ℌ, where

χ_{c}(x) := c^{−4} ∫_{|x−y|>cR_{Z}(c^{−1}x)} \frac{ϱ_{Z}(c^{−1}y)}{|x − y|} dy.

This implies that the first two terms on the right-hand side of (4.11), which we denote by Σ_{1}(Z, c) and Σ_{2}(Z, c), can be rewritten as

Σ_{1}(Z, c) = Z^{2}κ^{−2} ∑_{j=1/2}^{J−1} tr_{j} ([B_{κ}]_{−} − [S_{κ}]_{−}) ,

Σ_{2}(Z, c) = Z^{2}κ^{−2} ∑_{j=J}^{Z+1/2} tr_{j} ([B[κ|x|^{−1} − χ_{c}]]_{−} − [S[κ|x|^{−1} − χ_{c}]]_{−}) .

Inequality (2.6) and Theorem 3.1 guarantee that the terms in the first sum are non-negative and that the terms in both sums are bounded from above by a constant times κ^{4}j^{−2} independently of Z and c. Therefore, the first sum can be bounded from above by an absolutely convergent series,

Σ_{1}(Z, c) ≤ Z^{2}κ^{−2} ∑_{j=1/2}^{∞} tr_{j} ([B_{κ}]_{−} − [S_{κ}]_{−}) = Z^{2} s(κ).

By the same token

Σ_{2}(Z, c) ≤ const Z^{2}κ^{2} ∑_{j=J}^{∞} j^{−2} = O(Z^{17/9}),

uniformly in c. This concludes the proof of the upper bound on the energy difference.

4.2 Lower bound on the energy difference

Similarly to [55] we define one-particle density matrices d^{S} and d^{B} on ℌ as sums

d^{#} = d_{<}^{#} + d_{>}, # = S, B.  (4.12)

The contribution of small total angular momenta, d_{<}^{#} = ∑_{l<L} d_{l}^{#}, is defined in Appendix E.1. It comes from the eigenspinors of the atomic problems. The contribution of large angular momentum, d_{>} = ∑_{l=L}^{∞} d_{l}, is defined in Appendix E.2. It corresponds to the Macke orbitals of [55] and, in particular, coincides for the Schrödinger and Brown-Ravenhall case. The angular-momentum cut-off L will be chosen in a Z-dependent way, namely,

L := [Z^{1/12}].

Important properties of the density matrices, whose construction is explained in more detail in Appendix E, are:

• The densities

ρ^{#}(x) := tr_{ℂ^{2}} (d^{#}(x, x)) , ρ_{l}^{#}(x) := tr_{ℂ^{2}} (d_{l}^{#}(x, x)) ,
ρ_{<}^{#}(x) := ∑_{l<L} ρ_{l}^{#}(x), ρ_{>}(x) := ∑_{l≥L} ρ_{l}(x).

of d^{#}, d_{l}^{#}, and d_{>} are all spherically symmetric.

• The dimension of the ranges of the density matrices d^{S} and d^{B} is at most Z, in particular tr d^{#} ≤ Z. Moreover,

tr d_{l}^{#} = ∫ ρ_{l}^{#}(x) dx = 2(2l + 1)(K − l),  0 ≤ l < L,  (4.13)

with K = [const Z^{1/3}] and a suitable constant.

For a lower bound on the ground state energy in the Schrödinger case, we recall from [55] and [31, Proposition 4] the following

Proposition 4.4. For large Z,

E^{S}(Z) = tr [S[Z|x|^{−1}] d^{S}] + D(ρ^{S}, ρ^{S}) + O(Z^{47/24}).

To obtain an upper bound on the ground state energy in the Brown-Ravenhall case, we use the reduced Hartree-Fock variational principle. It involves the density

ρ_{U}^{B}(x) := tr_{ℂ^{2}} (U_{c}(d^{B})(x, x))

of the twisted density matrix U_{c}(d^{B}). For further reference, we also set

ρ_{U,l}^{B}(x) := tr_{ℂ^{2}} (U_{c}(d_{l}^{B})(x, x)) , ρ_{U,<}^{B}(x) := ∑_{l<L} ρ_{U,l}^{B}(x),
ρ_{U,>}(x) := ∑_{l≥L} ρ_{U,l}(x).

Applying to (1.1) the Hartree-Fock variational principle – in the strengthened version of Lieb [42] (see also Bach [2]) – and omitting the manifestly negative exchange energy we arrive at

Proposition 4.5. For all Z and c,

E_{c}^{B}(Z) ≤ tr[B_{c}[Z|x|^{−1}] d^{B}] + D(ρ_{U}^{B}, ρ_{U}^{B}).

Combining Propositions 4.4 and 4.5 we find

E_{c}^{B}(Z) − E^{S}(Z)

≤ tr[B_{c}[Z|x|^{−1}]d^{B}] − tr[S[Z|x|^{−1}]d^{S}] + D(ρ_{U}^{B} − ρ^{S}, ρ_{U}^{B} + ρ^{S}) + const Z^{47/24}.

Now we use the inequality p^{2} ≥ 2c^{2}(E(p/c) − 1) for the kinetic energy corresponding to d_{>}. Moreover, we write

D(ρ_{U}^{B} − ρ^{S}, ρ_{U}^{B} + ρ^{S}) = D(ρ_{U,>} − ρ_{>}, ρ_{U,>} + ρ_{>}) + 2 D(ρ_{U,<}^{B}, ρ_{U}^{B} + ρ^{S})

− D(ρ_{U,<}^{B} + ρ_{<}^{S}, ρ_{U,<}^{B} + ρ_{<}^{S}) − 2 D(ρ_{U,<}^{B} + ρ_{<}^{S}, ρ_{>})

and drop the two negative terms on the right hand side. We arrive at

E_{c}^{B}(Z) − E^{S}(Z) ≤ tr [B_{c}[\frac{Z}{|x|}]d_{<}^{B}] − tr [S[\frac{Z}{|x|}]d_{<}^{S}] + \underbrace{tr [(\frac{Z}{|x|} − U_{c}(\frac{Z}{|x|})) d_{>}]}_{=:R_{1}}

+ \underbrace{D(ρ_{U,>} − ρ_{>}, ρ_{U,>} + ρ_{>})}_{=:R_{2}} +2 \underbrace{D(ρ_{U,<}^{B}, ρ_{U}^{B} + ρ^{S})}_{=:R_{3}} +const Z^{47/24}. (4.14)

As we shall see, the first two terms will yield the Scott correction. In the following subsections we prove that R_{1}, R_{2}, and R_{3} are relatively small remainder terms. Hence, we wish to control the effects of the twisting operation U_{c}, which stems from the electronic projection, on the electrostatic Coulomb energy.

4.2.1 Controlling the electron projection for high angular momenta

Our task in this subsection is to prove that for large angular momenta, the twisted and untwisted electrostatic energy are asymptotically equal. We start by comparing the electric potential energy with or without electron projection for large angular momentum. This will imply that the term R_{1} in (4.14) is relatively small.

Lemma 4.6. In the limit Z → ∞ one has uniformly in κ = Z/c ∈ (0, κ^{B}]

∫ (ρ_{>}(x) − ρ_{U,>}(x)) \frac{dx}{|x|} = tr [(|x|^{−1} − U_{c}(|x|^{−1})) d_{>}] = O(Z^{11/12}).

Proof. Let {ψ_{α}} stand for the Macke orbitals building up d_{>} which we label by α = (j, l, m, n); see (E.1) and preceding equations in Appendix E.2. By the scaling x ↦ x/c one has the relation

⟨ψ_{α}, [|x|^{−1} − U_{c}(|x|^{−1})] ψ_{α}⟩ = c ⟨ψ_{α}^{(c)}, [|x|^{−1} − U_{1}(|x|^{−1})] ψ_{α}^{(c)}⟩

where ψ_{α}^{(c)}(x) := c^{−3/2}ψ_{α}(x/c). Assuming that α corresponds to a fixed (large) (j, l) we may use Lemma 2.8 to estimate the right-hand side by a constant times

\frac{c}{l^{2}} ⟨ψ_{α}^{(c)}, p^{2}ψ_{α}^{(c)}⟩ = \frac{1}{l^{2} c} ⟨ψ_{α}, p^{2}ψ_{α}⟩ .

Using that Z/c ≤ κ^{B} we obtain the estimate

tr [(|x|^{−1} − U_{c}(|x|^{−1})) d_{>}] ≤ const \frac{κ^{B}}{Z} ∑_{l=L}^{∞} \frac{1}{l^{2}} ∑_{j=l±1/2} tr_{j,l} [p^{2}d_{>}] .

The proof is completed using Lemma E.1 from Appendix E.3.

Next, we estimate the difference of Coulomb energies corresponding to large total angular momenta. This shows that the term R_{2} in (4.14) may be neglected.

Lemma 4.7. In the limit Z → ∞,one has uniformly in κ = Z/c ∈ (0, κ^{B}]

R_{2} = D(ρ_{U,>} − ρ_{>}, ρ_{>} + ρ_{U,>}) = O(Z^{5/3}).

Proof. We define v := (ρ_{>} + ρ_{U,>}) ∗ | · |^{−1} to be the electric potential generated by ρ_{>} + ρ_{U,>} which is obviously spherically symmetric and obeys

v(0) = tr [d_{>} (|x|^{−1} + U_{c}(|x|^{−1}))]

= 2 tr [d_{>}|x|^{−1}] − tr [d_{>} (|x|^{−1} − U_{c}(|x|^{−1}))] .

According to [55] (see also (E.8)) the first term on the right side is O(Z^{4/3}). Moreover the second term is O(Z^{11/12}) by Lemma 4.6, hence much smaller than the first term. Now,

D(ρ_{>} − ρ_{U,>}, ρ_{>} + ρ_{U,>}) = \frac{1}{2} tr [d_{>} (v − U_{c}(v))] .  (4.15)

Decomposing the trace in (4.15) into the orbitals contributing to d_{>} and scaling x ↦ x/c enables us to employ Lemma 2.9 to obtain the bound

tr [d_{>} (v − U_{c}(v))] ≤ \frac{const}{c^{2}} v(0) tr [d_{>}p^{2}] .

This concludes the proof, since again from [55] (see (E.8)) we conclude that the trace on the right-hand side is O(Z^{7/3}).

4.2.2 Contribution from low angular momenta to the Coulomb energy

We now show that the term R_{3} in (4.14) is negligible.

Lemma 4.8. In the limit Z → ∞ one has uniformly in κ = Z/c ∈ (0, κ^{B}]

R_{3} = D(ρ_{U,<}^{B}, ρ_{U}^{B} + ρ^{S}) = O(Z^{11/6} log Z).

Proof. We first treat the term D(ρ_{U,<}^{B}, ρ_{U,>}+ρ_{>}). By construction the densities ρ_{U,j}^{B} are spherically symmetric and satisfy according to (4.13)

∫ ρ_{U,l}^{B}(x) dx = ∫ ρ_{l}^{B}(x) dx = 2(2l + 1)(K − l), 0 ≤ l < L.  (4.16)

Recalling the choice of K and L we see that

∫ ρ_{U,<}^{B}(x) dx = O(Z^{1/2}).  (4.17)

It follows from (E.8) and Lemma 4.6 that

∫ \frac{ρ_{U,>}(x) + ρ_{>}(x)}{|x|}dx = O(Z^{4/3}).

Hence Newton’s theorem [49] yields

D(ρ_{U,<}^{B}, ρ_{U,>} + ρ_{>}) ≤ \frac{1}{2} ∫ ρ_{U,<}^{B}(x)dx ∫ \frac{ρ_{U,>}(y) + ρ_{>}(y)}{|y|}dy = O(Z^{11/6}).

In the remainder of the proof we are concerned with the term D(ρ_{U,<}^{B}, ρ_{U,<}^{B} + ρ_{<}^{S}). Noting that

D(ρ_{U,<}^{B}, ρ_{U,<}^{B} + ρ_{<}^{S}) ≤ \frac{3}{2}D(ρ_{U,<}^{B}, ρ_{U,<}^{B}) + \frac{1}{2}D(ρ_{<}^{S}, ρ_{<}^{S}).

Documenta Mathematica 14 (2009) 463–516

and that according to [55, Prop. 3.5] D(ρ_{<}^{S}, ρ_{<}^{S}) = O(Z^{11/6}), it suffices to consider D(ρ_{U,<}^{B}, ρ_{U,<}^{B}). We split the lowest angular momentum corresponding to l ≤ 2Z/c − 1/4 =: l_{0} off and define

d_{⊢}^{B} := ∑_{l≤l_{0}} d_{l}^{B}, d_{⊣}^{B} := ∑_{l>l_{0}}^{L−1} d_{l}^{B},

and

ρ_{U, ⊢}^{B} := tr_{ℂ^{2}} (U_{c}(d_{⊢}^{B})(x, x)) , ρ_{U, ⊣}^{B} := tr_{ℂ^{2}} (U_{c}(d_{⊣}^{B})(x, x)) .

Note that in case l_{0} < 0 there is no need for this procedure. Accordingly, we estimate

D(ρ_{U,<}^{B}, ρ_{U,<}^{B}) ≤ 2 D(ρ_{U, ⊢}^{B}, ρ_{U, ⊢}^{B}) + 2 D(ρ_{U,⊣}^{B}, ρ_{U,⊣}^{B}).

For an estimate of the second part corresponding to l_{0} < l < L, we apply the following angular momentum barrier inequality

B_{c}[0] ≥ U_{c} (\frac{2Z}{|x|} χ_{\{|x|≤r_{l}\}})  (4.18)

on ℌ_{j,l}, where r_{l} = ((l + 1/2)^{2}c^{2} − 4Z^{2}) /(4Zc^{2}) and l > 2Z/c. This bound follows by applying U_{1} to the inequality in [31, Lemma 2.6] with R_{l} = [(l + 1/2)^{2} − 4κ^{2}]/(4κ) and scaling x ↦ x/c. Inequality (4.18) implies

tr [U_{c} (|x|^{−1}) d_{l}^{B}] ≤ \frac{1}{2Z} tr [B_{c}[0] d_{l}^{B}] + tr [U_{c} (|x|^{−1}χ_{\{|x|>r_{l}\}}) d_{l}^{B}]

≤ \frac{1}{2} tr [U_{c} (|x|^{−1}) d_{l}^{B}] + \frac{4Z}{(l + 1/2)^{2} − 4Z^{2}/c^{2}} tr[d_{l}^{B}].

Here the last inequality used the fact that eigenfunctions of d_{l}^{B} are eigenfunctions of B_{c}[Z|x|^{−1}] with negative eigenvalue. Now, note that

(l + 1/2)^{2} − 4Z^{2}/c^{2} = (l + 1/2 + 2Z/c)(l + 1/2 − 2Z/c) ≥ const (l + 1/2)^{2}

for l ≥ l_{0}. Hence, using (4.16) and summing over l we obtain

∫ \frac{ρ_{U,⊣}^{B}(x)}{|x|}dx = ∑_{l>l_{0}}^{L−1} tr [U_{c}(|x|^{−1})d_{l}^{B}]

≤ const Z ∑_{l=0}^{L−1}(l + 1/2)^{−2} ∫ ρ_{l}^{B}(x) dx = O(Z^{4/3} log Z).

Accordingly, Newton’s theorem and (4.17) yield

D(ρ_{U,⊣}^{B}, ρ_{U,⊣}^{B}) ≤ \frac{1}{2} ∫ ρ_{U,⊣}^{B}(x) dx ∫ \frac{ρ_{U,⊣}^{B}(x)}{|x|} dx = O(Z^{11/6} log Z).

Finally, we consider the contribution from l ≤ l_{0}. Note that then l ≤ 2κ^{B} − 1/4 < 2. We claim that the electrostatic energy corresponding to the electrons in this subspace is bounded by

D(ρ_{U, ⊢}^{B}, ρ_{U, ⊢}^{B}) ≤ const cK^{2}.  (4.19)

Since by the choice of l_{0} one has 2Z/c ≥ l + 1/4 ≥ 1/4, estimate (4.19) will imply that D(ρ_{U, ⊢}^{B}, ρ_{U, ⊢}^{B}) ≤ const ZK^{2} = O(Z^{5/3}) and hence complete the proof of Lemma 4.8. By scaling it suffices to prove (4.19) for c = 1, which we will assume in the sequel. The Hardy-Littlewood-Sobolev inequality (see, e.g., [43, Thm. 4.3]) implies that

D(ρ_{U, ⊢}^{B}, ρ_{U, ⊢}^{B}) ≤ const ∥ρ_{U, ⊢}^{B}∥_{6/5}^{2}.  (4.20)

The triangle inequality together with the definition of U and (B.2) yields

∥ρ_{U, ⊢}∥_{6/5} ≤ ∑_{α∈A} ∑_{ν=0,1} ∥Φ_{ν}ψ_{α}∥_{12/5}^{2},  (4.21)

where {ψ_{α}|α ∈ A} stands for the collection of normalized eigenfunctions building up d_{⊢}^{B}, i.e., the corresponding sum ranges over all indices (j, l, m, n). We further estimate with the help of Lemma B.1 and Theorem 2.4,

∥Φ_{ν}ψ_{α}∥_{12/5}^{2} ≤ const ∥ψ_{α}∥_{12/5}^{2} ≤ const .

This, together with (4.20), (4.21) and the fact that the number of indices in A is bounded by a constant times K proves (4.19).

4.2.3 Finishing the proof

We repeat (4.14),

E^{S}(Z) − E_{c}^{B}(Z) ≥ tr[S[\frac{Z}{|x|}]d_{<}^{S}] − tr[B_{c}[\frac{Z}{|x|}]d_{<}^{B}] − R_{1} − R_{2} − R_{3} − const Z^{\frac{47}{24}} .

By Lemmata 4.6, 4.7, and 4.8 we have uniformly in κ = Z/c ∈ (0, κ^{B}]

R_{1} = O(Z^{23/12}), R_{2} = O(Z^{5/3}), R_{3} = O(Z^{11/6} log Z),

so these terms are of lower order than Z^{47/24}. Next, we scale x ↦ x/c and obtain

tr[S[Z|x|^{−1}]d_{<}^{S}] − tr[B_{c}[Z|x|^{−1}]d_{<}^{B}] = Z^{2}s(κ) − R_{4}

where s(κ) is introduced in (1.8) and

R_{4} := Z^{2}κ^{−2} ∑_{l=0}^{L−1}(2l + 1) ∑_{j=l±1/2} ∑_{n=K−l+1}^{∞} (λ_{n}(s_{l}(κ)) − λ_{n}(b_{j,l}(κ)) .

By Theorem 2.1 there is a constant such that for all 0 < κ ≤ κ^{B}

0 ≤ R_{4} ≤ Z^{2}κ^{−2} ∑_{l=0}^{L−1}(2l + 1) ∑_{j=l±1/2} ∑_{n=K−l+1}^{∞} |λ_{n}(b_{j,l}(κ)|

≤ const Z^{2} ∑_{l=0}^{L−1}(2l + 1) ∑_{n=K−l+1}^{∞}(n + l)^{−2}

≤ const Z^{2}L^{2}K^{−1} = O(Z^{11/6}).

This concludes the proof of the lower bound and hence of our main result.

Acknowledgments. We thank Elliott Lieb and Robert Seiringer for supportive discussions. H.S. thanks the Departments of Mathematics and Physics of Princeton University and R.F. and S.W. thank the Department of Mathematics of LMU Munich for hospitality while parts of this work were done. We also thank Volker Bach for stimulating questions resulting in several improvements. The work has been partially supported by the Deutscher Akademischer Austauschdienst, grant D/06/49117 (R.F.), the U.S. National Science Foundation, grant PHY 01 39984 (H.S.), the Deutsche Forschungsgemeinschaft, grant SI 348/13-1 (H.S.), and a Sloan Fellowship (S.W.).

Note added in proof. After this paper has been submitted, one of us (R. F.) found an easier proof of a stronger inequality than (2.19). This proof is based on an inequality from [65] as well as Lemma 2.6 in the present paper; see [29].

A Partial wave analysis

For the convenience of the reader and for normalization of the notation we gather some fact on the partial wave analysis of the Brown-Ravenhall operator. We denote by Y_{l,m} the normalized spherical harmonics on the unit sphere 𝕊^{2} (see, e.g., [48], p. 421) with the convention that Y_{l,m} ≡ 0 if |m| > l, and we define for j ∈ ℕ_{0} + \frac{1}{2}, l ∈ ℕ_{0}, and m = −j, …, j the spherical spinors

Ω_{j,l,m}(ω) := \begin{cases}(\begin{matrix}\sqrt{\frac{j+m}{2j}} Y_{l,m− \frac{1}{2}} (ω) \\ \sqrt{\frac{j−m}{2j}} Y_{l,m+ \frac{1}{2}} (ω) \end{matrix}) & if j = l + \frac{1}{2}, \\ (\begin{matrix}−\sqrt{\frac{j−m+1}{2j+2}} Y_{l,m− \frac{1}{2}} (ω) \\ \sqrt{\frac{j+m+1}{2j+2}} Y_{l,m+ \frac{1}{2}} (ω) \end{matrix}) & if j = l − \frac{1}{2}. \end{cases}  (A.1)

The set of admissible indices is I := {(j, l, m) : j ∈ ℕ − 1/2, l = j ± 1/2, m = −j, ..., j}. It is known that the functions Ω_{j,l,m}, (j, l, m) ∈ I, form an orthonormal basis of the Hilbert space L^{2}(𝕊^{2}; ℂ^{2}). They are joint eigenfunctions of J^{2}, J_{3}, and L^{2} with eigenvalues given by j(j + 1), l(l + 1), and m. The subspace ℌ_{j,l,m} corresponding to the joint eigenspace of total angular momentum J^{2} with eigenvalue j(j + 1) and angular momentum L^{2} with eigenvalue l(l + 1) is then given by

ℌ_{j,l,m} = span{x ↦ |x|^{−1} f(|x|) Ω_{j,l,m}(ω_{x}) | f ∈ L^{2}(ℝ_{+})}

where ω_{x} := x/|x|. This leads to the orthogonal decomposition

ℌ = ⨁_{j∈ℕ_{0}+ \frac{1}{2}} ⨁_{l=j±1/2} ℌ_{j,l},  ℌ_{j,l} = ⨁_{m=−j}^{j} ℌ_{j,l,m},  (A.2)

of the Hilbert space of two spinors. We note that the Fourier transform,

ψ̂(p) := (2π)^{−3/2} ∫_{ℝ^{3}} e^{−ip·x}ψ(x) dx,  (A.3)

leaves the spaces ℌ_{j,l} invariant. Namely, if we decompose ψ according to (A.2),

ψ(x) = ∑_{(j,l,m)∈I} r^{−1}ψ_{j,m,l}(r) Ω_{j,l,m}(ω_{x}),

then

ψ̂(p) = ∑_{(j,l,m)∈I} p^{−1} (F_{l}ψ_{j,m,l}) (p) Ω_{j,l,m}(ω_{p})  (A.4)

with the Fourier-Bessel transform

(F_{l}f)(p) = i^{−l}\sqrt{\frac{2}{π}} ∫_{0}^{∞} f(r)j_{l}(rp)rp dr.  (A.5)

Here j_{l} is a spherical Bessel function. Moreover,

∥ψ∥^{2} = ∑_{(j,l,m)∈I} ∫_{0}^{∞} |ψ_{j,m,l}(r)|^{2}dr = ∑_{(j,l,m)∈I} ∫_{0}^{∞} |(F_{l}ψ_{j,m,l})(p)|^{2}dp = ∥ψ̂∥^{2}.

B Properties of the twisting operators

We define the helicity operator H = ω_{p} · σ on ℌ by

Ĥψ(p) := σ · ω_{p}ψ̂(p).  (B.1)

It follows from the pointwise identity

(ω_{p} · σ)Ω_{j,l,m}(ω_{p}) = −Ω_{j,2j−l,m}(ω_{p}),  (B.2)

see, e.g., Greiner [33, p. 171, (12)], that H is an isomorphism between ℌ_{j,l} and ℌ_{j,2j−l}. Moreover, since (σ · a) (σ · b) = a · b + iσ · (a × b) for any a, b ∈ ℝ^{3}, we infer that H is an involution on ℌ, i.e., H = H^{−1}. We shall need to consider H on L^{p} spaces with p ≠ 2. The relevant properties are summarized in the next lemma, together with those of the operators

Φ̂_{ν}ψ(p) := Φ_{ν}(p) ψ̂(p),  (B.3)

introduced in (1.4). Note that while Φ_{0} acts trivially on the spin, Φ_{1} involves the helicity H.

Lemma B.1 (L^{p}-properties of H and Φ_{ν}). The operators H and Φ_{ν}, ν = 0, 1, extend to bounded operators from L^{p}(ℝ^{3}, ℂ^{2}) to L^{p}(ℝ^{3}, ℂ^{2}) for any p ∈ (1, ∞).

Proof. The L^{p}-boundedness of H follows from that of the Riesz transformation, see [68, Ch. II-III]. Therefore, to prove the statement about the operators Φ_{ν}, it suffices to consider the operators ϕ_{ν} defined analogously as in (B.3) on L^{2}(ℝ^{3}). Since p ↦ ϕ_{ν}(p) is smooth away from the origin and p^{k}∂^{k}ϕ_{ν} is bounded for k = 0, 1, 2, the Hörmander-Mihlin multiplier theorem [68, Thm. IV.3] implies that ϕ_{ν} extend to bounded operators from L^{p}(ℝ^{3}) to L^{p}(ℝ^{3}) for any p ∈ (1, ∞).

Lemma B.2. For all p, q ∈ ℝ^{3}

1 − Φ_{0}(p)Φ_{0}(q) − Φ_{1}(p)Φ_{1}(q)

= \frac{1}{2} ∑_{ν=0}^{1} (Φ_{ν}(p) − Φ_{ν}(q))^{2} + \frac{1}{2} (Φ_{1}(q)Φ_{1}(p) − Φ_{1}(p)Φ_{1}(q)) . (B.4)

and furthermore

|Φ_{0}(p) − Φ_{0}(q)|^{2} ≤ \frac{|p − q|^{2}}{8E(p)^{2}E(q)^{2}}

|Φ_{1}(p) − Φ_{1}(q)|^{2} ≤ \frac{|p − q|^{2}}{E(p)E(q)}

|Φ_{1}(q)Φ_{1}(p) − Φ_{1}(p)Φ_{1}(q)| ≤ \frac{\sqrt{|p||q|}|p − q|}{E(p)E(q)}

Proof. The first equality is an immediate consequence of the definition of Φ_{0} and Φ_{1}. From this definition we also conclude by an explicit calculation that

|Φ_{0}(p) − Φ_{0}(q)|^{2} = (ϕ_{0}(p) − ϕ_{0}(q))^{2} ≤ \frac{|p − q|^{2}}{8E(p)^{2}E(q)^{2}} .  (B.5)

Moreover, for a proof of the next inequality we write

|Φ_{1}(p) − Φ_{1}(q)|^{2} = (ϕ_{1}(p) − ϕ_{1}(q))^{2} + ϕ_{1}(p)ϕ_{1}(q)|ω_{p} − ω_{q}|^{2},

and estimate the last two terms with the help of the inequalities

(ϕ_{1}(p) − ϕ_{1}(q))^{2} ≤ \frac{(|p| − |q|)^{2}}{2E(p)^{2}E(q)^{2}} ≤ \frac{|p − q|^{2}}{2E(p)^{2}E(q)^{2}} ,  (B.6)

and

ϕ_{1}(p) ≤ \frac{1}{\sqrt{2}} \frac{|p|}{E(p)} and |ω_{p} − ω_{q}|^{2} ≤ \frac{|p − q|^{2}}{|p||q|} .  (B.7)

Finally, for a proof of the last inequality we use

|Φ_{1}(q)Φ_{1}(p) − Φ_{1}(p)Φ_{1}(q)| = 2ϕ_{1}(p)ϕ_{1}(q) |σ · (ω_{p} × ω_{q})|
≤ 2ϕ_{1}(p)ϕ_{1}(q)|ω_{p} − ω_{q}|.

Using again (B.7) concludes the proof of the third inequality.

C Basics of relativistic hydrogenic operators

In this section we collect – following [17] – some basic properties of the operators B_{κ} and C_{κ} which describe hydrogenic atoms in the Brown-Ravenhall respectively Chandrasekhar model. For pedagogical reasons we first discuss their massless analogues,

B_{κ}^{(0)} := |p| − \frac{κ}{2} (|x|^{−1} + ω_{p} · σ |x|^{−1} ω_{p} · σ) , C_{κ}^{(0)} := |p| − κ|x|^{−1}. (C.1)

C.1 Massless case

Expanding ψ̂ as in (A.4) and using (B.2) yields [17] the following partial diagonalization of the massless operators,

⟨ψ, B_{κ}^{(0)}ψ⟩ = ∑_{(l,m,s)∈I}⟨F_{l}ψ_{j,m,l}, b_{j}^{(0)}(κ) F_{l}ψ_{j,m,l}⟩,  (C.2)
⟨ψ, C_{κ}^{(0)}ψ⟩ = ∑_{(l,m,s)∈I}⟨F_{l}ψ_{j,m,l}, c_{l}^{(0)}(κ) F_{l}ψ_{j,m,l}⟩.  (C.3)

Here the operators b_{j}^{(0)}(κ) and c_{l}^{(0)}(κ) are densely defined in L^{2}(ℝ_{+}) through their quadratic forms,

⟨f, b_{j}^{(0)}(κ)f⟩ := ∫_{0}^{∞} p |f(p)|^{2}dp − κ ∫_{0}^{∞} ∫_{0}^{∞} \overline{f(p)} k_{j}^{B}(p, q) f(q) dq dp,
⟨f, c_{l}^{(0)}(κ)f⟩ := ∫_{0}^{∞} p |f(p)|^{2}dp − κ ∫_{0}^{∞} ∫_{0}^{∞} \overline{f(p)} k_{l}^{C}(p, q) f(q) dq dp,

with maximal form domain denoted by 𝔔(b_{j}^{(0)}(κ)) and 𝔔(c_{l}^{(0)}(κ)). In the above expression, the integral kernels k_{j}^{B} and k_{l}^{C} are given by

k_{j}^{B}(p, q) := \frac{1}{2π} [Q_{j−1/2} (\frac{1}{2} (\frac{p}{q} + \frac{q}{p})) + Q_{j+1/2} (\frac{1}{2} (\frac{p}{q} + \frac{q}{p}))] ,  (C.4)
k_{l}^{C}(p, q) := \frac{1}{π} Q_{l} (\frac{1}{2} (\frac{p}{q} + \frac{q}{p})) ,  (C.5)

where Q_{l} are the Legendre functions of the second kind, i.e.,

Q_{l}(z) = \frac{1}{2} ∫_{−1}^{1} P_{l}(t)(z − t)^{−1}dt  (C.6)

with P_{l} standing for Legendre polynomials; see Stegun [67] for the notation and some properties of these special functions. It was proved in [17] and [40, Eq. (5.33)] that the operators (C.1) are selfadjoint and lower bounded if and only if κ ≤ κ^{#}, # = B, C, cf. (2.3). More can be said about the reduced operators b_{j}^{(0)}(κ) and c_{l}^{(0)}(κ). They are lower bounded (in fact, non-negative) if and only if

\frac{1}{κ} ≥ \frac{1}{κ_{j}^{B}} := ∫_{0}^{∞} k_{j}^{B}(1, t) \frac{dt}{t},  (C.7)
\frac{1}{κ} ≥ \frac{1}{κ_{l}^{C}} := ∫_{0}^{∞} k_{l}^{C}(1, t) \frac{dt}{t}.  (C.8)

This follows by the same lines of reasoning as in [17]. Since [67, (8.4)] P_{0}(t) = 1, P_{1}(t) = t, we have

Q_{0}(t) = \frac{1}{2} log \frac{t + 1}{t − 1}, Q_{1}(t) = \frac{t}{2} log \frac{t + 1}{t − 1} − 1,  (C.9)

such that κ_{0}^{C} = 2/π, κ_{1}^{C} = π/2 and thus κ_{1/2}^{B} = 2/(2/π + π/2). The critical coupling constants κ_{j}^{B} and κ_{l}^{C} are strictly increasing in j and l and, in particular, κ_{1/2}^{B} = κ^{B} and κ_{0}^{C} = κ^{C}. This follows from the pointwise monotonicity

Q_{l}(t) ≥ Q_{l′}(t)  for l′ ≥ l and t > 1  (C.10)

which, in turn, is evident from the integral representation

Q_{l}(x) = ∫_{x+\sqrt{x^{2}−1}}^{∞} \frac{z^{−l−1}}{\sqrt{1 − 2xz + z^{2}}} dz, x > 1;

see Whittaker and Watson [77, p. 334, Chap. X, Sec. 3.2].

C.2 Massive case

Similarly as in the previous subsection, one obtains the following partial diagonalization of the massive hydrogenic Brown-Ravenhall and Chandrasekhar operators,

⟨ψ, B_{κ} ψ⟩ = ∑_{(l,m,s)∈I}⟨F_{l}ψ_{j,m,l}, b_{j,l}(κ) F_{l}ψ_{j,m,l}⟩,  (C.11)
⟨ψ, C_{κ}ψ⟩ = ∑_{(j,l,m)∈I}⟨F_{l}ψ_{j,m,l}, c_{l}(κ) F_{l}ψ_{j,m,l}⟩.  (C.12)

Here the operators b_{j,l}(κ) and c_{l}(κ) are densely defined in L^{2}(ℝ_{+}) through their quadratic forms,

⟨f, b_{j,l}(κ)f⟩
:= ∫_{0}^{∞}(E(p) − 1)|f(p)|^{2}dp − κ ∫_{0}^{∞} ∫_{0}^{∞} \overline{f(p)} k_{j,l}^{B}(p, q) f(q) dq dp, (C.13)
⟨f, c_{l}(κ)f⟩
:= ∫_{0}^{∞}(E(p) − 1)|f(p)|^{2}dp − κ ∫_{0}^{∞} ∫_{0}^{∞} \overline{f(p)} k_{l}^{C}(p, q) f(q) dq dp  (C.14)

with maximal form domain denoted by 𝔔(b_{j,l}(κ)) and 𝔔(c_{l}(κ)), cf. [17]. In the above expression, the integral kernel k_{j,l}^{B} depends, in contrast to the massless case, on both j and l and is given by

k_{j,l}^{B}(p, q)

:= \frac{1}{π} [ϕ_{0}(p)Q_{l} (\frac{1}{2} (\frac{p}{q} + \frac{q}{p})) ϕ_{0}(q) + ϕ_{1}(p)Q_{2j−l} (\frac{1}{2} (\frac{p}{q} + \frac{q}{p})) ϕ_{1}(q)] .

The form (C.13) defines a self-adjoint semi-bounded operator b_{j,l}(κ) if and only if κ ≤ κ_{j}^{B} (Evans et al. [17]). A trivially modified argument shows that (C.14) defines a self-adjoint semi-bounded operator c_{l}(κ) if and only if κ ≤ κ_{l}^{C}. In fact, the semiboundedness of the massive cases and the massless cases are equivalent, since the differences of the massive and massless forms are bounded (Tix [75, Thm. 1]). (One even knows that b_{j,l}(κ)+1 is non-negative (Tix [76]).)

D Critical Chandrasekhar operator on a finite domain

Lieb and Yau [46] have shown that the critical Chandrasekhar operator |p| − κ^{C}|x|^{−1} when restricted to a ball has only discrete spectrum with eigenvalues accumulating at infinity at the rate predicted by the semiclassical result for |p| alone. This is remarkable since the semiclassical phase-space volume corresponding to |p| − κ|x|^{−1} is infinite. We aim at proving an analogous result for the Chandrasekhar operator restricted to a ball and restricted to the subspace of fixed angular momentum. In the proof of Theorem 2.1 it is essential to handle coupling constants which are larger than κ^{C}, all the way up to and including κ_{1}^{C}. In order to define the above operator we consider for R > 0 and l ∈ ℕ the Hilbert space

𝔉_{l}(R) := {f ∈ L^{2}(0, ∞) | (F_{l}^{−1}f) (r) = 0 for all r ≥ R} ,

where F_{l} denotes the Fourier-Bessel transformation, cf. (A.5). The quadratic form given by ⟨f, c_{l}^{(0)}(κ)f⟩ with domain 𝔉_{l}(R)∩𝔔(c_{l}^{(0)}(κ)) defines for all κ ≤ κ_{l}^{C} a self-adjoint, non-negative operator in 𝔉_{l}(R) which we will denote by c_{l}^{(0)}(κ, R).

Lemma D.1. Let l ∈ ℕ. There is a constant such that for all R > 0, μ > 0, and κ ≤ κ_{l}^{C}

tr (c_{l}^{(0)}(κ, R) − μ)_{−} ≤ const μ^{2} R.  (D.1)

We have not tried to track the l-dependence of the constant, since the cases l = 0, 1 will be enough for our purpose.

Proof. For a proof of (D.1) we basically follow the argument in [46]. The starting point is the following reduction to a simpler variational problem involving only functions. Namely, for any non-negative function h : ℝ_{+} → ℝ_{+}, let

t(p) := \frac{κ_{l}^{C}}{πh(p)} ∫_{0}^{∞} Q_{l}(\frac{1}{2}(\frac{p}{q} + \frac{q}{p}))h(q) dq .

Then

− tr (c_{l}(κ_{l}^{C}, R) − μ)_{−} ≥ inf {∫_{0}^{∞} σ(p) (p − μ − t(p)) dp | 0 ≤ σ ≤ M_{l}} (D.2)

where M_{l} := R sup_{r>0}(2/π)r^{2}j_{l}^{2}(r). The proof of (D.2) is analogous to the one of [46, Eq. (7.8)]. We merely replace the Fourier transformation in ℝ^{3} by the Fourier-Bessel transformation F_{l} in ℝ_{+}. From now on we assume that l ≥ 1 and comment on the necessary changes in case l = 0 at the end. We choose h of the form

h(p) = \{ \begin{matrix}p^{−1} − (A/2)p^{−2} & if p > A, \\ (2A)^{−1} & if p ≤ A. \end{matrix}

Below we shall show that the constant A can be picked in such a way that for some δ > 0

p − μ − t(p) ≥ \{ \begin{matrix}0 & if p ≥ δ^{−1}A, \\ −const A^{−1}μ^{2} & if p < δ^{−1}A. \end{matrix}  (D.3)

In view of (D.2) this will prove the result, since then

inf {∫_{0}^{∞} σ(p) (p − μ − t(p)) dp | 0 ≤ σ ≤ M_{l}}

≥ − const A^{−1}μ^{2}M_{l} ∫_{0}^{δ^{−1}A}dp = −const δ^{−1}μ^{2}M_{l}.

To prove (D.3) we recall that ∫_{0}^{∞} Q_{l}(\frac{1}{2}(t + \frac{1}{t})) \frac{dt}{t} = π(κ_{l}^{C})^{−1}, cf. (C.8), and hence by a straightforward calculation

p − t(p) = p \frac{κ_{l}^{C}}{π} ∫_{0}^{∞} Q_{l}(\frac{1}{2}(t + \frac{1}{t})) (\frac{1}{t} − \frac{h(tp)}{h(p)}) dt

= p \frac{κ_{l}^{C}}{π} \begin{cases}\frac{A/2p}{1−A/2p} (F (1) − F (A/p)) & if p ≥ A, \\ (−F (1) + F (p/A)) & if p < A. \end{cases}

Here for 0 < s ≤ 1 we have set

F (s) := ∫_{0}^{s} Q_{l}(\frac{1}{2}(t + \frac{1}{t})) (\frac{1}{t} − \frac{1}{s})^{2} dt .

Since Q_{l}(τ) ≤ Q_{1}(τ), which vanishes like a constant times τ^{−2} as τ → ∞, one has F (s) → 0 as s → 0. Choosing δ ∈ (0, 1) such that F (s) ≤ \frac{1}{2}F (1) for all 0 < s ≤ δ, we have shown that for all p ≥ δ^{−1}A one has

p − t(p) ≥ \frac{A F (1)}{8(1 − A/2p)} ≥ A \frac{F (1)}{8} .

For A ≤ p < δ^{−1}A we use the monotonicity, dF/ds ≥ 0, to bound

p − t(p) ≥ 0.

Finally, for 0 ≤ p < A we drop the term F (p/A) ≥ 0 to obtain

p − t(p) ≥ −\frac{p}{2}F (1) ≥ −A\frac{F (1)}{2}.

Choosing A := 8μ/F (1) yields the claimed inequality (D.3). In case l = 0, the function h can be chosen as before. However, the corresponding expressions F (1) − F (s) should be interpreted as a single integral, and estimated with slightly more care.

Corollary D.2. Let l ∈ ℕ. Then there is a constant such that for all 0 < κ ≤ κ_{l}^{C}, all μ > 0 and all functions χ on ℝ_{+} with χ = 0 on [R, ∞) for some R > 0 one has

N_{l}(0, χ (|p| − κ|x|^{−1} − μ) χ) ≤ const μR.

Proof. The variational principle implies that

N_{l}(0, χ (|p| − κ|x|^{−1} − μ) χ) ≤ N(μ, c_{l}^{(0)}(κ, R)).

Indeed, if V_{l} is the negative spectral subspace of χ (|p| − κ|x|^{−1} − μ) χ with fixed l, then any f ∈ F_{l}χV_{l} ⊂ 𝔉_{l}(R) satisfies ⟨f, (c_{l}^{(0)}(κ, R) − μ) f⟩ < 0. Hence, in order to prove the assertion, it suffices to show that

N(μ, c_{l}^{(0)}(κ, R)) ≤ const μR.

For a proof, we note that the elementary inequality χ_{(−∞,μ)}(E) ≤ \frac{(E−λ)_{−}}{λ−μ}, valid for any μ < λ, together with Lemma D.1 implies that

N(μ, c_{l}^{(0)}(κ, R)) ≤ (λ − μ)^{−1} tr(c_{l}^{(0)}(κ, R) − λ)_{−} ≤ const (λ − μ)^{−1}λ^{2}R.

The proof is completed by optimizing over λ.

E The trial density matrix

In this section we define the density matrices d^{S} and d^{B} that we use to bound the Schrödinger energy, respectively the Brown-Ravenhall energy, from above. Both density matrices are split into two parts corresponding to low and high angular momenta

d^{S} := d_{<}^{S} + d_{>}, d^{B} := d_{<}^{B} + d_{>}.

Low angular momenta correspond to orbits whose perinucleon is close to the nucleus, while high angular momenta ensure that the orbits are never close to the nucleus. We will cut between these two at L := [Z^{1/12}].

E.1 Low angular momenta

In the vicinity of the nucleus the nuclear attraction dominates the interaction with the other electrons. This motivates to choose the orbitals as the ones of the Bohr atom, i.e., as the eigenfunctions of the unscreened operator with nuclear charge Z. The corresponding density matrices d_{<}^{#} are of the form

d_{<}^{#} = ∑_{l=0}^{L−1} d_{j}^{#},  d_{l}^{#} = ∑_{j=l±1/2, j≥1/2} d_{j,l}^{#}

and

d_{j,l}^{#} = ∑_{m=−j}^{j} ∑_{n=1}^{K−l} |ψ_{j,l,m,n}^{#}⟩⟨ψ_{j,l,m,n}^{#}|.

Here K = [const Z^{1/3}] with some positive constant, i.e., on the order of the last occupied shell of the Bohr atom. We now turn to the definition of the orbitals ψ_{j,l,m,n}^{#} for which we consider the cases # = B, S separately. In the Brown-Ravenhall case we choose ψ_{j,l,m,n}^{B} such that its Fourier transform is

ψ̂_{j,l,m,n}^{B}(p) = p^{−1}f_{j,l,n}^{B}(p)Ω_{j,l,m}(ω_{p}),

where f_{j,l,n}^{B} is the n-th eigenfunction of the operator V_{c} b_{j,l}(Z/c) V_{c}^{∗} in L^{2}(ℝ_{+}). Here the unitary scaling operator V_{c} is defined by (V_{c}f)(p) := c^{−1/2}f(p/c) and we recall that the operator b_{j,l}(κ) was defined in Subsection C.2. The operators V_{c} b_{j,l}(Z/c) V_{c}^{∗} appear as the angular momentum reductions of B_{c}[Z|x|^{−1}]. Indeed, by (C.11) and scaling one has

⟨ψ, B_{c}[Z|x|^{−1}] ψ⟩ = c^{2} ∑_{(j,l,m)∈I}⟨ψ̂_{j,m,l}, V_{c} b_{j,l}(Z/c) V_{c}^{∗} ψ̂_{j,m,l}⟩.

In the Schrödinger case we choose

ψ_{j,l,m,n}^{S}(x) = r^{−1}f_{l,n}^{S}(r)Ω_{j,l,m}(ω_{x}),

where f_{l,n}^{S} is the n-th eigenfunction of −\frac{1}{2} \frac{d^{2}}{dr^{2}} + \frac{l(l+1)}{2r^{2}} − \frac{Z}{r} in L^{2}(ℝ_{+}) with Dirichlet boundary conditions.

E.2 High angular momenta

For large angular momenta, the electrons are sufficiently far from the center moving – classically speaking – slowly. This motivates to pick non-relativistic orbitals in both in the relativistic and non-relativistic case. Moreover, for large quantum numbers the correspondence principle would predict quasi-classical behavior (in the quantum sense) as well. This motivates the following choice which we take – with slight modifications – from [55]:

d_{>} := ∑_{l≥L} d_{l}, d_{l} := ∑_{j=l±1/2} ∑_{m=−j}^{j} ∑_{n∈ℤ} w_{n,l}|φ_{n,l}Ω_{j,l,m}⟩⟨φ_{n,l}Ω_{j,l,m}|. (E.1)

We repeat at this point the construction of the Macke orbitals φ_{n,l} and their weights w_{n,l}. We will also present a new estimate not directly given in that paper. The semi-classical mean-field in which the electrons move is the Thomas-Fermi potential ϕ_{TF} (see (4.3)). According to Hellmann [36] the semi-classical electron density for fixed angular momentum is

σ_{l}^{H}(r) := \frac{2(2l + 1)}{π}\sqrt{2 [n_{Z}ϕ_{TF}(r) − \frac{(l + \frac{1}{2})^{2}}{2r^{2}} ]_{+}} ,  (E.2)

where we added the factor n_{Z} = (1 − aZ^{−1/2})^{2/3} for normalization purposes with some fixed positive a and where we replaced the self-generated field of the sum of the radial densities σ_{l} by the Thomas-Fermi potential. We will write ρ_{l}^{H} for the functions σ_{l}^{H} when a = 0, i.e., no normalization factor occurs. In passing we note that the densities ρ_{l}^{H} are the minimizers of the Hellmann functional with external potential given by the Thomas-Fermi density and no other interaction between the electrons (see [61]). The functions σ_{l}^{H} vanish for large l and we define

k′ := min{l ∈ ℕ | σ_{l}^{H} ≡ 0}.

By scaling, k′ is of the order Z^{1/3}. Moreover, since the function r ↦ ϕ_{TF}(r)r^{2} has exactly one maximum, the support of σ_{l}^{H} is an interval [r_{1}(l), r_{2}(l)]. We cannot use the density σ_{l}^{H} directly in defining semi-classical orbitals, since the derivative of its square root is not square integrable. Thus we pick two points,

x_{1}(l) := r_{1}(l) + T (l + \frac{1}{2})Z^{−1}, x_{2}(l) := r_{2}(l) − SZ^{−2/3}  (E.3)

for some positive S and T ∈ (0, 4), and set

ρ_{l}(r) := \begin{cases}2(2l + 1)α^{2}r^{2l+2}, & r ∈ [0, x_{1}(l)], \\ σ_{l}^{H}(r), & r ∈ [x_{1}(l), x_{2}(l)], \\ 2(2l + 1)β^{2} exp(−2^{3/2}Z^{2/3}r), & r ∈ [x_{2}(l), ∞). \end{cases}  (E.4)

The constants α and β are chosen such that ρ_{l} is continuous. We suppress their dependence on l in the notation. Next, we define for l < k′ and n ∈ ℤ the Macke orbitals

φ_{n,l}(r) := \frac{\sqrt{ζ′_{l}(r)}}{r}e^{iπk_{n,l}ζ_{l}(r)}  (E.5)

where ζ_{l} : [0, ∞) → [0, 1) is the Macke transform

ζ_{l}(r) := \frac{∫_{0}^{r} ρ_{l}(t)dt}{∫_{0}^{∞} ρ_{l}(t)dt}.  (E.6)

For l ≥ k′ we set φ_{n,l} :≡ 0. The integral

N_{j,l,m} := \frac{1}{2(2l + 1)} ∫_{0}^{∞} ρ_{l}(r)dr,

which is independent of j and m, will represent the number of electrons in the angular momentum channel (j, l, m). Moreover, we set ε_{l} := N_{j,l,m} − [N_{j,l,m}]. If [N_{j,l,m}] is odd, we pick k_{n,l} = 2n, otherwise k_{n,l} = 2n − 1. The weights are chosen as

w_{n,l} := \begin{cases}1 & |k_{n,l}| ≤ [N_{j,l,m}] − 1 \\ ε_{l}/2 & |k_{n,l}| = [N_{j,l,m}] + 1 \\ 0 & otherwise \end{cases}  (E.7)

which guarantees that ∑_{n∈ℤ} w_{n,l} = N_{j,l,m}. Strictly speaking, our trial density matrix differs from the one used in [55], since we label the orbitals by the modulus of total angular momentum, by the third component of total angular momentum, and by the orbital angular momentum. This, however, is merely a minor rearrangement of terms. We also adapt to atomic units used in this paper which changes the value of the Thomas-Fermi constant and gives a factor 1/2 in front of all three kinetic energy terms in the Hellmann-Weizsäcker functional.

E.3 Energy estimates for high angular momenta

For the convenience of the reader, we gather from [55] (based on the construction in [60]) two estimates on the order of the average kinetic and potential energy of the Schrödinger operator associated with the semi-classical density matrix d_{>},

tr(p^{2}d_{>}) = O(Z^{7/3}),  tr(|x|^{−1}d_{>}) = O(Z^{4/3}).  (E.8)

We also need a more detailed estimate on the kinetic energy.

Lemma E.1. Let L = [Z^{1/12}]. Then for large Z,

∑_{l=L}^{∞} l^{−2} tr(p^{2}d_{l}) = O(Z^{2}/L).  (E.9)

Proof. The definition of d_{l} implies (cf. [55, (2.3)]) that for angular momenta l < k′ one has

tr(p^{2}d_{l}) = ∫_{0}^{∞} [\sqrt{ρ_{l}} ^{′2} + \frac{α_{l}}{3} ρ_{l}^{3} + \frac{l(l + 1)}{r^{2}}ρ_{l}] dr + F_{l}  (E.10)

where we set

F_{l} := \frac{α_{l}}{3} (\frac{−1 + 6ε_{l} − 3ε_{l}^{2}}{N_{j,l,m}^{2}} + \frac{2ε_{l}^{3} − 6ε_{l}^{2} + 4ε_{l}}{N_{j,l,m}^{3}} ) ∫_{0}^{∞} ρ_{l}^{3}dr,

α_{l} := \frac{π^{2}}{4(2l + 1)^{2}} ,

and emphasize that α_{l} should not be confused with α from (E.4). According to [55, Proposition 3.6]

∑_{l=L}^{∞} l^{−2} F_{l} ≤ ∑_{l=L}^{∞} F_{l} ≤ const Z^{5/3}

where L = [Z^{1/12}]. The first term on the right-hand side of (E.10) is estimated according to

∫_{0}^{∞} [\sqrt{ρ_{l}} ^{′2} + \frac{α_{l}}{3} ρ_{l}^{3} + \frac{l(l + 1)}{r^{2}}ρ_{l}] dr

≤ ∫_{0}^{∞} [\frac{α_{l}}{3} ρ_{l}^{H}(r)^{3} + \frac{(l + \frac{1}{2})^{2}}{r^{2}} ρ_{l}^{H}(r)] dr + G_{l} + H_{l} + I_{l}.  (E.11)

with

G_{l} := ∫_{0}^{x_{1}(l)} [\sqrt{ρ_{l}} ^{′2} + \frac{α_{l}}{3} ρ_{l}^{3} + \frac{(l + \frac{1}{2})^{2}}{r^{2}} ρ_{l}] dr ≤ const Z^{2} (l + \frac{1}{2})^{−3/2} ,

H_{l} := ∫_{x_{2}(l)}^{∞} [\sqrt{ρ_{l}} ^{′2} + \frac{α_{l}}{3} ρ_{l}^{3} + \frac{(l + \frac{1}{2})^{2}}{r^{2}} ρ_{l}] dr ≤ const Z^{7/6} (l + \frac{1}{2})

where the inequalities were obtained by integration as in [55, (3.4)]. Inequality [55, (3.9)] for the gradient term in the middle region reads

I_{l} := ∫_{x_{1}(l)}^{x_{2}(l)} \sqrt{ρ_{l}} ^{′2}dr ≤ const (l + \frac{1}{2})

× [Z^{2} (l + \frac{1}{2})^{−3} + Z + Z^{2} (l + \frac{1}{2})^{−5/2} + \frac{Z^{5/3} (l + \frac{1}{2})^{−1/2}}{min\{l + \frac{1}{2}, Z^{1/4}\}}] .

Documenta Mathematica 14 (2009) 463–516

This implies

∑_{l=L}^{∞} l^{−2} G_{l} ≤ const Z^{2} ∑_{l=L}^{∞} l^{−7/2} ≤ const Z^{2}L^{−5/2},

∑_{l=L}^{∞} l^{−2} H_{l} ≤ ∑_{l=L}^{k′} l^{−2} H_{l} ≤ const Z^{7/6} log k′ ≤ const Z^{7/6} log Z

∑_{l=L}^{∞} l^{−2} I_{l} ≤ const [Z^{2}L^{−3} + Z log Z + Z^{2}L^{−5/2} + Z^{5/3}L^{−3/2}]
≤ const Z^{43/24}.

It thus remains to estimate the sum of the first terms on the right-hand side of (E.11). We begin with the first summand,

∑_{l=L}^{∞} \frac{1}{l^{2}} ∫_{0}^{∞} \frac{α_{l}}{3} ρ_{l}^{H}(r)^{3}dr ≤ const ∑_{l=L}^{∞} \frac{1}{l} ∫_{0}^{∞}(Z/r − l^{2}/r^{2})_{+}^{3/2}dr

=const Z^{2} ∑_{l≥L} \frac{1}{l^{2}} ∫_{0}^{∞} r^{−3/2}(1 − r^{−1})_{+}^{3/2}dr = O(Z^{2}/L)

where we used that the Thomas-Fermi potential is bounded from above by Z/r. This leaves the second summand,

∑_{l=L}^{∞} \frac{1}{l^{2}} ∫_{0}^{∞} \frac{(l + \frac{1}{2})^{2}}{r^{2}} ρ_{l}^{H}(r)dr ≤ const ∑_{l=L}^{∞} l ∫_{0}^{∞} r^{−2}(Z/r − l^{2}/r^{2})_{+}^{1/2}dr

= const Z^{2} ∑_{l=L}^{∞} \frac{1}{l^{2}} ∫_{0}^{∞} r^{−5/2}(1 − r^{−1})_{+}^{1/2}dr = O(Z^{2}/L),

which completes the proof of Lemma E.1.

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Rupert L. Frank Dept. of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA rlfrank@math.princeton.edu Heinz Siedentop Mathematisches Institut LMU München Theresienstraße 39 80333 München, Germany h.s@lmu.de

Simone Warzel TU München Zentrum Mathematik - M7 Boltzmannstraße 3 85747 Garching, Germany warzel@ma.tum.de