Localization of the Essential Spectrum

for Relativistic N-Electron Ions and Atoms

D. H. Jakubassa-Amundsen

Received: October 12, 2005 Revised: November 4, 2005

Communicated by Heinz Siedentop

Abstract. The HVZ theorem is proven for the pseudorelativistic Nelectron Jansen-Hess operator (2 ≤ N ≤ Z) which acts on the spinor Hilbert space A(H_{1}(ℝ^{3}) ⊗ ℂ^{4})^{N} where A denotes antisymmetrization with respect to particle exchange. This ’no pair’ operator results from the decoupling of the electron and positron degrees of freedom up to second order in the central potential strength γ = Ze^{2}.

2000 Mathematics Subject Classification: 81Q10, 81V45

Introduction

We consider N interacting electrons in a central Coulomb field generated by a point nucleus of charge number Z which is infinitely heavy and located at the origin. For stationary electrons where the radiation field and pair creation can be neglected, the N + 1 particle system is described by the Coulomb-Dirac operator, introduced by Sucher [23]. The Jansen-Hess operator used in the present work, which acts on the positive spectral subspace of N free electrons, is derived from the Coulomb-Dirac operator by applying a unitary transformation scheme [12, 13] which is equivalent to the Douglas-Kroll transformation scheme [6]. The transformed operator is represented as an infinite series of operators which do not couple the electron and positron degrees of freedom. For N = 1, each successive term in this series is of increasing order in the strength γ of the central field. The series has been shown to be convergent for subcritical potential strength (γ < γ_{c} = 0.3775, corresponding to Z < 52 [21]). For N > 1 the expansion parameter is e^{2}, which comprises the central field strength Ze^{2} and the strength e^{2} of the electron-electron interaction. A numerical investigation of the cases N = 1, Z − 1 and Z across the periodic table has revealed [27] that the ground-state energy of an N-electron system is already quite well represented if the series is truncated after the second-order term. This approximation defines the Jansen-Hess operator (see (3.1) below). In the present work we provide the localization of the essential spectrum of this operator. Recently [14] we have proven the HVZ theorem (which dates back to Hunziker [10], van Winter [26] and Zhislin [28] for the Schrödinger operator and to Lewis, Siedentop and Vugalter [16] for the scalar pseudorelativistic Hamiltonian) for the two-particle Brown-Ravenhall operator [2] which is the first-order term in the above mentioned series of operators. Now we extend this proof successively to the multiparticle Brown-Ravenhall operator (section 1), to the two-electron Jansen-Hess operator (section 2) and finally to the Nelectron Jansen-Hess operator. We closely follow the earlier work [14] where the details can be found. A quite different proof of the HVZ theorem for the multiparticle Brown-Ravenhall operator is presently under investigation [18].

1 Multiparticle Brown-Ravenhall case

For N electrons of mass m in a central field, generated by a point nucleus which is infinitely heavy and fixed at the origin, the Brown-Ravenhall operator is given by (in relativistic units, ℏ = c = 1)

H^{BR} = Λ_{+,N} (∑_{k=1}^{N}(D_{0}^{(k)} + V^{(k)}) + ∑_{k>l=1}^{N} V^{(kl)}) Λ_{+,N}  (1.1)

where D_{0}^{(k)} = α^{(k)}p_{k} + β^{(k)}m is the free Dirac operator of electron k, V^{(k)} = −γ/x_{k} is the central potential with strength γ = Ze^{2}, and V^{(kl)} = e^{2}/|x_{k} − x_{l}| is the electron-electron interaction, e^{2} ≈ 1/137.04 being the fine structure constant and x_{k} = |x_{k}| the distance of electron k from the origin. Further,

Λ_{+,N} = Λ_{+}^{(1)} ⋯ Λ_{+}^{N} (as shorthand for ⨂_{k=1}^{N} Λ_{+}^{(k)}) is the (tensor) product of the

single-particle projectors Λ_{+}^{(k)} = \frac{1}{2}(1 + D_{0}^{(k)}/E_{p_{k}}) onto the positive spectral subspace of D_{0}^{(k)}. H^{BR} acts in the Hilbert space A(L_{2}(ℝ^{3}) ⊗ ℂ^{4})^{N}, and is well-defined in the form sense and positive on A(H_{1/2}(ℝ^{3}) ⊗ ℂ^{4})^{N} for γ < γ_{BR} = \frac{2}{π/2+2/π} ≈ 0.906 (see (1.10) below). For the multi-nucleus case the Brown-Ravenhall operator was shown to be positive if γ < 0.65 [9]. An equivalent operator, which is defined in a reduced spinor space by means of (ψ_{+}, H^{BR} ψ_{+}) = (φ, h^{BR} φ) with ψ_{+} ∈ Λ_{+,N}(A(H_{1/2}(ℝ^{3}) ⊗ ℂ^{4})^{N}) and φ ∈ A(H_{1/2}(ℝ^{3}) ⊗ ℂ^{2})^{N}, is [7]

h^{BR} = ∑_{k=1}^{N}(T^{(k)} + b_{1m}^{(k)}) + ∑_{k>l=1}^{N} v^{(kl)}.  (1.2)

N-Electron HVZ Theorem 419

Explicitly, with A_{k} := A(p_{k}) = (\frac{E_{p_{k}} +m}{2E_{p_{k}}})^{1/2} and G_{k} := σ^{(k)}p_{k}g(p_{k}), g(p_{k}) = (2E_{p_{k}}(E_{p_{k}} + m))^{−1/2}, one has [14]

T^{(k)} := E_{p_{k}} = \sqrt{p_{k}^{2} + m^{2}}, b_{1m}^{(k)} = −γ (A_{k} \frac{1}{x_{k}} A_{k} + G_{k} \frac{1}{x_{k}} G_{k})

v^{(kl)} = A_{k}A_{l} \frac{e^{2}}{|x_{k} − x_{l}|} A_{k}A_{l} + A_{k}G_{l} \frac{e^{2}}{|x_{k} − x_{l}|} A_{k}G_{l}  (1.3)

+ G_{k}A_{l} \frac{e^{2}}{|x_{k} − x_{l}|} G_{k}A_{l} + G_{k}G_{l} \frac{e^{2}}{|x_{k} − x_{l}|} G_{k}G_{l}.

Let us consider the two-cluster decompositions {C_{1j}, C_{2j}} of the N-electron atom, obtained by moving electron j far away from the atom or by separating the nucleus from all electrons. Denote by C_{1j} the cluster located near the origin (containing the nucleus), while C_{2j} contains either one electron (j = 1, ..., N) or all electrons (j = 0). Correspondingly, h^{BR} is split into

h^{BR} = T + a_{j} + r_{j},  j = 0, 1, ..., N,  (1.4)

with T := ∑_{k=1}^{N} T^{(k)}, while a_{j} denotes the interaction of the particles located all

in cluster C_{1j} or all in C_{2j}. The remainder r_{j} collects the interactions between particles sitting in different clusters and is supposed to vanish when C_{2j} is moved to infinity. Define for j ∈ {0, 1, ..., N}

Σ_{0} := min_{j} inf σ(T + a_{j}).  (1.5)

Then we have

Theorem 1 (HVZ theorem for the multiparticle Brown-Ravenhall operator). Let h^{BR} be the Brown-Ravenhall operator for N > 2 electrons in a central field of strength γ < γ_{BR} = \frac{2}{π/2+2/π} , and let (1.4) be its two-cluster decompositions. Then the essential spectrum of h^{BR} is given by

σ_{ess}(h^{BR}) = [Σ_{0}, ∞).  (1.6)

In fact, the assertion (1.6) holds even in a more general case. For K ≥ 2 introduce K-cluster decompositions d := {C_{1}, ..., C_{K}} of the N + 1 particles, and split h^{BR} = T + a_{d} + r_{d} accordingly (where T + a_{d} describes the infinitely separated clusters while r_{d} comprises all interactions between particles sitting in two different clusters). Let

Σ_{1} := min_{#d≥2} inf σ(T + a_{d}).  (1.7)

Then σ_{ess}(h^{BR}) = [Σ_{1}, ∞) with Σ_{1} = Σ_{0}. This result, known from the Schrödinger case [20, p.122], relies on the fact that the electron-electron interaction is repulsive (V^{(kl)} ≥ 0 respective v^{(kl)} ≥ 0) and can be proved as follows.

First consider K-cluster decompositions of the form #C_{1} = N + 1 − (K − 1) and #C_{i} = 1, i = 2, ..., K (i.e. one ion and K − 1 separated electrons). For any j ∈ {1, ..., N} we use for the two-cluster decompositions the notation T + a_{j} = h_{N−1}^{BR} + T^{(j)}, where the subscript on h^{BR} denotes the number of electrons in the central field, and assume (1.6) to hold. Then

inf σ(h_{N}^{BR}) ≤ inf σ_{ess}(h_{N}^{BR}) ≤ inf σ(h_{N−1}^{BR} + T^{(j)})
= inf σ(h_{N−1}^{BR}) + m.  (1.8)

By induction (corresponding to successive removal of an electron) we get

inf σ(h_{N−1}^{BR}) ≤ inf σ(h_{N−1−N′}^{BR}) + N′m,  0 ≤ N′ < N − 1.  (1.9)

Since for a K-cluster decomposition of this specific form one has T + a_{d} = h_{N−(K−1)}^{BR}+T^{(1)}+...+T^{(K−1)}, it follows that inf σ(T +a_{d}) = inf σ(h_{N−(K−1)}^{BR}) + (K − 1)m ≥ inf σ(h_{N−1}^{BR}) + m ≥ Σ_{0}. Assume now cluster decompositions d with #C_{1} = N + 1 − (K − 1) fixed (K ∈ {3, ..., N}) but where #C_{i} > 1 for at least one i > 1. Then T + a_{d} is increased by (nonnegative) electron-electron interaction terms v^{(kl)} as compared to the K-cluster decompositions considered above, such that inf σ(T + a_{d}) is higher (or equal) than for the case #C_{i} = 1, i = 2, ..., K. Therefore, cluster decompositions with #C_{i} > 1 (for some i > 1) do not contribute to Σ_{1}, such that, together with (1.9), Σ_{1} = Σ_{0} is proven.

Let us embark on the proof of Theorem 1. The required lemmata will bear the same numbers as in [14]. We say that an operator O is \frac{1}{R}-bounded if O is bounded by \frac{c}{R} with some constant c > 0.

(a) In order to prove the ’hard part’ of the HVZ theorem, σ_{ess}(h^{BR}) ⊂ [Σ_{0}, ∞), we start by noting that the potential of h^{BR} is T -form bounded with form bound c < 1 if γ < γ_{BR}. With ψ_{+} ∈ Λ_{+,N}A(H_{1/2}(ℝ^{3})⊗ℂ^{4})^{N}, this follows from the estimates [4, 25, 13] (using that V^{(k)} ≤ 0 and V^{(kl)} ≥ 0),

(ψ_{+}, (∑_{k=1}^{N} V^{(k)} + ∑_{k>l=1}^{N} V^{(kl)}) ψ_{+}) ≤ ∑_{k>l=1}^{N} \frac{e^{2}}{γ_{BR}} (ψ_{+}, E_{p_{1}} ψ_{+})

= \frac{N − 1}{2} \frac{e^{2}}{γ_{BR}} (ψ_{+}, T ψ_{+})

(ψ_{+}, (∑_{k=1}^{N} V^{(k)} + ∑_{k>l=1}^{N} V^{(kl)}) ψ_{+}) ≥ −\frac{γ}{γ_{BR}} ∑_{k=1}^{N}(ψ_{+}, E_{p_{1}} ψ_{+})  (1.10)

= −\frac{γ}{γ_{BR}} (ψ_{+}, T ψ_{+})

such that c := max\{\frac{γ}{γ_{BR}} , \frac{N−1}{2} \frac{e^{2}}{γ_{BR}} \}. c < 1 requires γ < γ_{BR} for all physical values of N (N < 250). From (1.10), h^{BR} ≥ 0 for γ ≤ γ_{BR}. In order to establish Persson’s theorem (proven in [5] for Schrödinger operators and termed Lemma 2 in [14]),

inf σ_{ess}(h^{BR}) = lim_{R→∞} inf_{∥φ∥=1}(φ, h^{BR} φ)  (1.11)

if φ ∈ A(C_{0}^{∞}(ℝ^{3N}\B_{R}(0)) ⊗ ℂ^{2N}) where B_{R}(0) ⊂ ℝ^{3N} is a ball of radius R centered at the origin, we need the fact that the Weyl sequence φ_{n} for a λ in the essential spectrum of h^{BR} can be chosen such that it is supported outside a ball B_{n}(0):

Lemma 1. Let h^{BR} = T + V, let V be relatively form bounded with respect to T. Then λ ∈ σ_{ess}(h^{BR}) iff there exists a sequence of functions φ_{n} ∈ A(C_{0}^{∞}(ℝ^{3N}\B_{n}(0)) ⊗ ℂ^{2N}) with ∥φ_{n}∥ = 1 such that

∥(h^{BR} − λ) φ_{n}∥ ⟶ 0  as n → ∞.  (1.12)

If such φ_{n} exist they form a Weyl sequence because φ_{n} converge weakly to zero [14]. For the proof of the converse direction, let λ ∈ σ_{ess}(h^{BR}) be characterized by a Weyl sequence ψ_{n} ∈ A(C_{0}^{∞}(ℝ^{3}) ⊗ ℂ^{2})^{N}, ∥ψ_{n}∥ = 1, with ψ_{n} ⇀^{w} 0 and ∥(h^{BR} − λ)ψ_{n}∥ → 0 as n → ∞. Let x = (x_{1}, ..., x_{N}) ∈ ℝ^{3N} be the coordinates of the N electrons and define a smooth symmetric auxiliary function χ_{0} ∈ C_{0}^{∞}(ℝ^{3N}) mapping to [0, 1] by means of

χ_{0}(\frac{x}{n}) = \{ \begin{matrix}1, & x ≤ n \\ 0, & x > 2n \end{matrix}  (1.13)

where x = |x| = \sqrt{x_{1}^{2} + ... + x_{N}^{2}} . Then we set χ_{n}(x) := 1−χ_{0}(x/n) and claim that a subsequence of the sequence φ_{n} := ψ_{n}χ_{n} ∈ A(C_{0}^{∞}(ℝ^{3N}\B_{n}(0)) ⊗ ℂ^{2N}) satisfies the requirements of Lemma 1. In order to show that ∥(h^{BR} − λ)φ_{n}∥ = ∥χ_{n}(h^{BR} − λ)ψ_{n} + [h^{BR}, χ_{0}]ψ_{n}∥ → 0 for n → ∞, we have to estimate the single-particle contributions ∥[T^{(k)}, χ_{0}]ψ_{n}∥ and ∥[b_{1m}^{(k)}, χ_{0}]ψ_{n}∥. With b_{1m}^{(k)} of the form B_{k} \frac{1}{x_{k}} B_{k} where B_{k} ∈ \{A_{k}, G_{k}\} is a bounded multiplication operator in momentum space, we have to consider commutators of the type p_{k}[B_{k}, χ_{0}] which are multiplied by bounded operators. These commutators are shown to be \frac{1}{n}-bounded in the same way as for N = 2 [14], by working in momentum space and introducing the N-dimensional Fourier transform (marked by a hat) of the Schwartz function χ_{0},

\binom{}{χ_{0}(\frac{·}{n})} (p) = \frac{1}{(2π)^{3N/2}} ∫_{ℝ^{3N}} dx e^{−ipx} χ_{0}(\frac{x}{n}) = n^{3N} χ̂_{0}(p_{1}n, ..., p_{N}n),

(1.14)

where p = (p_{1}, ..., p_{N}), and by using the mean value theorem to estimate the difference |B_{k}(p_{k}) − B_{k}(p_{k′})| respective |T^{(k)}(p_{k}) − T^{(k)}(p_{k′})|. The twoparticle contributions ∥[v^{(kl)}, χ_{0}] ψ_{n}∥ can, according to the representation (1.3) of v^{(kl)}, also be split into single-particle commutators p_{k}[B_{k}, χ_{0}] multiplied by bounded operators. Their estimate as well as the remaining parts of the proof of Lemma 1 for N > 2 (in particular the normalizability of φ_{n} for sufficiently large n which relies on the relative form boundedness of the total potential) can be mimicked from the case of N = 2. Our aim is a generalization of the localization formula of Lewis et al [16] to the operator h^{BR}. We introduce the Ruelle-Simon [22] partition of unity (ϕ_{j})_{j=0,...,N} ∈ C^{∞}(ℝ^{3N}) which is subordinate to the two-cluster decompositions (1.4). It is defined on the unit sphere in ℝ^{3N} and has the following properties (see e.g. [5, p.33], [24])

∑_{j=0}^{N} ϕ_{j}^{2} = 1, ϕ_{j}(λx) = ϕ_{j}(x) for x = 1 and λ ≥ 1,

supp ϕ_{j} ∩ ℝ^{3N}\B_{1}(0) ⊆ {x ∈ ℝ^{3N}\B_{1}(0) : |x_{k} − x_{l}| ≥ Cx for all k ∈ C_{1j}

(1.15)

and l ∈ C_{2j}, and x_{k} ≥ Cx for all k ∈ C_{2j}}, j = 0, 1, ..., N

where C is a constant and it is again assumed that the nucleus belongs to cluster C_{1j}. Then we have

Lemma 3. Let h^{BR} = T + a_{j} + r_{j}, (ϕ_{j})_{j=0,...,N} be the Ruelle-Simon partition of unity and φ ∈ A(C_{0}^{∞}(ℝ^{3N}\B_{R}(0)) ⊗ ℂ^{2N}) with R > 1. Then, with some constant c,

|(ϕ_{j}φ, r_{j} ϕ_{j}φ)| ≤ \frac{c}{R} ∥φ∥^{2},  j = 0, .., N.  (1.16)

There are two possibilities. r_{j} may (a) consist of terms b_{1m}^{(k)} for some k ∈ C_{2j}, or (b) of terms v^{(kl)} with particles k and l in different clusters. For the proof, all summands of r_{j} are estimated separately. For each summand of r_{j} (to a given cluster decomposition j), a specific smooth auxiliary function χ mapping to [0, 1] is introduced which is unity on the support of ϕ_{j}φ, such that ϕ_{j}φχ = ϕ_{j}φ. In case (a) we have supp ϕ_{j}φ ⊂ ℝ^{3N}\B_{R}(0) ∩ {x_{k} ≥ Cx}, i.e. x_{k} ≥ CR. Therefore we define the (single-particle) function

χ_{k}(\frac{x_{k}}{R}) := \{ \begin{matrix}0, & x_{k} < CR/2 \\ 1, & x_{k} ≥ CR \end{matrix} .  (1.17)

With b_{1m}^{(k)} of the form B_{k} \frac{1}{x_{k}} B_{k} we have to consider (ϕ_{j}φ, B_{k} \frac{1}{x_{k}} B_{k} χ_{k}ϕ_{j}φ) = (ϕ_{j}φ, B_{k} \frac{1}{x_{k}} χ_{k}B_{k} ϕ_{j}φ) +(ϕ_{j}φ, B_{k} \frac{1}{x_{k}} [B_{k}, 1−χ_{k}] ϕ_{j}φ). The first term is uniformly 2/R-bounded by the choice (1.17) of χ_{k}, whereas the second term can be estimated in momentum space as in the two-electron case (respective in the proof of Lemma 1). In case (b) we have supp ϕ_{j}φ ⊂ ℝ^{3N}\B_{R}(0) ∩ {|x_{k} −x_{l}| ≥ Cx}, i.e. |x_{k} −x_{l}| ≥ CR. Accordingly, we take

χ_{kl}(\frac{x_{k} − x_{l}}{R}) := \{ \begin{matrix}0, & |x_{k} − x_{l}| < CR/2 \\ 1, & |x_{k} − x_{l}| ≥ CR \end{matrix} .  (1.18)

With the representation (1.3) of v^{(kl)}, we have to estimate commutators of the type p_{k}[B_{k} B_{l}, 1 − χ_{kl}]. The proof of their uniform 1/R-boundedness can be copied from the two-electron case. The second ingredient of the localization formula is an estimate for the commutator of ϕ_{j} with h^{BR}:

Lemma 4. Let h^{BR} from (1.2) and (ϕ_{j})_{j=0,...,N} be the Ruelle-Simon partition of unity. Then for φ ∈ A(C_{0}^{∞}(ℝ^{3N}\B_{R}(0)) ⊗ ℂ^{2N}) and R > 2 one has

(a)  | ∑_{j=0}^{N}(ϕ_{j}φ, [T, ϕ_{j}] φ)| ≤ \frac{c}{R^{2}} ∥φ∥^{2}
(b)  |(ϕ_{j}φ, [b_{1m}^{(k)}, ϕ_{j}] φ)| ≤ \frac{c}{R} ∥φ∥^{2}  (1.19)
(c)  |(ϕ_{j}φ, [v^{(kl)}, ϕ_{j}] φ)| ≤ \frac{c}{R} ∥φ∥^{2}

where c is a generic constant.

Item (a) is proven in [16]. For items (b) and (c) we define the smooth auxiliary N-particle function χ mapping to [0, 1],

χ(\frac{x}{R}) := \{ \begin{matrix}0, & x < R/2 \\ 1, & x ≥ R \end{matrix} .  (1.20)

Then ϕ_{j}φ = ϕ_{j}φχ on supp φ, and therefore (ϕ_{j}φ, [b_{1m}^{(k)}, ϕ_{j}] φ) = (ϕ_{j}φ, [b_{1m}^{(k)}, ϕ_{j}χ] φ). The \frac{1}{R}-estimate, claimed in (1.19), relies on the scaling property ϕ_{j}(x)χ(\frac{x}{R}) = ϕ_{j}(\frac{x}{R/2})χ(\frac{x}{R}) which holds for R > 2 since supp χ (and hence supp ϕ_{j}χ) is outside B_{R/2}(0). Thus, working in coordinate space and using the mean value theorem, we get the estimate

|(ϕ_{j}χ)(x_{1}, ..., x_{k}, ..., x_{N}) − (ϕ_{j}χ)(x_{1}, ..., x′_{k}, ..., x_{N})| ≤ |x_{k} − x′_{k}| \frac{c_{0}}{R} (1.21)

(only the k-th coordinate in the second entry of the l.h.s. is primed). Since (1.21) holds for arbitrary k ∈ {1, ..., N}, the proof of (b) and (c) can be carried out in the same way as done in the two-electron case, by estimating the kernel of B_{k} ∈ {A_{k}, G_{k}} in coordinate space by c/|x_{k} − x′_{k}|^{3} (using asymptotic analysis [19]) and subsequently proving the uniform \frac{1}{R}-boundedness of [B_{k}, ϕ_{j}χ] \frac{1}{x_{k}} respective [B_{k}, ϕ_{j}χ] \frac{1}{|x_{k}−x_{l}|}. With Lemmata 3 and 4 we obtain the desired localization formula for h^{BR},

(φ, h^{BR}φ) = ∑_{j=0}^{N}(ϕ_{j}φ, (T + a_{j}) ϕ_{j}φ)

+ ∑_{j=0}^{N}(ϕ_{j}φ, r_{j} ϕ_{j}φ) − ∑_{j=0}^{N}(ϕ_{j}φ, [h^{BR}, ϕ_{j}] φ)  (1.22)

= ∑_{j=0}^{N}(ϕ_{j}φ, (T + a_{j}) ϕ_{j}φ) + O(\frac{1}{R}) ∥φ∥^{2}

for R > 2. From Persson’s theorem (1.11) and the definition (1.5) of Σ_{0} we therefore get

inf σ_{ess}(h^{BR}) = lim_{R→∞} inf_{∥φ∥=1} ∑_{j=0}^{N}(ϕ_{j}φ, (T + a_{j}) ϕ_{j}φ)

≥ Σ_{0} ∑_{j=0}^{N}(ϕ_{j}φ, ϕ_{j}φ) = Σ_{0}  (1.23)

which proves the inclusion σ_{ess}(h^{BR}) ⊂ [Σ_{0}, ∞).

(b) We now turn to the ’easy part’ of the proof where we have to verify [Σ_{0}, ∞) ⊂ σ_{ess}(h^{BR}). We start by showing that for every j ∈ {0, 1, ..., N}, σ(T + a_{j}) is continuous, i.e. for any λ ∈ [inf σ(T + a_{j}), ∞) one has λ ∈ σ(T + a_{j}). If the cluster C_{2j} consists of a single electron j, then T + a_{j} = T^{(j)} + h_{N−1}^{BR} where h_{N−1}^{BR} does not contain any interaction with electron j. The continuity of σ(T^{(j)} + h_{N−1}^{BR}) then follows from the continuity of σ(T^{(j)}) in the same way as for N = 2. In the case j = 0 where C_{2j} contains N electrons, the total momentum p_{0} of

C_{2j} is well-defined and commutes with its Hamiltonian h_{0} := T + ∑_{k>l=1}^{N} v^{(kl)} =

T + a_{0}. This follows from the absence of any central potential in h_{0} and from the symmetry of v^{(kl)},

[(−i∇_{x_{k}}−i∇_{x_{l}}), \frac{1}{|x_{k}−x_{l}|}]ψ(x) = (−i∇_{x_{k}}−i∇_{x_{l}})(\frac{1}{|x_{k}−x_{l}|}) ψ(x) = 0·ψ(x) = 0.

(1.24)

Thus the eigenfunctions to h_{0} can be chosen as eigenfunctions of p_{0}. For p_{0} ≥ 0 the associated center of mass energy of C_{2j} is continuous. Therefore, inf σ(h_{0}) is attained for p_{0} = 0 and σ(h_{0}) is continuous. Let λ ∈ [Σ_{0}, ∞). We have Σ_{0} = inf σ(T +a_{j}) for a specific j ∈ {0, ..., N}. Then λ ∈ σ(T + a_{j}), i.e. there exists a defining sequence φ_{n}(x) ∈ C_{0}^{∞}(ℝ^{3N}) ⊗ ℂ^{2N} with ∥φ_{n}∥ = 1 and ∥(T + a_{j} − λ) φ_{n}∥ → 0 for n → ∞. Assume that l electrons belong to cluster C_{2j} which we will enumerate by N − l + 1, ..., N, and follow [10] to define the unitary translation operator T_{a} by means of

T_{a} ψ(x_{1}, ..., x_{N}) = ψ(x_{1}, ..., x_{N−l}, x_{N−l+1} − a, ..., x_{N} − a)  (1.25)

with |a| = a and let a_{l} := (a, ..., a) ∈ ℝ^{3l}. Hence cluster C_{2j} moves to infinity as a → ∞.

Let ψ_{n}^{(a)} := T_{a}φ_{n} and Aψ_{n}^{(a)} be the antisymmetric function constructed from ψ_{n}^{(a)}. We claim that Aψ_{n}^{(a)} is a defining sequence for λ ∈ σ(h^{BR}). It is sufficient (as shown below) to prove that ψ_{n}^{(a)} has this property. We have trivially ∥ψ_{n}^{(a)}∥ = ∥φ_{n}∥ and we have to show that ∥(h^{BR} − λ) ψ_{n}^{(a)}∥ → 0 for n → ∞ and a suitably large a. We have

∥(h^{BR} − λ) ψ_{n}^{(a)}∥ ≤ ∥(T + a_{j} − λ) ψ_{n}^{(a)}∥ + ∥r_{j} ψ_{n}^{(a)}∥.  (1.26)

T commutes with T_{a} because T is a multiplication operator in momentum space. Since the central potentials contained in a_{j} are not affected by T_{a} (because T_{a} does not act on the particle coordinates of cluster C_{1j}), we also have [T_{a}, a_{j}] = 0. In fact, assuming e.g. that electrons k and l are in cluster C_{2j} and using the representation (1.3) for v^{(kl)} we have with T_{a}^{∗}T_{a} = 1 and B_{k}, B̃_{k} ∈ {A_{k}, G_{k}},

T_{a}^{∗}B_{k}B̃_{l} \frac{1}{|x_{k} − x_{l}|} B_{k}B̃_{l} T_{a} = B_{k}B̃_{l}T_{a}^{∗} \frac{1}{|x_{k} − x_{l}|} T_{a} B_{k}B̃_{l}

= B_{k}B̃_{l} \frac{1}{|x_{k} + a − (x_{l} + a)|} B_{k}B̃_{l} = B_{k}B̃_{l} \frac{1}{|x_{k} − x_{l}|} B_{k}B̃_{l}  (1.27)

such that [T_{a}, v^{(kl)}] = 0. Then, given some ϵ > 0, the first term of (1.26) reduces to

∥(T +a_{j} −λ) T_{a}φ_{n}∥ = ∥T_{a} (T +a_{j} −λ) φ_{n}∥ ≤ ∥T_{a}∥ ∥(T +a_{j} −λ) φ_{n}∥ < ϵ/2

(1.28)

if n > N_{0} for N_{0} sufficiently large. For the second term in (1.26) we note that r_{j} consists of terms b_{1m}^{(k)} with k ∉ C_{1j} and terms v^{(kk′)} with k ∈ C_{i}, k′ ∈ C_{i′}, i ≠ i′. Moreover, since a_{j} does not contain any intercluster interactions, we can choose φ_{n} = φ_{1}^{(n)}·φ_{2}^{(n)} as a product of functions (φ_{1}^{(n)} ∈ C_{0}^{∞}(ℝ^{3(N−l)} ⊗ ℂ^{2(N−l)}), φ_{2}^{(n)} ∈ C_{0}^{∞}(ℝ^{3l} ⊗ ℂ^{2l})) each of which describing the electrons in cluster C_{1j} respective C_{2j}. Let supp φ_{i}^{(n)} ⊂ B_{R_{i}}(0) for a suitable R_{i}. Consider ∥b_{1m}^{(k)}ψ_{n}^{(a)}∥ with k ∈ C_{2j}. We have supp T_{a}φ_{2}^{(n)} ⊂ B_{R_{2}}(a_{l}). Let a > 2R_{2}. For all k′ ∈ C_{2j}, on the support of T_{a}φ_{2}^{(n)} we have R_{2} > |x_{k′} −a| ≥ a−x_{k′} and thus x_{k′} > a−R_{2}. Therefore we can write supp T_{a}φ_{2}^{(n)} ⊂ ℝ^{3l}\B_{|a_{l}|−R_{2}}(0) ∩ {x_{k′} > a − R_{2} ∀ k′ ∈ C_{2j}}. Assume we can prove

Lemma 5. Let φ ∈ C_{0}^{∞}(Ω) ⊗ ℂ^{2l} with Ω := {x = (x_{1}, ..., x_{l}) ∈ ℝ^{3l} : x_{i} > R ∀ i = 1, ..., l} and R > 1. Then for k ∈ {1, ..., l} and some constant c,

∥b_{1m}^{(k)} φ∥ ≤ \frac{c}{R} ∥φ∥.  (1.29)

Since b_{1m}^{(k)} acts only on T_{a}φ_{2}^{(n)} we obtain

∥b_{1m}^{(k)} T_{a}φ_{n}∥ = ∥φ_{1}^{(n)}∥ ∥b_{1m}^{(k)} T_{a}φ_{2}^{(n)}∥ ≤ \frac{c}{a − R_{2}} ∥T_{a} φ_{2}^{(n)}∥ < \frac{2c}{a} (1.30)

because the φ_{i}^{(n)} are normalized. As a consequence, for any k ∈ C_{2j}, the l.h.s. of (1.30) can be made smaller than ϵ/4l for sufficiently large a. For the proof of Lemma 5 or, equivalently, of |(ϕ, b_{1m}^{(k)}φ)| ≤ \frac{c}{R} ∥φ∥ ∥ϕ∥ for all ϕ ∈ (C_{0}^{∞}(ℝ^{3}) ⊗ ℂ^{2})^{l}, we note that the basic difference to the respective assertion for N = 2 lies in the possible multiparticle nature of ϕ and φ. However, the property of the domain Ω of φ allows for the introduction of the (single-particle) smooth auxiliary function (mapping to [0, 1]),

χ(\frac{x_{k}}{R}) := \{ \begin{matrix}0, & x_{k} < R/2 \\ 1, & x_{k} ≥ R \end{matrix} ,  (1.31)

such that φχ = φ. Then the proof can be copied from the two-electron case. For the two-particle interaction contained in r_{j}, one has

Lemma 6. Let ψ_{n}^{(a)} = T_{a}φ_{n} = T_{a} φ_{1}^{(n)}φ_{2}^{(n)} as defined above. Then for all φ ∈ (C_{0}^{∞}(ℝ^{3}) ⊗ ℂ^{2})^{N} and a > 4R,

|(φ, v^{(kk′)}ψ_{n}^{(a)})| ≤ \frac{c_{0}}{a − 2R} ∥φ∥ ∥ψ_{n}^{(a)}∥  (1.32)

with some positive constants c_{0} and R, provided particles k and k′ belong to two different clusters.

For the proof of Lemma 6, we need again a suitable auxiliary function χ. Let k′ ∈ C_{1j}, k ∈ C_{2j}. We have supp φ_{1}^{(n)}φ_{2}^{(n)} ⊂ B_{R_{1}}(0) × B_{R_{2}}(0) and supp T_{a}φ_{1}^{(n)}φ_{2}^{(n)} ⊂ B_{R_{1}}(0)×B_{R_{2}}(a_{l}). Hence x_{k′} < R_{1} and x_{k} > a−R_{2}. So the inter-electron separation can be estimated by |x_{k}−x_{k′}| ≥ x_{k}−x_{k′} > a−R_{2}−R_{1}. Let R := max{R_{1}, R_{2}} and ã := a − 2R. Define

χ_{kk′}(\frac{x_{k} − x_{k′}}{ã}) := \{ \begin{matrix}0, & |x_{k} − x_{k′}| < ã/2 \\ 1, & |x_{k} − x_{k′}| ≥ ã \end{matrix} .  (1.33)

Then χ_{kk′} is unity on the support of ψ_{n}^{(a)}, such that χ_{kk′}ψ_{n}^{(a)} = ψ_{n}^{(a)}. With this function, the proof of Lemma 6 is done exactly as in the two-electron case. Collecting results, we obtain for n > N_{0} and a > 4R sufficiently large

∥(h^{BR} − λ) ψ_{n}^{(a)}∥ ≤ ∥(T + a_{j} − λ) φ_{n}∥ + l \frac{2c}{a}

+ Ñ \frac{2c_{0}}{a} ∥ψ_{n}^{(a)}∥ < ϵ  (1.34)

where Ñ is the total number of two-electron intercluster interactions. This proves that λ ∈ σ(h^{BR}). Since λ ∈ [Σ_{0}, ∞) was chosen arbitrarily, we therefore have [Σ_{0}, ∞) ⊂ σ(h^{BR}), indicating that σ(h^{BR}) has to be continuous in [Σ_{0}, ∞). Consequently, [Σ_{0}, ∞) ⊂ σ_{ess}(h^{BR}) which completes the proof of Theorem 1.

We are left to show that the defining sequence for λ can be chosen to be antisymmetric. We write Aψ_{n}^{(a)} = c_{1} ∑_{σ∈P} sign(σ) ψ_{n,σ}^{(a)} where ψ_{n,σ}^{(a)} =

ψ_{n}^{(a)}(x_{σ(1)}, ..., x_{σ(N)}) with P the permutation group of the numbers 1, ..., N, and c_{1} is a normalization constant. Since h^{BR} is symmetric upon particle exchange we have

∥(h^{BR} − λ) Aψ_{n}^{(a)}∥ ≤ c_{1} ∑_{σ∈P} ∥(h^{BR} − λ) ψ_{n,σ}^{(a)}∥ = c_{1} (#σ) ∥(h^{BR} − λ) ψ_{n}^{(a)}∥.

(1.35)

By (1.34) this can be made smaller than ϵ since the number #σ of permutations is finite. It remains to prove that Aψ_{n}^{(a)} is normalizable. Without restriction we can assume in the factorization φ_{n} = φ_{n}^{(1)}·φ_{n}^{(2)} that φ_{n}^{(1)} and φ_{n}^{(2)} are antisymmetric, such that σ can be restricted to the permutation of coordinates relating to different clusters. We claim that scalar products of the form

(φ_{1}^{(n)}(x_{1}, ..., x_{N−l}) φ_{2}^{(n)}(x_{N−l+1}−a, ..., x_{N} −a), φ_{1}^{(n)}(x_{σ(1)}, ..., x_{σ(N−l)}) (1.36)
· φ_{2}^{(n)}(x_{σ(N−l+1)} − a, ..., x_{σ(N)} − a) )

where ∃ k ∈ {1, ..., N − l} and k′ ∈ {N − l + 1, ..., N} such that σ(k) ∈ {N − l + 1, ..., N} and σ(k′) ∈ {1, ..., N − l}, can be made arbitrarily small for a suitably large a. In fact, since x_{σ(k′)} < R_{1} on supp φ_{1}^{(n)} and |x_{σ(k′)} − a| < R_{2}

on supp φ_{2}^{(n)}, we have ∫_{ℝ^{3}} dx_{σ(k′)}\overline{φ}_{1}^{(n)}(..., x_{σ(k′)}, ...) φ_{2}^{(n)}(..., x_{σ(k′)} − a, ...) = 0 if

a > R_{1} + R_{2}. Thus we get

∥Aψ_{n}^{(a)}∥^{2} = c_{1}^{2} ∑_{σ∈P}(ψ_{n,σ}^{(a)}, ψ_{n,σ}^{(a)} )  (1.37)

since all cross terms vanish for sufficiently large a. This guarantees the normalizability of Aψ_{n}^{(a)}.

2 The two-electron Jansen-Hess operator

The Jansen-Hess operator includes the terms which are quadratic in the fine structure constant e^{2}. We restrict ourselves in this section to the two-electron ion and write the Jansen-Hess operator H^{(2)} in the following form [11]

H^{(2)} = H_{2}^{BR} + Λ_{+,2} (∑_{k=1}^{2} B_{2m}^{(k)} + C^{(12)}) Λ_{+,2}  (2.1)

B_{2m}^{(k)} := \frac{γ^{2}}{8π^{2}} \{\frac{1}{x_{k}} (1 − \frac{α^{(k)}p_{k}+β^{(k)}m}{E_{p_{k}}})V_{10,m}^{(k)} + V_{10,m}^{(k)}(1 − \frac{α^{(k)}p_{k}+β^{(k)}m}{E_{p_{k}}})\frac{1}{x_{k}}\}

Documenta Mathematica 10 (2005) 417–445

V_{10,m}^{(k)} := 2π^{2} ∫_{0}^{∞} dt e^{−tE_{p_{k}}} \frac{1}{x_{k}} e^{−tE_{p_{k}}}

where H_{2}^{BR} is the Brown-Ravenhall operator from (1.1) indexed by 2 (for N = 2), Λ_{+,2} = Λ_{+}^{(1)}Λ_{+}^{(2)} and V_{10,m}^{(k)} is a bounded single-particle integral operator. The two-particle second-order contribution C^{(12)} is given by

C^{(12)} := ∑_{k=1}^{2}(V^{(12)} Λ_{−}^{(k)} F_{0}^{(k)} + F_{0}^{(k)} Λ_{−}^{(k)} V^{(12)})  (2.2)

F_{0}^{(k)} := −\frac{1}{2π} ∫_{−∞}^{∞} dη \frac{1}{D_{0}^{(k)} + iη} V^{(k)} \frac{1}{D_{0}^{(k)} + iη}

and Λ_{−}^{(k)} = 1 − Λ_{+}^{(k)}. Also F_{0}^{(k)} is a bounded single-particle integral operator. In the same way as for the Brown-Ravenhall operator, an equivalent operator h^{(2)} acting on the reduced spinor space A(L_{2}(ℝ^{3}) ⊗ ℂ^{2})^{2}, can be defined,

h^{(2)} = h_{2}^{BR} + ∑_{k=1}^{2} b_{2m}^{(k)} + c^{(12)}  (2.3)

with

b_{2m}^{(k)} = \frac{γ^{2}}{8π^{2}} \{A_{k} \frac{1}{x_{k}} V_{10,m}^{(k)}A_{k} − G_{k} \frac{1}{x_{k}} \frac{σ^{(k)}p_{k}}{E_{p_{k}}} V_{10,m}^{(k)}A_{k} − A_{k} \frac{1}{x_{k}} \frac{m}{E_{p_{k}}} V_{10,m}^{(k)}A_{k}

(2.4)

+ G_{k} \frac{1}{x_{k}} V_{10,m}^{(k)}G_{k} − A_{k} \frac{1}{x_{k}} \frac{σ^{(k)}p_{k}}{E_{p_{k}}} V_{10,m}^{(k)}G_{k} + G_{k} \frac{1}{x_{k}} \frac{m}{E_{p_{k}}} V_{10,m}^{(k)}G_{k} + h.c.\}

where A_{k}, G_{k} are defined below (1.2) and h.c. means Hermitean conjugate (such that b_{2m}^{(k)} is a symmetric operator). Note that, due to the presence of the projector Λ_{+}^{(k)} in (2.1), b_{2m}^{(k)} contains only even powers in σ^{(k)}. In a similar way, c^{(12)} is derived from C^{(12)}. The particle mass m is assumed to be nonzero throughout (for m = 0, the spectrum of the single-particle Jansen-Hess operator is absolutely continuous with infimum zero [11]). For potential strength γ < 0.89 (slightly smaller than γ_{BR}), it was shown [13] that the total potential of H^{(2)} (and hence also of h^{(2)}) is relatively form bounded (with form bound smaller than 1) with respect to the kinetic energy operator. Therefore, h^{(2)} is well-defined in the form sense and is a selfadjoint operator by means of the Friedrichs extension of the restriction of h^{(2)} to A(C_{0}^{∞}(ℝ^{3}) ⊗ ℂ^{2})^{2}. The above form boundedness guarantees the existence of a μ > 0 such that h^{(2)} + μ > 0 for γ < 0.89. If γ < 0.825, one can even choose μ = 0 [13]. Let us introduce the operator h̃^{(2)} by means of h^{(2)} =: h̃^{(2)} + c^{(12)} and define in analogy to (1.4) the two-cluster decompositions of h̃^{(2)} for j=0,1,2,

h̃^{(2)} = T + a_{j} + r_{j}  (2.5)

as well as

Σ_{0} := min_{j} inf σ(T + a_{j}).  (2.6)

The aim of this section is to prove

Theorem 2 (HVZ theorem for the two-electron Jansen-Hess operator).

Let h^{(2)} = ∑_{k=1}^{2}(T^{(k)}+b_{1m}^{(k)}+b_{2m}^{(k)}) +v^{(12)}+c^{(12)} = h̃^{(2)}+c^{(12)} be the two-electron

Jansen-Hess operator with potential strength γ < 0.66 (Z ≤ 90). Let (2.5) be the two-cluster decompositions of h̃^{(2)} and Σ_{0} from (2.6). Then the essential spectrum of h^{(2)} is given by

σ_{ess}(h^{(2)}) = [Σ_{0}, ∞).  (2.7)

We start by noting that the two-particle second-order potential c^{(12)} does not change the essential spectrum of h^{(2)} :

Proposition 1. Let h^{(2)} = h̃^{(2)}+c^{(12)} be the two-electron Jansen-Hess operator with potential strength γ < 0.66. Then one has

σ_{ess}(h^{(2)}) = σ_{ess}(h̃^{(2)}).  (2.8)

Proof. The proof is performed for the equivalent operator H^{(2)} =: H̃^{(2)} + Λ_{+,2}C^{(12)}Λ_{+,2}. The resolvent difference

R_{μ} := (H^{(2)} + μ)^{−1} − (H̃^{(2)} + μ)^{−1}  (2.9)

is bounded for μ ≥ 0 since H^{(2)} as well as H̃^{(2)} are positive for γ < 0.825 which exceeds the critical γ of Proposition 1. We will show that R_{μ} is compact. Then, following the argumentation of [7], one can use Lemma 3 of [20, p.111] together with the strong spectral mapping theorem ([20, p.109]) to prove that the essential spectra of H^{(2)} and H̃^{(2)} coincide.

Let T_{0} := Λ_{+,2} ∑_{k=1}^{2} D_{0}^{(k)} Λ_{+,2} which is a positive operator (for m ≠ 0) on

the positive spectral subspace Λ_{+,2}A(H_{1}(ℝ^{3}) ⊗ ℂ^{4})^{2}. (The negative spectral subspace is disregarded throughout because H^{(2)} = 0 on that subspace.) With the help of the second resolvent identity, one decomposes R_{μ} into

R_{μ} = −(H̃^{(2)} + μ)^{−1} Λ_{+,2} C^{(12)} Λ_{+,2} (H^{(2)} + μ)^{−1}  (2.10)
= − [(H̃^{(2)} + μ)^{−1}(T_{0} + μ)] · {(T_{0} + μ)^{−1} Λ_{+,2} C^{(12)} Λ_{+,2} (T_{0} + μ)^{−1}}
· [(T_{0} + μ)(H^{(2)} + μ)^{−1}] .

One can show (see [11, proof b) of Theorem II.1 with T replaced by T_{0}]) that for γ < 0.66, the two operators in square brackets are bounded. This relies on the relative boundedness of the total potential of H^{(2)} (respective H̃^{(2)}) with respect to T_{0}, with (operator) bound less than one for m = 0 ([11]; Appendix B). Due to scaling (for μ = 0), the boundedness of the operators in square brackets holds for all m. The operator in curly brackets is shown to be compact. To this aim it is written as

(T_{0} + μ)^{−1}Λ_{+,2} C^{(12)}Λ_{+,2} (T_{0} + μ)^{−1}  (2.11)
= (T_{0} + μ)^{−1}(T + μ) Λ_{+,2}W_{2} Λ_{+,2} (T + μ)(T_{0} + μ)^{−1}

with W_{2} := (T + μ)^{−1}C^{(12)}(T + μ)^{−1}. According to Herbst [8], W_{2} is decomposed into W_{2n} + R_{n} where (W_{2n})_{n∈ℕ} is a sequence of Hilbert-Schmidt operators satisfying ∥W_{2n} − W_{2}∥ → 0 for n → ∞. It follows that W_{2n} is compact such that also W_{2} is compact (see e.g. [15, III.4.2,V.2.4]). W_{2n} is defined by regularizing the Coulomb potential by means of \frac{1}{x} e^{−ϵx} and by introducing convergence generating functions e^{−ϵp} in momentum space, where ϵ := \frac{1}{n} > 0 is a small quantity. Details of the proof are found in [11]. The adjacent factors of W_{2} in (2.11) are easily seen to be bounded for μ = 0. Since Λ_{+,2} = Λ_{+,2}^{2} commutes with T, one has e.g. Λ_{+,2}T T_{0}^{−1} = (Λ_{+,2}T Λ_{+,2})T_{0}^{−1} = T_{0}T_{0}^{−1} = 1. Therefore, the operator in curly brackets and hence R_{μ} is proven to be compact for μ = 0.

Proof of Theorem 2. With Proposition 1 at hand, it remains to prove the HVZ theorem for the operator h̃^{(2)}, which in fact holds for all γ < γ_{BR}. We proceed along the same lines as done in the proof of the HVZ theorem for the Brown-Ravenhall operator. It is thus only necessary to extend Lemmata 1,3,4 and 5 to the operator h̃^{(2)} which is obtained from h^{BR} by including the single-particle second-order potentials b_{2m}^{(k)}, k = 1, 2. We start with the lemmata required for the ’hard part’ of the proof.

a) In the formulation of Lemma 1 we simply replace h^{BR} by h̃^{(2)} throughout (and take N = 2). In order to prove ∥[h̃^{(2)}, χ_{0}] ψ_{n}∥ ≤ \frac{c}{n} ∥ψ_{n}∥ with ψ_{n} ∈ A(C_{0}^{∞}(ℝ^{3}) ⊗ ℂ^{2})^{2} a Weyl sequence for λ ∈ σ_{ess}(h̃^{(2)}) and χ_{0} from (1.13) with x := (x_{1}, x_{2}), we have to show in addition to the Brown-Ravenhall case,

|(ϕ, [b_{2m}^{(1)}, χ_{0}] ψ_{n})| ≤ \frac{c}{n} ∥ϕ∥ ∥ψ_{n}∥  (2.12)

for all ϕ ∈ C_{0}^{∞}(ℝ^{6}) ⊗ ℂ^{4}. Due to the symmetry property of ψ_{n}, the same bound holds also for [b_{2m}^{(2)}, χ_{0}]. The operator b_{2m}^{(1)} defined in (2.4) is a sum of terms of the structure B(p_{1})\frac{1}{x_{1}} B_{λ}(p_{1}) V_{10,m}^{(1)}B_{μ}(p_{1}) where B(p_{1}) ∈ \{A_{1}, G_{1}\}

like for b_{1m}^{(1)} whereas B_{λ}(p_{1}), B_{μ}(p_{1}) ∈ \{1, \frac{σ^{(1)}p_{1}}{E_{p_{1}}} , \frac{m}{E_{p_{1}}} , A_{1}, G_{1}\} are all analytic, bounded multiplication operators in momentum space. We pick for the sake of demonstration the second term of (2.4) and decompose

[G_{1} \frac{1}{x_{1}} \frac{σ^{(1)}p_{1}}{E_{p_{1}}} V_{10,m}^{(1)}A_{1}, χ_{0}] = [G_{1}, χ_{0}] p_{1} · \frac{1}{p_{1}x_{1}} \frac{σ^{(1)}p_{1}}{E_{p_{1}}} V_{10,m}^{(1)}A_{1}

+ G_{1} \frac{1}{x_{1}p_{1}} · p_{1} [\frac{σ^{(1)}p_{1}}{E_{p_{1}}} , χ_{0}] V_{10,m}^{(1)}A_{1} + G_{1} \frac{1}{x_{1}p_{1}} \frac{σ^{(1)}p_{1}}{E_{p_{1}}} · p_{1} [V_{10,m}^{(1)}, χ_{0}] A_{1}

+ G_{1} \frac{1}{x_{1}p_{1}} \frac{σ^{(1)}p_{1}}{E_{p_{1}}} · p_{1} V_{10,m}^{(1)} \frac{1}{p_{1}} · p_{1} [A_{1}, χ_{0}].  (2.13)

We will show that the commutators (including the factor p_{1}) are \frac{1}{n}-bounded and the adjacent factors bounded. The latter is trivial (since also \frac{1}{x_{1}p_{1}} is bounded,

∥\frac{1}{x_{1}p_{1}} ∥ = 2, see e.g. [8]) except for the operator p_{1}V_{10,m}^{(1)} \frac{1}{p_{1}} in the last term. The boundedness of this operator is readily proved by invoking its kernel in momentum space. From (2.1) we have

∥ p_{1} V_{10,m}^{(1)} \frac{1}{p_{1}} ∥ = 2π^{2} ∥ ∫_{0}^{∞} dt e^{−tE_{p_{1}}} p_{1} \frac{1}{x_{1}p_{1}} e^{−tE_{p_{1}}} ∥

≤ 2π^{2} ∥ ∫_{0}^{∞} dt e^{−tE_{p_{1}}} p_{1} ∥ · ∥\frac{1}{x_{1}p_{1}} ∥ · ∥e^{−tE_{p_{1}}} ∥ ≤ 4π^{2}  (2.14)

where ∫_{0}^{∞}dt e^{−tE_{p_{1}}} = 1/E_{p_{1}} has been used.

The commutators p_{1} [A_{1}, χ_{0}] and p_{1} [G_{1}, χ_{0}] have already been dealt with in the context of the Brown-Ravenhall operator. p_{1} [\frac{σ^{(1)}p_{1}}{E_{p_{1}}} , χ_{0}] is of the same type, because for any B_{λ}, one has the estimate |B_{λ}(p_{1}) − B_{λ}(p′_{1})| = |p_{1} − p′_{1}| |∇_{p_{1}} B_{λ}(ξ)| ≤ |p_{1} − p′_{1}| \frac{c_{0}}{1+p_{1}} from the mean value theorem, where ξ is

some point between p_{1} and p′_{1}. For the commutator with V_{10,m}^{(1)} we have

p_{1} [V_{10,m}^{(1)}, χ_{0}] = 2π^{2} ∫_{0}^{∞} dt p_{1} [e^{−tE_{p_{1}}} , χ_{0}] \frac{1}{x_{1}} e^{−tE_{p_{1}}}

+ 2π^{2} ∫_{0}^{∞} dt p_{1} e^{−tE_{p_{1}}} \frac{1}{x_{1}} [e^{−tE_{p_{1}}} , χ_{0}].  (2.15)

The proof of its \frac{1}{n}-boundedness proceeds with the help of the Lieb and Yau formula [17], derived from the Schwarz inequality (see also [14, Lemma 7]), in momentum space. Explicitly, in the estimate

|(ϕ̂, { \atop O ψ_{n}})| ≤ (∫_{ℝ^{6}} dp |ϕ̂(p)|^{2} I(p))^{\frac{1}{2}} (∫_{ℝ^{6}} dp′ |ψ̂_{n}(p′)|^{2} J(p′))^{\frac{1}{2}} (2.16)

where O := p_{1}[V_{10,m}^{(1)}, χ_{0}] and k_{O} its kernel, one has to show that the integrals I and J obey

I(p) := ∫_{ℝ^{6}} dp′ |k_{O}(p, p′)| \frac{f(p)}{f(p′)} ≤ \frac{c}{n}

J(p′) := ∫_{ℝ^{6}} dp |k_{O}(p, p′)| \frac{f(p′)}{f(p)} ≤ \frac{c}{n}  (2.17)

with some constant c (independent of p, p′) for a suitably chosen nonnegative convergence generating function f. We use the two-dimensional (N = 2) Fourier transform (1.14) of χ_{0} and the momentum representation of \frac{1}{x_{1}} to write for the first term in (2.15),

(∫_{0}^{∞}dt{ \atop p_{1}[e^{−tE_{p_{1}}} , χ_{0}]\frac{1}{x_{1}} e^{−tE_{p_{1}}} φ})(p_{1}, p_{2}) =∫_{ℝ^{6}}dqdp′_{2}k_{1}(p_{1}, p_{2}, q, p′_{2})φ̂(q, p′_{2})

k_{1}(p_{1}, p_{2}, q, p′_{2}) := \frac{1}{(2π)^{3}} \frac{1}{2π^{2}} p_{1} n^{6} ∫_{0}^{∞} dt ∫_{ℝ^{3}} dp′_{1} χ̂_{0}(n(p_{1} −p′_{1}), n(p_{2} −p′_{2}))

(2.18)

·(e^{−tE_{p_{1}}} − e^{−tE_{p′_{1}}} ) \frac{1}{|p′_{1} − q|^{2}} e^{−tE_{q}}.

From the mean value theorem we get

|e^{−tE_{p_{1}}} − e^{−tE_{p′_{1}}} | = |p_{1} − p′_{1}| |te^{−tE_{ξ}} \frac{ξ}{E_{ξ}} | ≤ |p_{1} − p′_{1}| t e^{−tE_{ξ}}  (2.19)

with ξ = λp′_{1} +(1−λ)p_{1} for some λ ∈ [0, 1]. We have to show that the integral over the modulus of the kernel of (2.18), with a suitable convergence generating function f, is \frac{1}{n}-bounded. We choose f(p) = p and make the substitution y_{i} := n(p_{i} − p′_{i}) for p′_{i}, i = 1, 2. Then

I(p_{1}, p_{2}) := ∫_{ℝ^{6}} dq dp′_{2} |k_{1}(p_{1}, p_{2}, q, p′_{2})| \frac{f(p_{1})}{f(q)}

≤ \frac{1}{(2π)^{4}π} p_{1} ∫_{0}^{∞} dt ∫_{ℝ^{3}} dq ∫_{ℝ^{6}} dy_{1} dy_{2} |χ̂_{0}(y_{1}, y_{2})|  (2.20)

· \frac{y_{1}}{n} t e^{−tE_{ξ}} \frac{1}{|q − (p_{1} − y_{1}/n)|^{2}} e^{−tE_{q}} · \frac{p_{1}}{q} .

The t-integral can be carried out, ∫_{0}^{∞}dt te^{−(E_{ξ}+E_{q})} = (E_{ξ} + E_{q})^{−2} with ξ =

p_{1} − \frac{λ}{n}y_{1}. Define q_{1} := p_{1} − y_{1}/n and consider

S := p_{1}^{2} ∫_{ℝ^{3}} dq \frac{1}{|q − q_{1}|^{2}} \frac{1}{q} \frac{1}{(E_{ξ} + E_{q})^{2}} .  (2.21)

Estimating the last factor by \frac{1}{E_{ξ}} · \frac{1}{q} and performing the angular integration, one obtains

S ≤ p_{1}^{2} \frac{2π}{q_{1}} \frac{1}{E_{ξ}} ∫_{0}^{∞} \frac{dq}{q} ln |\frac{q + q_{1}}{q − q_{1}} | = π^{3} \frac{p_{1}}{|p_{1} − y_{1}/n|} \frac{p_{1}}{\sqrt{(p_{1} − \frac{λ}{n}y_{1})^{2} + m^{2}}} .

(2.22)

Documenta Mathematica 10 (2005) 417–445

Insertion into (2.20) gives

I(p_{1}, p_{2}) ≤ \frac{1}{(4π)^{2}} \frac{1}{n} ∫_{ℝ^{6}} dy_{1} dy_{2} |χ̂_{0}(y_{1}, y_{2})| y_{1}

·\frac{p_{1}}{|p_{1} − y_{1}/n|} \frac{p_{1}}{\sqrt{(p_{1} − \frac{λ}{n}y_{1})^{2} + m^{2}}} ≤ \frac{c}{n}  (2.23)

because the singularity at y_{1} = np_{1} is integrable and the integral is finite for all p_{1} ≥ 0 due to χ̂_{0} ∈ S(ℝ^{6}). Since the kernel k_{1} is not symmetric in p_{1}, q, the estimate of J(q, p′_{2}) := ∫_{ℝ^{6}} dp_{1} dp_{2} |k_{1}(p_{1}, p_{2}, q, p′_{2})| \frac{f(q)}{f(p_{1})} is needed

too. The \frac{1}{n}-boundedness of J(q, p′_{2}) can be shown along the same lines, using (E_{ξ′} + E_{q})^{−2} ≤ ξ^{′−2} with ξ′ := p′_{1} + λ(p_{1} − p′_{1}). We still have to estimate the second term in (2.15). Its kernel is

k_{2}(p_{1}, p_{2}, q, p′_{2}) := \frac{1}{(2π)^{3}} \frac{1}{2π^{2}} p_{1} n^{6} ∫_{0}^{∞} dt e^{−tE_{p_{1}}}

∫_{ℝ^{3}} dp′_{1} \frac{1}{|p_{1} − p′_{1}|^{2}} χ̂_{0}(n(p′_{1} − q), n(p_{2} − p′_{2})) (e^{−tE_{p′_{1}}} − e^{−tE_{q}}). (2.24)

With (2.19) the t-integral can be carried out as before. Making the substitution y_{1} := n(p′_{1} − q), y_{2} := n(p_{2} − p′_{2}) for q and p′_{2}, respectively, one gets with the choice f(p) = p^{\frac{1}{2}} ,

Ĩ(p_{1}, p_{2}) := ∫_{ℝ^{6}} dq dp′_{2} |k_{2}(p_{1}, p_{2}, q, p′_{2})| \frac{f(p_{1})}{f(q)}  (2.25)

≤ \frac{1}{(2π)^{4}π} p_{1}∫_{ℝ^{6}}dy_{1}dy_{2}|χ̂_{0}(y_{1}, y_{2})|∫_{ℝ^{3}}dp′_{1} \frac{1}{|p_{1}−p′_{1}|^{2}} \frac{y_{1}}{n} \frac{1}{(E_{p_{1}}+Eξ̃)^{2}} · \frac{p1^{\frac{1}{2}}}{|p′_{1}− \frac{y_{1}}{n} |^{\frac{1}{2}}}

with ξ̃ := λq + (1 − λ)p′_{1} = p′_{1} − \frac{λ}{n}y_{1}. We estimate (E_{p_{1}} + Eξ̃)^{−2} ≤ p_{1}^{− \frac{1}{2}} E− \frac{3}{2} ; ξ̃ .

Then the integral over p′_{1} reduces to

S̃ := p_{1} ∫_{ℝ^{3}} dp′_{1} \frac{1}{|p_{1} − p′_{1}|^{2}} \frac{1}{|p′_{1} − \frac{y_{1}}{n} |^{\frac{1}{2}}} \frac{1}{[(p′_{1} − \frac{λ}{n}y_{1})^{2} + m^{2}]^{\frac{3}{4}}} .  (2.26)

Even when the two singularities coincide (for y_{1} = np_{1}), they are integrable. Since the integrand behaves like p_{1}^{′−2} for p′_{1} → ∞, S̃ is finite for all 0 ≤ p_{1} < ∞. It remains to estimate S̃ for p_{1} → ∞. We substitute p_{1}x := p′_{1} − p_{1}, such that with e_{p_{1}} := p_{1}/p_{1},

S̃ = ∫_{ℝ^{3}} \frac{dx}{x^{2}} \frac{1}{|x + e_{p_{1}} − \frac{y_{1}}{np_{1}} |^{\frac{1}{2}}} \frac{1}{[(x + e_{p_{1}} − \frac{λ}{np_{1}} y_{1})^{2} + \frac{m^{2}}{p_{1}^{2}} ]^{\frac{3}{4}}}

⟶ ∫_{ℝ^{3}} \frac{dx}{x^{2}} \frac{1}{|x + e_{p_{1}}|^{2}} = π^{3}  as p_{1} → ∞.  (2.27)

Documenta Mathematica 10 (2005) 417–445

Therefore, Ĩ is \frac{1}{n}-bounded for all p_{1} ≥ 0. It is easy to prove that also

J̃(q, p′_{2}) := ∫_{ℝ^{6}} dp_{1} dp_{2} |k_{2}(p_{1}, p_{2}, q, p′_{2})| \frac{q^{\frac{1}{2}}}{p \frac{1}{2} 1} is \frac{1}{n}-bounded, using the estimate

(E_{p_{1}} + E_{|q+λ(p′_{1}−q)|})^{−2} ≤ p_{1}^{−2}. Collecting results, this shows the \frac{1}{n}-boundedness of p_{1}[V_{10,m}^{(1)}, χ_{0}]. With the same tools, the \frac{1}{n}-boundedness of the commutator of χ_{0} with the remaining contributions from (2.4) to b_{2m}^{(1)} is established. The second item of Lemma 1, the normalizability of the sequence φ_{n} := (1 − χ_{0})ψ_{n}, follows immediately from the proof concerning the Brown-Ravenhall operator, because of the relative form boundedness of the total potential of h̃^{(2)} with form bound smaller than one for γ < γ_{BR} ( see [13] and Lemma 7).

b) In the formulation of Lemma 3, the only change is again the replacement of h^{BR} with h̃^{(2)} (and N = 2). We consider the case j = 1 where r_{1} = b_{1m}^{(1)} + b_{2m}^{(1)} + v^{(12)}, and we have to show in addition to the Brown-Ravenhall case that

|(ϕ_{1}φ, b_{2m}^{(1)} ϕ_{1}φ)| ≤ \frac{c}{R} ∥φ∥^{2}  (2.28)

provided φ ∈ A(C_{0}^{∞}(ℝ^{6}\B_{R}(0)) ⊗ ℂ^{4}) and R > 1. We note that every summand of b_{2m}^{(1)} in (2.4) is of the form B_{1} \frac{1}{x_{1}} W_{1} or W_{1} \frac{1}{x_{1}} B_{1} where B_{1} is a bounded multiplication operator in momentum space, while W_{1} is a bounded integral operator. For operators of the first type we take the smooth auxiliary function χ_{1}(\frac{x_{1}}{R}) from (1.17) which is unity on the support of ϕ_{1}φ and decompose

(χ_{1}ϕ_{1}φ, B_{1} \frac{1}{x_{1}} W_{1} ϕ_{1}φ) = (ϕ_{1}φ, B_{1}χ_{1} \frac{1}{x_{1}} W_{1} ϕ_{1}φ) + (ϕ_{1}φ, [χ_{1}, B_{1}] \frac{1}{x_{1}} W_{1} ϕ_{1}φ).

(2.29)

Since supp χ_{1} ⊂ ℝ^{3}\B_{CR/2}(0) we have

|(B_{1}ϕ_{1}φ, χ_{1} \frac{1}{x_{1}} W_{1}ϕ_{1}φ)| ≤ \frac{2}{CR}∫_{ℝ^{6}} dx_{1}dx_{2}|(B_{1}ϕ_{1}φ)(x_{1}, x_{2})| χ_{1}|(W_{1}ϕ_{1}φ)(x_{1}, x_{2})|

≤ \frac{2}{CR} ∥B_{1}∥ ∥φ∥ ∥W_{1}∥ ∥φ∥ ≤ \frac{c_{0}}{R} ∥φ∥^{2}.  (2.30)

For the second contribution to (2.29), we have to estimate [χ_{1,0}, B_{1}]p_{1} with χ_{1,0} := 1 − χ_{1} in momentum space. Since B_{1} ∈ {A_{1}, G_{1}} we use the relation |(φ̃, [χ_{1,0}, B_{1}]p_{1} ψ̃)| = |(ψ̃, p_{1}[B_{1}, χ_{1,0}] φ̃)| (for suitable φ̃, ψ̃), the uniform \frac{1}{R}- boundedness of which has already been proven in the context of the Brown-Ravenhall case. The second operator, W_{1} \frac{1}{x_{1}} B_{1} is treated in the same way, using W_{1} \frac{1}{x_{1}} B_{1}χ_{1} ϕ_{1}φ = W_{1} \frac{1}{x_{1}} χ_{1}B_{1} ϕ_{1}φ + W_{1} \frac{1}{x_{1}} [B_{1}, χ_{1}] ϕ_{1}φ.

In the case j = 0 we have r_{0} = ∑_{k=1}^{2}(b_{1m}^{(k)} + b_{2m}^{(k)}), and since supp ϕ_{0} requires

x_{1} ≥ Cx as well as x_{2} ≥ Cx, x = (x_{1}, x_{2}), the auxiliary function can be taken from (1.17) for k = 1 or k = 2. The further proof of the lemma is identical to the one for j = 1.

c) Lemma 4 (formulated for h̃^{(2)} in place of h^{BR}) which is needed for the localization formula, has to be supplemented with the following estimate

(d)  |(ϕ_{j}φ, [b_{2m}^{(k)}, ϕ_{j}] φ)| ≤ \frac{c}{R} ∥φ∥^{2}  (2.31)

for φ ∈ A(C_{0}^{∞}(ℝ^{6}\B_{R}(0)) ⊗ ℂ^{4}) and R > 2.

The proof is carried out in coordinate space as are the proofs of the Brown-Ravenhall items of Lemma 4. We split the commutator in the same way as in the proof of Lemma 1. In order to show how to proceed, we pick again the second term of (2.4), take k = 1 and decompose

[G_{1} \frac{1}{x_{1}} \frac{σ^{(1)}p_{1}}{E_{p_{1}}} V_{10,m}^{(1)} A_{1}, ϕ_{j}] = [G_{1}, ϕ_{j}] \frac{1}{x_{1}} · \frac{σ^{(1)}p_{1}}{E_{p_{1}}} V_{10,m}^{(1)} A_{1}

+ G_{1} \frac{1}{x_{1}} [\frac{σ^{(1)}p_{1}}{E_{p_{1}}} , ϕ_{j}] V_{10,m}^{(1)} A_{1} + G_{1} \frac{1}{x_{1}} \frac{σ^{(1)}p_{1}}{E_{p_{1}}} x_{1} · \frac{1}{x_{1}} [V_{10,m}^{(1)}, ϕ_{j}] A_{1} (2.32)

+ G_{1} \frac{1}{x_{1}} \frac{σ^{(1)}p_{1}}{E_{p_{1}}} V_{10,m}^{(1)} x_{1} · \frac{1}{x_{1}} [A_{1}, ϕ_{j}].

We have to prove the \frac{1}{R}-boundedness of the commutators (including the factor \frac{1}{x_{1}} ) and to assure the boundedness of the adjacent operators. The commutators with G_{1} and A_{1} have already been dealt with in the Brown-Ravenhall case. As concerns [\frac{σ^{(1)}p_{1}}{E_{p_{1}}} , ϕ_{j}]\frac{1}{x_{1}} , we have to show that its kernel obeys the estimate

|ǩ_{σ^{(1)}p_{1} _{\frac{1}{Ep_{1}}}} (x_{1}, x′_{1})| ≤ \frac{c}{|x_{1} − x′_{1}|^{3}}  (2.33)

with some constant c. When dealing with the Brown-Ravenhall operator, we have shown the corresponding estimate for the kernel of the operator σ^{(1)}p_{1}g(p_{1}) with g(p_{1}) = [2(p_{1}^{2} + m^{2} + m\sqrt{p_{1}^{2} + m^{2}} )]^{− \frac{1}{2}} . Replacing g(p_{1}) with (p_{1}^{2} + m^{2})^{− \frac{1}{2}} does neither change the analyticity property of the kernel nor its behaviour as |x_{1} − x′_{1}| tends to 0 or infinity, from which (2.33) is established [14]. For the further proof of the \frac{1}{R}-boundedness of the commutator, we can substitute ϕ_{j} with ϕ_{j}χ where χ(\frac{x}{R}) is defined in (1.20) with x = (x_{1}, x_{2}) (see the discussion below (1.20)). Thus we can use the estimate (1.21) (for k = 1 and N = 2) derived from the mean value theorem and mimic the proof of the two-electron Brown-Ravenhall case. For the treatment of the remaining commutator, [V_{10,m}^{(1)}, ϕ_{j}χ]\frac{1}{x_{1}} , we set ψ_{j} := ϕ_{j}χ and decompose

[V_{10,m}^{(1)}, ψ_{j}] \frac{1}{x_{1}} = [V_{10,m}^{(1)} \frac{1}{x_{1}} , ψ_{j}]  (2.34)

= 2π^{2} ∫_{0}^{∞} dt [e^{−tE_{p_{1}}} , ψ_{j}] \frac{1}{x_{1}} e^{−tE_{p_{1}}} \frac{1}{x_{1}} + 2π^{2} ∫_{0}^{∞} dt e^{−tE_{p_{1}}} \frac{1}{x_{1}} [e^{−tE_{p_{1}}} , ψ_{j}] \frac{1}{x_{1}} .

The kernel of e^{−tE_{p_{1}}} in coordinate space is given by [17]

ǩ_{e^{−tEp_{1}}} (x_{1}, x′_{1}, t) = ǩ_{e^{−tEp_{1}}} (x̃, t) = \frac{t}{2π^{2}} \frac{m^{2}}{x̃^{2} + t^{2}} K_{2}(m\sqrt{x̃^{2} + t^{2}} ) (2.35)

where K_{2} is a modified Bessel function of the second kind and x̃ := x_{1} − x′_{1}. Making use of the analyticity of K_{2}(z) for z > 0 and its behaviour K_{2}(z) ∼ \frac{2}{z^{2}} for z → 0 and K_{2}(z) ∼ \sqrt{\frac{π}{2z}} e^{−z} for z → ∞ [1, p.377] we have

|K_{2}(z)| ≤ \frac{2c_{1}}{z^{2}}  (2.36)

and therefore we can estimate ǩ_{e^{−tEp_{1}}} by the corresponding kernel for m = 0,

|ǩ_{e^{−tEp_{1}}} (x̃, t)| ≤ \frac{t}{π^{2}} \frac{c_{1}}{(x̃^{2} + t^{2})^{2}} = c_{1} ǩ_{e^{−tp_{1}}} (x̃, t).  (2.37)

Thus we obtain for the kernel of the second contribution to (2.34), using (2.37) and (1.21),

S_{0} := |∫_{0}^{∞} dt ǩ_{e^{−tEp_{1}} \frac{1}{x_{1}} [e^{−tEp_{1}} ,ψ_{j}] \frac{1}{x_{1}}} (x_{1}, y_{1}, x_{2})|  (2.38)

= |∫_{0}^{∞} dt ∫_{ℝ^{3}} dx′_{1} \frac{t}{2π^{2}} \frac{m^{2}}{(x_{1} − x′_{1})^{2} + t^{2}} K_{2}(m\sqrt{(x_{1} − x′_{1})^{2} + t^{2}} ) \frac{1}{x′_{1}}

·\frac{t}{2π^{2}} \frac{m^{2}}{(x′_{1} − y_{1})^{2} + t^{2}} K_{2}(m\sqrt{(x′_{1} − y_{1})^{2} + t^{2}} ) \frac{1}{y_{1}} (ψ_{j}(y_{1}, x_{2}) − ψ_{j}(x′_{1}, x_{2}))|

≤ \frac{c_{1}^{2}}{π^{4}} ∫_{0}^{∞} t^{2} dt ∫_{ℝ^{3}} dx′_{1} \frac{1}{[(x_{1} − x′_{1})^{2} + t^{2}]^{2}} \frac{1}{x′_{1}} \frac{1}{[(x′_{1} − y_{1})^{2} + t^{2}]^{2}} \frac{1}{y_{1}} ·|x′_{1}−y_{1}| \frac{c_{0}}{R} .

With the help of the estimate \frac{1}{[(x′_{1}−y_{1})^{2}+t^{2}]^{2}} ≤ \frac{1}{t} \frac{1}{|x′_{1}−y_{1}|^{3}} , the t-integral can be carried out,

∫_{0}^{∞} dt \frac{t}{[(x_{1} − x′_{1})^{2} + t^{2}]^{2}} = \frac{1}{2 |x_{1} − x′_{1}|^{2}} .  (2.39)

According to the Lieb and Yau formula (2.16) in coordinate space, the \frac{1}{R}- boundedness of S_{0} integrated over y_{1}, respectively over x_{1}, with a suitably chosen convergence generating function f, has to be shown (in analogy to (2.17)). With the choice f(x) = x^{α} and (2.39) we have

I(x_{1}) := ∫_{ℝ^{3}} dy_{1} S_{0} \frac{f(x_{1})}{f(y_{1})}

≤ \frac{c̃}{R} ∫_{ℝ^{3}} dx′_{1} \frac{1}{|x_{1} − x′_{1}|^{2}} \frac{1}{x′_{1}} ∫_{ℝ^{3}} dy_{1} \frac{1}{y_{1}} \frac{1}{|x′_{1} − y_{1}|^{2}} · \frac{x_{1}^{α}}{y_{1}^{α}} .  (2.40)

With the substitutions y_{1} =: x′_{1}z and then x′_{1} =: x_{1}ξ the two integrals separate such that (with e_{x} := x/x)

I(x_{1}) ≤ \frac{c̃}{R} ∫_{ℝ^{3}} \frac{dξ}{ξ^{1+α}} \frac{1}{|e_{x_{1}} − ξ|^{2}} · ∫_{ℝ^{3}} \frac{dz}{z^{1+α}} \frac{1}{|e_{x′_{1}} − z|^{2}} ≤ \frac{C}{R}  (2.41)

if 0 < α < 2 [3]. In the same way it is shown that J(y_{1}) := ∫_{ℝ^{3}} dx_{1} S_{0} \frac{y_{1}^{α}}{x_{1}^{α}} ≤ \frac{C}{R}

for 1 < α < 3. Thus α = 3/2 assures the 1/R-boundedness of I and J. The first contribution to (2.34) is treated along the same lines. This proves the 1/R-boundedness of [V_{10,m}^{(1)}, ψ_{j}]\frac{1}{x_{1}} .

Finally the boundedness of the two operators occurring in (2.32), \frac{1}{x_{1}} \frac{σ^{(1)}p_{1}}{E_{p_{1}}} x_{1}

and \frac{1}{x_{1}} V_{10,m}^{(1)}x_{1}, has to be shown. We use the fact that for any bounded operator O, \frac{1}{x_{1}} Ox_{1} = \frac{1}{x_{1}} [O, x_{1}] + O, such that for the first operator, only

the boundedness of \frac{1}{x_{1}} [\frac{σ^{(1)}p_{1}}{E_{p_{1}}} , x_{1}] has to be established. We use the estimate (2.33) to write

|ǩ_{\frac{1}{x_{1}} [ \frac{σ^{(1)}p_{1}}{Ep_{1}} ,x_{1}]}(x_{1}, x′_{1})|

= |\frac{1}{x_{1}} ǩ_{\frac{σ^{(1)}p_{1}}{Ep_{1}}} (x_{1}, x′_{1}) · (x_{1} − x′_{1})| ≤ \frac{c̃}{x_{1}} \frac{1}{|x_{1} − x′_{1}|^{2}}  (2.42)

and with the choice f(x) = x^{3/2}, (2.42) multiplied by f(x_{1})/f(x′_{1}) and integrated over dx′_{1}, respective multiplied by f(x′_{1})/f(x_{1}) and integrated over dx_{1}, is finite. This proves the desired boundedness. Concerning the operator \frac{1}{x_{1}} V_{10,m}^{(1)}x_{1} we decompose

\frac{1}{x_{1}} V_{10,m}^{(1)} x_{1} = 2π^{2} ∫_{0}^{∞} dt \frac{1}{x_{1}} e^{−tE_{p_{1}}} \frac{1}{x_{1}} \{[e^{−tE_{p_{1}}} , x_{1}] + x_{1}e^{−tE_{p_{1}}} \} . (2.43)

In the second contribution the t-integral can be carried out, ∫_{0}^{∞}dt \frac{1}{x_{1}} e^{−2tE_{p_{1}}} =

\frac{1}{2x_{1}E_{p_{1}}} which is a bounded operator. For the first contribution, we can again use the estimate (2.36) for the Bessel function together with the estimate for the t-dependence, resulting in (2.39), such that

S̃_{0} := |∫_{0}^{∞} dt ǩ_{\frac{1}{x_{1}} e^{−tEp_{1}} \frac{1}{x_{1}} [e^{−tEp_{1}} ,x_{1}]}(x_{1}, y_{1})|

= |∫_{0}^{∞} dt \frac{1}{x_{1}} ∫_{ℝ^{3}} dx′_{1} \frac{t}{2π^{2}} \frac{m^{2}}{|x_{1} − x′_{1}|^{2} + t^{2}} K_{2}(m\sqrt{(x_{1} − x′_{1})^{2} + t^{2}} ) \frac{1}{x′_{1}}

(2.44)

Documenta Mathematica 10 (2005) 417–445

· \frac{t}{2π^{2}} \frac{m^{2}}{(x′_{1} − y_{1})^{2} + t^{2}} K_{2}(m\sqrt{(x′_{1} − y_{1})^{2} + t^{2}} ) (y_{1} − x′_{1})|

≤ c̃_{0} \frac{1}{x_{1}} ∫_{ℝ^{3}} dx′_{1} \frac{1}{|x_{1} − x′_{1}|^{2}} \frac{1}{x′_{1}} \frac{1}{|x′_{1} − y_{1}|^{2}} .

With α = 3/2, in the same way as shown in the step from (2.40) to (2.41), one

obtains Ĩ(x_{1}) := ∫_{ℝ^{3}} dy_{1} S̃_{0} \frac{x_{1}^{α}}{y_{1}^{α}} ≤ c and J̃(y_{1}) := ∫_{ℝ^{3}} dx_{1} S̃_{0} \frac{y_{1}^{α}}{x_{1}^{α}} ≤ c. Thus the

boundedness of \frac{1}{x_{1}} V_{10,m}^{(1)}x_{1} is shown.

In the remaining contributions to [b_{2m}^{(1)}, ϕ_{j}χ] the terms not treated so far are [\frac{m}{E_{p_{1}}} , ϕ_{j}χ]\frac{1}{x_{1}} , the \frac{1}{R}-boundedness of which follows from the estimate

|ǩ_{\frac{m}{Ep_{1}}} (x_{1}, x′_{1})| ≤ c/|x_{1} − x′_{1}|^{3} (which is proven in the same way as the corre-

sponding Brown-Ravenhall estimate for g̃(p_{1}) := \frac{m}{\sqrt{2}} (E_{p_{1}} + \sqrt{E_{p_{1}}^{2} + mE_{p_{1}}} )^{−1}

in place of m/E_{p_{1}} [14]). The boundedness of the additional term \frac{1}{x_{1}} \frac{m}{E_{p_{1}}} x_{1}

(respective \frac{1}{x_{1}} [\frac{m}{E_{p_{1}}} , x_{1}]) follows from (2.42) formulated for ǩ_{\frac{m}{Ep_{1}}} . This completes the proof of Lemma 4.

We now turn to the ’easy part’ of the HVZ theorem, where we have to assure that [Σ_{0}, ∞) ⊂ σ_{ess}(h̃^{(2)}). We use the method of proof applied to the multiparticle Brown-Ravenhall operator (see section 1 (b)). The proof of continuity of σ(T +a_{j}) for j ∈ {0, ..., N} with a_{j} from (2.5) does not depend on the choice of the single-particle potential and hence also holds true for the Jansen-Hess operator. With T_{a} from (1.25) for N = 2 and φ_{n} ∈ C_{0}^{∞}(ℝ^{6}) ⊗ ℂ^{4} a defining sequence for λ ∈ σ(T + a_{j}) we have (according to (1.26)) to show that ∥r_{j} T_{a}φ_{n}∥ < ϵ for n and a sufficiently large, where r_{j} now includes the terms b_{2m}^{(k)} with k ∉ C_{1j}.

(d) Lemma 5 has therefore to be supplemented with the conjecture

∥b_{2m}^{(k)} φ∥ ≤ \frac{c}{R} ∥φ∥  (2.45)

where φ ∈ C_{0}^{∞}(Ω) ⊗ ℂ^{2l} with Ω := {x = (x_{1}, ..., x_{l}) ∈ ℝ^{3l} : x_{i} > R ∀ i = 1, ..., l}, R > 1 and k ∈ {1, ..., l} where l is the number of electrons in cluster C_{2j}. The domain Ω allows for the introduction of the auxiliary function χ from (1.31) which is unity on the support of φ. As discussed in the proof of Lemma 3, b_{2m}^{(k)} consists (for k = 1) of terms like W_{1} \frac{1}{x_{1}} B_{1} (and its Hermitean conjugate), such that the idea of (2.29) can be used,

|(ϕ, W_{1} \frac{1}{x_{1}} B_{1} φ)| ≤ |(W_{1}ϕ, \frac{1}{x_{1}} χ B_{1} φ)| + |(W_{1}ϕ, \frac{1}{x_{1}} [χ, B_{1}] φ)|. (2.46)

Therefore, the proof of Lemma 3 establishes the validity of (2.45), too. Thus the proof of Theorem 2 is complete. □

We remark that the two-particle potentials of h̃^{(2)} coincide with those of h^{BR} and hence are nonnegative. Therefore, as demonstrated in section 1 (below (1.9)), j = 0 (corresponding to the cluster decomposition where the nucleus is separated from all electrons) can be omitted in the determination of Σ_{0}. Thus the infimum of the essential spectrum of h^{(2)} is given by the first ionization threshold (i.e. the infimum of the spectrum of the operator describing an ion with one electron less) increased by the electron’s rest energy m.

3 The multiparticle Jansen-Hess operator

Let

H_{N}^{(2)} := H^{BR} + Λ_{+,N} (∑_{k=1}^{N} B_{2m}^{(k)} + ∑_{k>l=1}^{N} C^{(kl)}) Λ_{+,N}  (3.1)

=: H̃_{N}^{(2)} + Λ_{+,N} ∑_{k>l=1}^{N} C^{(kl)} Λ_{+,N}

with H^{BR} from (1.1) and the second-order potentials from (2.1) and (2.2). According to section 1, the proofs of the required lemmata to assure the HVZ theorem for H̃_{N}^{(2)} are easily generalized to the N-electron case (with the exception of Lemma 1). For Lemma 1 to hold, we have to establish the form boundedness of the total potential W̃_{0} of H̃_{N}^{(2)} with respect to the multiparticle kinetic energy T_{0}. We can prove (see Appendix A)

Lemma 7. Let H̃_{N}^{(2)} =: T_{0} + W̃_{0} be (as defined in (3.1) with T_{0} :=

Λ_{+,N} ∑_{k=1}^{N} D_{0}^{(k)} Λ_{+,N}) the N-electron Jansen-Hess operator without the second-

order two-electron interaction terms, acting on A(H_{1}(ℝ^{3}) ⊗ ℂ^{4})^{N}. Then W̃_{0} is relatively form bounded with respect to the kinetic energy operator T_{0},

|(ψ, W̃_{0}ψ)| ≤ c_{1} (ψ, T_{0} ψ) + C_{1} (ψ, ψ)  (3.2)

with c_{1} < 1 for γ < γ_{BR} irrespective of the electron number N (for N ≤ Z).

We remark that the relative form boundedness of the total potential W_{0} := H_{N}^{(2)} − T_{0} holds only for a smaller critical γ. Using the estimate

|(ψ_{+}, ∑_{k>l=1}^{N} C^{(kl)}ψ_{+})| ≤ γ \frac{e^{2}π^{2}}{2} \frac{N−1}{2} (ψ, T_{0}ψ) with ψ_{+} := Λ_{+,N}ψ [13], we found

γ < 0.454 (Z ≤ 62) for N = Z. The proof of Lemma 1 for the N-electron operator H̃_{N}^{(2)} is then done in the same way as for the Brown-Ravenhall operator in section 1 (using the estimates for the second-order single-particle interaction from section 2 (a)). For the proof of Proposition 1 formulated for the N-electron case we note that the resolvent R_{N,μ} := (H_{N}^{(2)} + μ)^{−1} − (H̃_{N}^{(2)} + μ)^{−1} can be written as a finite sum of compact operators of the type (2.10). In place of W_{2}, we have W_{kl} := (T + μ)^{−1}C^{(kl)}(T + μ)^{−1} with the N-particle kinetic energy T. Since, however, (T + μ)^{−1} ≤ (T^{(k)} + T^{(l)} + μ)^{−1}, the compactness proof for W_{kl} can be copied from the N = 2 case. In addition, we have to assure the relative operator boundedness of the total potential of H_{N}^{(2)}:

Lemma 8. Let H_{N}^{(2)} =: T_{0} + W_{0} be the N-electron Jansen-Hess operator. For γ < γ_{1} the total potential W_{0} is bounded by the kinetic energy operator,

∥W_{0} ψ∥ ≤ c_{0} ∥T_{0} ψ∥  (3.3)

with c_{0} < 1. For N = Z (and m = 0), γ_{1} = 0.285 (Z ≤ 39).

The proof is given in Appendix B. A consequence of this proof is the relative boundedness of the total potential of H̃_{N}^{(2)} (with bound < 1) for γ < γ_{1}. We note that the critical potential strength may well be improved by using more refined techniques for the estimate of W_{0}ψ in the case of large N. Collecting results, we have shown that the HVZ theorem holds also for the Nelectron Jansen-Hess operator, provided γ is below a critical potential strength (γ < 0.285 if N = Z).

Appendix A (Proof of Lemma 7)

When showing the relative form boundedness of the potential W̃_{0}, we can disregard the projectors Λ_{+,N} in (1.1) and (3.1). In fact, define the potential

W̃ by H̃_{N}^{(2)} = T_{0} + W̃_{0} =: Λ_{+,N}( ∑_{k=1}^{N} D_{0}^{(k)} + W̃) Λ_{+,N}. Assume we prove for

ψ_{+} := Λ_{+,N} ψ ∈ Λ_{+,N}(A(H_{1}(ℝ^{3}) ⊗ ℂ^{4})^{N}) an N-particle function in the positive spectral subspace and T = E_{p_{1}} + ... + E_{p_{N}} ,

|(ψ_{+}, W̃ ψ_{+})| ≤ c_{1} (ψ_{+}, T ψ_{+}) + C_{1} (ψ_{+}, ψ_{+})  (A.1)

with constants c_{1} < 1 and C_{1} ≥ 0. Then we get

(ψ, W̃_{0}ψ) = (ψ, Λ_{+,N} W̃ Λ_{+,N} ψ) = (ψ_{+}, W̃ ψ_{+}).  (A.2)

Noting that (ψ_{+}, T ψ_{+}) = (ψ_{+}, ∑_{k=1}^{N} D_{0}^{(k)}ψ_{+}) = (ψ, T_{0} ψ) and ∥Λ_{+,N} ψ∥ ≤

∥Λ_{+,N}∥ ∥ψ∥ ≤ ∥ψ∥ because ∥Λ_{+,N}∥ = ∥Λ_{+}^{(1)}∥ ⋯ ∥Λ_{+}^{(n)}∥ = 1, Lemma 7 is verified with the help of (A.1). In order to show (A.1) we start by estimating from below. We use V^{(kl)} ≥ 0, |(ψ_{+}, V^{(k)}ψ_{+})| ≤ \frac{γ}{γ_{BR}} (ψ_{+}, E_{p_{1}}ψ_{+}) (for γ ≤ γ_{BR}; [4, 13]) as well as

(ψ_{+}, B_{2m}^{(k)}ψ_{+}) ≥ −md_{0}γ^{2} (ψ_{+}, ψ_{+}) (for γ ≤ 4/π) with d_{0} := 8 + 12\sqrt{2} [3, 13]. Then

(ψ_{+}, W̃ ψ_{+}) ≥ − \frac{γ}{γ_{BR}} ∑_{k=1}^{N}(ψ_{+}, E_{p_{1}} ψ_{+}) − md_{0}γ^{2} ∑_{k=1}^{N}(ψ_{+}, ψ_{+})

= − \frac{γ}{γ_{BR}} (ψ_{+}, T ψ_{+}) − md_{0}Nγ^{2} (ψ_{+}, ψ_{+}).  (A.3)

For the estimate from above we use (ψ_{+}, (V^{(k)} + B_{2m}^{(k)}) ψ_{+}) ≤ m(d_{0}γ^{2} + \frac{3}{2}γ) (ψ_{+}, ψ_{+}) for γ ≤ 4/π [3], [11, Lemma II.8] as well as (ψ_{+}, V^{(kl)}ψ_{+}) ≤ \frac{e^{2}}{γ_{BR}} (ψ_{+}, E_{p_{1}}ψ_{+}) (for γ ≤ γ_{BR}) which is an immediate consequence of the estimate of V^{(k)}. Then

(ψ_{+}, W̃ ψ_{+}) ≤ m(d_{0}γ^{2} + \frac{3}{2}γ) N (ψ_{+}, ψ_{+}) + \frac{N − 1}{2} \frac{e^{2}}{γ_{BR}} (ψ_{+}, T ψ_{+}) (A.4)

such that (A.1) holds with c_{1} := max\{\frac{γ}{γ_{BR}} , \frac{N−1}{2} \frac{e^{2}}{γ_{BR}} \}. For N ≤ Z, one has c_{1} = \frac{γ}{γ_{BR}} which is smaller than one if γ < γ_{BR}.

Appendix B (Proof of Lemma 8)

For the proof of the relative boundedness of the total potential W_{0}, let H_{N}^{(2)} =

T_{0} + W_{0} =: Λ_{+,N}( ∑_{k=1}^{N} D_{0}^{(k)} + W)Λ_{+,N} where W denotes the total potential from (3.1). Assume that

∥W ψ_{+}∥ ≤ c_{0} ∥ ∑_{k=1}^{N} D_{0}^{(k)}ψ_{+}∥ = c_{0} ∥T ψ_{+}∥  (B.1)

with ψ_{+} = Λ_{+,N} ψ. Then

∥W_{0}ψ∥ = ∥Λ_{+,N}W Λ_{+,N}ψ∥ ≤ ∥Λ_{+,N}∥ ∥W ψ_{+}∥ ≤ c_{0} ∥ ∑_{k=1}^{N} D_{0}^{(k)}ψ_{+}∥

= c_{0} ∥Λ_{+,N} ∑_{k=1}^{N} D_{0}^{(k)}Λ_{+,N}ψ∥ = c_{0} ∥T_{0}ψ∥  (B.2)

since Λ_{+,N} = Λ_{+,N}^{2} commutes with D_{0}^{(k)}. In order to verify (B.1) we set W^{(k)} := V^{(k)} + B_{2m}^{(k)} and estimate

∥W ψ_{+}∥ ≤ ∥ ∑_{k=1}^{N}W^{(k)}ψ_{+}∥ + ∥∑_{k>l=1}^{N}V^{(kl)}ψ_{+}∥ + 2∥∑_{k>l=1}^{N}C_{a}^{(kl)}ψ_{+}∥ + 2∥∑_{k>l=1}^{N}C_{b}^{(kl)}ψ_{+}∥

(B.3)

where according to (2.2), C_{a}^{(kl)} := V^{(kl)}Λ_{−}^{(l)}F_{0}^{(l)} and C_{b}^{(kl)} := F_{0}^{(l)}Λ_{−}^{(l)}V^{(kl)} = C_{a}^{(kl)∗}, and the antisymmetry of ψ_{+} with respect to particle exchange was used to reduce the four contributions to C^{(kl)} to two.

From [11] it follows that ∥ ∑_{k=1}^{N} W^{(k)}ψ_{+}∥ ≤ \sqrt{c_{w}} ∥T ψ_{+}∥ with c_{w} := (\frac{4}{3}γ + \frac{2}{9}γ^{2})^{2}

and ∥V^{(kl)}ψ_{+}∥ ≤ \sqrt{c_{v}} ∥E_{p_{k}}ψ_{+}∥ with c_{v} := 4e^{4}. Likewise, using ∥Λ_{−}^{(l)}∥ = 1 and ∥F_{0}^{(l)}∥ ≤ \frac{γ}{π} (\frac{π^{2}}{4} − 1) [11], one has ∥C_{a}^{(kl)}ψ_{+}∥ ≤ \sqrt{c_{v}} ∥E_{p_{k}}(Λ_{−}^{(l)}F_{0}^{(l)}ψ_{+})∥ ≤ \sqrt{c_{v}} ∥F_{0}^{(l)}∥ ∥E_{p_{k}}ψ_{+}∥ ≤ \sqrt{c̃_{s}} ∥E_{p_{k}}ψ_{+}∥ and the same estimate for ∥C_{b}^{(kl)}ψ_{+}∥, with c̃_{s} := (\frac{γ}{π} (\frac{π^{2}}{4} −1))^{2}c_{v} (for m = 0). For the cross terms V^{(kl)}V^{(kl′)} (l ≠ l′) we substitute y_{l} := x_{l} − x_{k} and y_{l′} := x_{l′} − x_{k} for x_{l} and x_{l′}, respectively, and get

(ψ_{+}, V^{(kl)}V^{(kl′)}ψ_{+}) = ∫_{ℝ^{3N}} ( ∏_{{k′=1 \atop k′≠l,l′}}^{N}dx_{k′}) dy_{l} dy_{l′} \frac{e^{2}}{y_{l}} \overline{ψ}_{+}(..., y_{l} + x_{k}, y_{l′} + x_{k}, ...)

· \frac{e^{2}}{y_{l′}} ψ_{+}(..., y_{l} + x_{k}, y_{l′} + x_{k}, ...).  (B.4)

Keeping for the moment x_{k′} fixed and using the Fourier representation with respect to y_{l} and y_{l′} (setting φ_{+}(y_{l}, y_{l′}) := ψ_{+}(..., y_{l} + x_{k}, y_{l′} + x_{k}, ...)),

({ \atop \frac{1}{y_{l′}} φ_{+}})(p_{l}, p_{l′}) = \frac{1}{2π^{2}} ∫_{ℝ^{3}} dp′ \frac{1}{|p_{l′} − p′|^{2}} φ̂_{+}(p_{l}, p′),  (B.5)

the Lieb and Yau formula (2.16) with the convergence generating function f(p) = p^{3/2} gives

|∫_{ℝ^{6}} dy_{l} dy_{l′} \frac{e^{2}}{y_{l}} \overline{φ}_{+} \frac{e^{2}}{y_{l′}} φ_{+}| ≤ 4e^{4} ∫_{ℝ^{6}} dp_{l} dp_{l′} |φ̂_{+}(p_{l}, p_{l′})|^{2} p_{l}p_{l′} (B.6)

such that, using p ≤ E_{p}, |(ψ_{+}, V^{(kl)}V^{(kl′)}ψ_{+})| ≤ c_{v} (ψ_{+}, p_{l}p_{l′}ψ_{+}) ≤ c_{v} (ψ_{+}, E_{p_{l}}E_{p_{l′}} ψ_{+}). The same estimate holds for V^{(kl)}V^{(k′l′)} with distinct indices. The symmetry of ψ_{+} with

respect to particle exchange and ∑_{k>l=1}^{N} 1 = \frac{N(N−1)}{2} then leads to the result

∥ ∑_{k>l=1}^{N} V^{(kl)}ψ_{+}∥^{2} ≤ c_{v} · max\{\frac{N−1}{2}, \frac{1}{2}[\frac{N(N−1)}{2} − 1]\} ∥T ψ_{+}∥^{2}.

The remaining contribution to (B.3) can partly be reduced to the estimate of V^{(kl)}. Let k, l, k′, l′ be distinct indices and set φ_{l} := Λ_{−}^{(l)}F_{0}^{(l)}ψ_{+}. Then we obtain

for the cross terms of ( ∑_{k>l=1}^{N} C_{a}^{(kl)})^{2},

|(C_{a}^{(kl)}ψ_{+}, C_{a}^{(k′l′)}ψ_{+})| = |(Λ_{−}^{(l)}F_{0}^{(l)}ψ_{+}, V^{(kl)}V^{(k′l′)}Λ_{−}^{(l′)}F_{0}^{(l′)}ψ_{+})|
= |(φ_{l}, V^{(kl)}V^{(k′l′)}φ_{l′})| ≤ c_{v} (φ_{l}, p_{k}p_{k′}φ_{l})^{\frac{1}{2}} (φ_{l′}, p_{k}p_{k′}φ_{l′})^{\frac{1}{2}} .  (B.7)

Since l ≠ k′, p_{k′} commutes with Λ_{−}^{(l)}F_{0}^{(l)} such that

(φ_{l}, p_{k}p_{k′}φ_{l}) = ∥(p_{k}p_{k′})^{\frac{1}{2}} φ_{l}∥^{2} = ∥Λ_{−}^{(l)}F_{0}^{(l)}(p_{k}p_{k′})^{\frac{1}{2}} φ_{+}∥^{2}
≤ ∥F_{0}^{(l)}∥^{2} (ψ_{+}, p_{k}p_{k′}ψ_{+}).  (B.8)

The cross terms of ( ∑_{k>l=1}^{N} C_{b}^{(kl)})^{2} have the same estimate. In fact,

|(C_{b}^{(kl)}ψ_{+}, C_{b}^{(k′l′)}ψ_{+})| = |(ψ_{+}, V^{(kl)}Λ_{−}^{(l)}F_{0}^{(l)} F_{0}^{(l′)}Λ_{−}^{(l′)}V^{(k′l′)}ψ_{+})|
= |(φ_{l′}, V^{(kl)}V^{(k′l′)}φ_{l})| ≤ c_{v} ∥F_{0}^{(l)}∥^{2} (ψ_{+}, p_{k}p_{k′}ψ_{+}).  (B.9)

If any two indices coincide, we use for simplicity a weaker estimate, e.g.

|(C_{b}^{(kl)}ψ_{+}, C_{b}^{(k′l)}ψ_{+})| ≤ ∥C_{b}^{(kl)}ψ_{+}∥∥C_{b}^{(k′l)}ψ_{+}∥ ≤ c̃_{s}∥E_{p_{k}}ψ_{+}∥^{2} ≤ c̃_{s} \frac{1}{N} ∥T ψ_{+}∥^{2}

(B.10)

and similarly for C_{a}^{(kl)}.

Counting terms in the sum ∑_{k>l=1}^{N} C_{a}^{(kl)} ∑_{k′>l′=1}^{N} C_{a}^{(k′l′)} we have \frac{N(N−1)}{2} square

terms, \frac{1}{4}N(N − 1)(N − 2)(N − 3) terms with four distinct indices and N(N − 1)(N − 2) terms where two of the four indices agree (while the other two are distinct). For all terms of the last type, the estimate (B.10) is used whereas

for the other terms we proceed as in the case of ( ∑_{k>l=1}^{N} V^{(kl)})^{2}. This leads to

∥ ∑_{k>l=1}^{N} C_{a}^{(kl)}ψ_{+}∥^{2} ≤ c̃_{s} · max\{\frac{N − 1}{2}, \frac{(N − 2)(N − 3)}{4}\} ∥T ψ_{+}∥^{2}

+ c̃_{s} (N − 1)(N − 2) ∥T ψ_{+}∥^{2}.  (B.11)

Inserting our results into (B.3) we find ∥W ψ_{+}∥ ≤ c_{0}∥T ψ_{+}∥ with

c_{0} := \sqrt{c_{w}} + \sqrt{c_{v} · max\{\frac{N − 1}{2}, \frac{1}{2} [\frac{N(N − 1)}{2} − 1]\}}  (B.12)

+ 4 \sqrt{c̃_{s}} \sqrt{max\{\frac{N − 1}{2}, \frac{(N − 2)(N − 3)}{4}\} + (N − 1)(N − 2)}.

For N = Z we get c_{0} < 1 for γ < 0.285 which corresponds to Z ≤ 39. For N = 2, we need γ < 0.66 (Z ≤ 90) which slightly improves on our earlier estimate (Z ≤ 89 [11]), obtained by using (B.10)-type estimates for all twoparticle interaction cross terms.

Acknowledgment

I would like to thank A. Sobolev, H. Siedentop and S. Morozov for stimulating discussions and critical comments. Partial support by the EU network Analysis and Quantum (contract HPRN-CT-2002-00277) is gratefully acknowledged.

References

[1] Abramowitz M. and Stegun I. A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. New York, Dover Publications, 1965

[2] Brown G. E. and Ravenhall D. G.: On the interaction of two electrons. Proc. Roy. Soc. London A208, 552-559 (1951)

[3] Brummelhuis R., Siedentop H. and Stockmeyer E.: The ground-state energy of relativistic one-electron atoms according to Jansen and Hess. Doc. Math. 7, 167-182 (2002)

[4] Burenkov V. I. and Evans W. D.: On the evaluation of the norm of an integral operator associated with the stability of one-electron atoms. Proc. Roy. Soc. (Edinburgh) 128A, 993-1005 (1998)

[5] Cycon H. L., Froese R. G., Kirsch W. and Simon B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Text and Monographs in Physics, 1st Edition. Berlin, Springer-Verlag, 1987

[6] Douglas M. and Kroll N. M.: Quantum electrodynamical corrections to the fine structure of helium. Ann. Phys. (N. Y.) 82, 89-155 (1974)

[7] Evans W. D., Perry P. and Siedentop H.: The spectrum of relativistic oneelectron atoms according to Bethe and Salpeter. Commun. Math. Phys. 178, 733-746 (1996)

[8] I. W. Herbst, Spectral theory of the operator (p^{2}+m^{2})^{\frac{1}{2}} −Ze^{2}/r. Commun. Math. Phys. 53, 285-294 (1977).

[9] Hoever G. and Siedentop S.: Stability of the Brown-Ravenhall operator. Math. Phys. Electron. J. 5(6), 1-11 (1999)

[10] Hunziker W.: On the spectra of Schrödinger multiparticle Hamiltonians. Helv. Phys. Acta 39, 451-462 (1966)

[11] Jakubaßa-Amundsen D. H.: Spectral theory of the atomic Dirac operator in the no-pair formalism. Ph. D. thesis, University of Munich (2004)

[12] Jakubaßa-Amundsen D. H.: The projected single-particle Dirac operator for Coulombic potentials. Doc. Math. 10, 331-356 (2005)

[13] Jakubaßa-Amundsen D. H.: Pseudorelativistic operator for a two-electron ion. Phys. Rev. A71, 032105, 1-8 (2005)

[14] Jakubaßa-Amundsen D. H.: The HVZ theorem for a pseudo-relativistic operator. Submitted to Ann. Henri Poincaré (2005)

[15] Kato T.: Perturbation Theory for Linear Operators. Berlin, Springer-Verlag, 1980

[16] Lewis R. T., Siedentop H. and Vugalter S.: The essential spectrum of relativistic multi-particle operators. Ann. Inst. Henri Poincaré 67, No. 1, 1-28 (1997)

[17] Lieb E. H. and Yau H.-T.: The stability and instability of relativistic matter. Commun. Math. Phys. 118, 177-213 (1988)

[18] Morozov S. and Vugalter S.: Stability of atoms in the Brown-Ravenhall model. arXiv:math-ph/0511033 (2005)

[19] Murray J. D.: Asymptotic Analysis. Oxford, Clarendon Press, 1974, §2, 5

[20] Reed M. and Simon B.: Analysis of Operators. Vol. IV of Methods of Modern Mathematical Physics. New York, Academic Press, 1978

[21] Siedentop H. and Stockmeyer E.: An analytic Douglas-Kroll-Hess method. Phys. Lett. A341 473-478 (2005)

[22] Simon B.: Geometric methods in multiparticle quantum systems. Commun. Math. Phys. 55, 259-274 (1977)

[23] Sucher J.: Foundations of the relativistic theory of many-electron atoms. Phys. Rev. A22, 348-362 (1980)

[24] Teschl G.: Lecture Notes on Schrödinger Operators. Section 9.4, 2004.

http://www.mat.univie.ac.at/˜ gerald/ftp/index.html

[25] Tix C.: Strict positivity of a relativistic Hamiltonian due to Brown and Ravenhall. Bull. London Math. Soc. 30, 283-290 (1998)

[26] van Winter C.: Theory of finite systems of particles I. The Green function. Mat. Fys. Dan. Vid. Selsk. 2 No. 8, 1-60 (1964)

[27] Wolf A., Reiher M. and Hess B. A.: The generalized Douglas-Kroll transformation. J. Chem. Phys. 117, 9215-9226 (2002)

[28] Zhislin G. M.: A study of the spectrum of the Schrödinger operator for a system of several particles. Trudy Moskov. Mat. Obsc. 9, 81-120 (1960)

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