Documenta Math. 417 Localization of the Essential Spectrum for Relativistic N-Electron Ions and Atoms

The HVZ theorem is proven for the pseudorelativistic N- electron Jansen-Hess operator (2 ≤ N ≤ Z) which acts on the spinor Hilbert space A(H1(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 .


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].

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, = c = 1) where D 0 .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 = 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 v (kl) . (1. 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 with T := N k=1 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 = 2 π/2+2/π , and let (1.4) be its two-cluster decompositions.Then the essential spectrum of h BR is given by (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 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 BR N −1 + T (j) , where the subscript on h BR denotes the number of electrons in the central field, and assume (1.6) to hold.Then By induction (corresponding to successive removal of an electron) we get (1.9) Since for a K-cluster decomposition of this specific form one has 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 1 R -bounded if O is bounded by 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), γBR }. 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]), 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 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 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 where x = |x| = x 2 1 + ... + x 2 N .Then we set χ n (x) := 1 − χ 0 (x/n) and claim that a subsequence of the sequence 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 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 , (1.14) where p = (p 1 , ..., p N ), and by using the mean value theorem to estimate the difference (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]) 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 Then, with some constant c, |(φ j ϕ, r j φ j ϕ)| ≤ c R ϕ 2 , j = 0, .., N. (1.16) There are two possibilities.r j may (a) consist of terms b 1m 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 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 With the representation (1.3) of v (kl) , we have to estimate commutators of the type 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 : 2) and (φ j ) j=0,...,N be the Ruelle-Simon partition of unity.Then for 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], Then φ j ϕ = φ j ϕχ on supp ϕ, and therefore (φ j ϕ, [b ) 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 (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 (using asymptotic analysis [19]) and subsequently proving the uniform

With Lemmata 3 and 4 we obtain the desired localization formula for
for R > 2. From Persson's theorem (1.11) and the definition (1.5) of Σ 0 we therefore get (b) We now turn to the 'easy part' of the proof where we have to verify 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 BR N −1 where h BR N −1 does not contain any interaction with electron j.The continuity of σ(T (j) + h BR N −1 ) 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 T + a 0 .This follows from the absence of any central potential in h 0 and from the symmetry of v (kl) , (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 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 with |a| = a and let a l := (a, ..., a) ∈ R 3l .Hence cluster C 2j moves to infinity as a → ∞.

Let ψ (a)
n := T a ϕ n and Aψ  → 0 for n → ∞ and a suitably large a.We have (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 such that [T a , v (kl) ] = 0.Then, given some ǫ > 0, the first term of (1.26) reduces to (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 Moreover, since a j does not contain any intercluster interactions, we can choose 2 as a product of functions (ϕ are normalized.As a consequence, for any k ∈ C 2j , the l.h.s. of (1.30) can be made smaller than ǫ/4l for sufficiently large a.For the proof of Lemma 5 or, equivalently, of ) 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]), 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 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.
So the inter-electron separation can be estimated by n .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 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.

Documenta Mathematica 10 (2005) 417-445
We are left to show that the defining sequence for λ can be chosen to be antisymmetric.We write Aψ n (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 (1.35)By (1.34) this can be made smaller than ǫ since the number #σ of permutations is finite.It remains to prove that Aψ n and ϕ (2) n 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 .., N − l}, can be made arbitrarily small for a suitably large a.In fact, since since all cross terms vanish for sufficiently large a.This guarantees the normalizability of Aψ (a) n .

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] where H BR 2 is the Brown-Ravenhall operator from (1.1) indexed by 2 (for 10,m is a bounded single-particle integral operator.The two-particle second-order contribution C (12) is given by 2) 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, (12)  ( where A k , G k are defined below (1.2) and h.c.means Hermitean conjugate (such that b 2m is a symmetric operator).Note that, due to the presence of the projector Λ 2m 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 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 h(2) by means of h (2) =: h(2) + c (12) and define in analogy to (1.4) the two-cluster decompositions of h(2) for j=0,1,2, h(2) = T + a j + r j (2.6) The aim of this section is to prove Theorem 2 (HVZ theorem for the two-electron Jansen-Hess operator). Let 2m ) +v (12) +c (12) = h(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 h(2) and Σ 0 from (2.6).Then the essential spectrum of h (2) is given by (2.7) We start by noting that the two-particle second-order potential c (12) does not change the essential spectrum of h (2) : 2) +c (12) be the two-electron Jansen-Hess operator with potential strength γ < 0.66.Then one has Proof.
The resolvent difference is bounded for µ ≥ 0 since H (2) as well as H(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 H(2) coincide. Let 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 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 H(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 x e −ǫx and by introducing convergence generating functions e −ǫp in momentum space, where ǫ := 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 −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 h(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 h(2) which is obtained from h BR by including the single-particle second-order potentials b (k) 2m , 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 h(2) throughout (and take N = 2).In order to prove [ h (2) 2) ) and χ 0 from (1.13) with x := (x 1 , x 2 ), we have to show in addition to the Brown-Ravenhall case, Due to the symmetry property of ψ n , the same bound holds also for [b bounded multiplication operators in momentum space.We pick for the sake of demonstration the second term of (2.4) and decompose We will show that the commutators (including the factor p 1 ) are 1 n -bounded and the adjacent factors bounded.The latter is trivial (since also 1 x1p1 is bounded, , see e.g.[8]) except for the operator p 1 V (1) 10,m 1 p1 in the last term.The boundedness of this operator is readily proved by invoking its kernel in momentum space.From (2.1) we have 1+p1 from the mean value theorem, where ξ is some point between p 1 and p ′ 1 .For the commutator with V (1) 10,m we have The proof of its 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 (2. 16) where 10,m , χ 0 ] and k O its kernel, one has to show that the integrals I and J obey 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 1 x1 to write for the first term in (2.15), From the mean value theorem we get We have to show that the integral over the modulus of the kernel of (2.18), with a suitable convergence generating function f , is 1 n -bounded.We choose f (p) = p and make the substitution The t-integral can be carried out, Define q 1 := p 1 − y 1 /n and consider (2.21) Estimating the last factor by 1 E ξ • 1 q and performing the angular integration, one obtains Insertion into (2.20)gives because the singularity at y 1 = np 1 is integrable and the integral is finite for all p 1 ≥ 0 due to χ0 ∈ S(R 6 ).Since the kernel k 1 is not symmetric in n -boundedness of J(q, p ′ 2 ) can be shown along the same lines, using ).We still have to estimate the second term in (2.15).Its kernel is With (2.19) the t-integral can be carried out as before.Making the substitution for q and p ′ 2 , respectively, one gets with the choice . (2.26) Even when the two singularities coincide (for y 1 = np 1 ), they are integrable.Since the integrand behaves like p Therefore, Ĩ is 1 n -bounded for all p 1 ≥ 0. It is easy to prove that also J(q, p ′ 2 ) := is 1 n -bounded, using the estimate 10,m , χ 0 ].With the same tools, the 1 n -boundedness of the commutator of χ 0 with the remaining contributions from (2.4) to b (1) 2m 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 h(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 h(2) (and N = 2).We consider the case j = 1 where 2m + v (12) , and we have to show in addition to the Brown-Ravenhall case that 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 ( x1 R ) from (1.17) which is unity on the support of φ 1 ϕ and decompose (2.29) Since supp χ 1 ⊂ R 3 \B CR/2 (0) we have (2.30) For the second contribution to (2.29), we have to estimate ), the uniform 1 Rboundedness of which has already been proven in the context of the Brown-Ravenhall case.The second operator, W 1 1 ), 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 Documenta Mathematica 10 (2005) 417-445 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 h(2) in place of h BR ) which is needed for the localization formula, has to be supplemented with the following estimate 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 We have to prove the 1 R -boundedness of the commutators (including the factor , φ j ] 1 x1 , we have to show that its kernel obeys the estimate 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) 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 1 R -boundedness of the commutator, we can substitute φ j with φ j χ where χ( 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 (1) 10,m , φ j χ] 1 x1 , we set ψ j := φ j χ and decompose The kernel of e −tEp 1 in coordinate space is given by [17] ǩe where K 2 is a modified Bessel function of the second kind and x := x 1 − x ′ 1 .Making use of the analyticity of K 2 (z) for z > 0 and its behaviour and therefore we can estimate ǩe −tEp 1 by the corresponding kernel for m = 0, (2.37) Thus we obtain for the kernel of the second contribution to (2.34), using (2.37) and (1.21), With the help of the estimate (2.39) According to the Lieb and Yau formula (2.16) in coordinate space, the 1 Rboundedness 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 (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) In the same way it is shown that J(y 1 ) := , x 1 ] has to be established.We use the estimate (2.33) to write 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 1 x1 V (1) 10,m x 1 we decompose In the second contribution the t-integral can be carried out, 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  .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 ( h(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 < ǫ for n and a sufficiently large, where r j now includes the terms b x1 B 1 (and its Hermitean conjugate), such that the idea of (2.29) can be used, 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 h(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 =: 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 H N 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 W0 of H N with respect to the multiparticle kinetic energy T 0 .We can prove (see Appendix A) 0 Λ +,N ) the N -electron Jansen-Hess operator without the secondorder two-electron interaction terms, acting on A(H 1 (R 3 ) ⊗ C 4 ) N .Then W0 is relatively form bounded with respect to the kinetic energy operator T 0 , 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 − T 0 holds only for a smaller critical γ.Using the estimate , we found γ < 0.454 (Z ≤ 62) for N = Z.The proof of Lemma 1 for the N -electron operator H(2) N 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 γBR }.For N ≤ Z, one has c 1 = γ γBR 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 , 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 ψ + , with cs := ( γ π ( 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 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 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 ≈ 1/137.04being the fine structure constant and x k = |x k | the distance of electron k from the origin.Further, Λ +,N = Λ p k ) onto the positive spectral subspace of D (k)

n
be the antisymmetric function constructed from ψ (a) n .We claim that Aψ sequence for λ ∈ σ(h BR ).It is sufficient (as shown below) to prove that ψ (a) n has this property.We have trivially ψ (a) n = ϕ n and we have to show that (h BR − λ) ψ (a) n (a) n is normalizable.Without restriction we can assume in the factorization ϕ n = ϕ (1)

1 x1)
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 [ σ (1) p1 Ep 1

x 1 1 , φ j χ] 1 x1 , the 1 R
in the same way as shown in the step from (2.40) to (2.41), one obtains Ĩ(x 1 ) := is shown.In the remaining contributions to [b (1) 2m , φ j χ] the terms not treated so far are [ m Ep -boundedness of which follows from the estimate | ǩ m Ep 1 Lemma 5 has therefore to be supplemented with the conjecture b 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 (k) 2m consists (for k = 1) of terms like W 1 1