The Ground State Energy of Relativistic One-Electron Atoms According to Jansen and Heß

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1 Documenta Math. 67 The Ground State Energy of Relativistic One-Electron Atoms According to Jansen and Heß Raymond Brummelhuis, Heinz Siedentop, and Edgardo Stockmeyer Received: August 6, Revised: June 5, Communicated by Alfred K. Louis Abstract. Jansen and Heß correcting an earlier paper of Douglas and Kroll have derived a (pseudo-relativistic energy expression which is very successful in describing heavy atoms. It is an approximate no-pair Hamiltonian in the Furry picture. We show that their energy in the one-particle Coulomb case, and thus the resulting self-adjoint Hamiltonian and its spectrum, is bounded from below for αz.6. Mathematics Subject Classification: 8Q, 8V45 Introduction The energy of relativistic electrons in the electric field of a nucleus of charge Ze is described by the Dirac Operator D γ = cα i + mc β γ x ( with γ = Ze and α,β the four Dirac matrices. The constant m is the mass of the electron, c is the velocity of light, and is the rationalized Planck constant which we both take equal to one by a suitable choice of units. This operator describes both electrons and positrons. In low energy processes as, e.g., in quantum chemistry, there occur, however, only electrons. Brown and Ravenhall [] proposed to project the positrons out and to use the electronic degrees of freedom only. They originally took the electrons and positrons given by the free Dirac operator D. Later it was observed that it might be suitable Documenta Mathematica 7 ( 67 8

2 68 R. Brummelhuis, H. Siedentop, and Edgardo Stockmeyer to define electrons directly by their external field (Furry picture. (See Sucher [7] for a review. This strategy, however, meets immediate difficulties, since the projection χ (, (D γ is much harder to find for positive γ than for γ =. To handle this problem Douglas and Kroll [4] used an approximate Foldy-Wouthuysen transform to decouple the positive and negative spectral subspaces of D γ. Their approximation is perturbative of second order in the coupling constant γ. Jansen and Heß [] correcting a sign mistake in [4] wrote down pseudo-relativistic one- and multi-particle operators to describe the energy which were successfully used to describe heavy relativistic atoms (see, e.g., []. This derivation yields the operator (see [], Equation (7 where HD ext = βe + E + [W,O], ( e(p := p + m, (3 E := A(V + RV RA, (4 O := βa[r,v ]A, (5 A(p := ( e(p + m, e(p (6 R(p := α p e(p + m, (7 W(p,p = β O(p,p e(p + e(p. (8 (Note that we write p for p. Here V is the external potential which in the case at hand is the Coulomb potential, and in configuration space it is multiplication by γ/ x. This operator which acts on four spinors is then sandwiched by the projection onto the first two components, namely ( +β/. The resulting upper left corner matrix operator J γ : C (R 3 C L (R 3 C is with J γ := B γ + γ K = e (γ/(π K + γ K. (9 K(p,p = (e(p + m(e(p + m + (p σ(p σ n(p p p n(p ( where n(p := (e(p(e(p + m /, i.e., B γ is the Brown-Ravenhall operator []. (See also Bethe and Salpeter[] and Evans et al. [5]. The last summand in (9 is given by the kernel K(p,p = dp [W(p,p P(p,p + P(p,p W(p,p ] ( Documenta Mathematica 7 ( 67 8

3 with and Relativistic One-Electron Atoms 69 P(p,p = σ p(e(p + m (e(p + mσ gp π n(p p p n(p ( W(p,p = P(p,p e(p + e(p. (3 Introducing b(p := p/n(p and a(p := ((e(p + m/e(p / we get more explicitly K(p,p ( = (π dp p p p p e(p + e(p + e(p + e(p [ (ωp σ (ω p σ b(pa(p b(p (ω p σ (ω p σb(pb(p a(p a(p + a(pb(p a(p (ω p σ (ω p σ a(pb(p a(p b(p ]. (4 (For later use we name the expression in the first line of the integrand in (4 C and the four terms in the square bracket T,...,T 4. The corresponding energy in a state u C (R 3 C is J (u := (u,j γ u = B(u + γ (u, Ku (5 with B(u = dp e(p u(p γ R π dp dp u(p K(p,p u(p (6 3 R 3 R 3 It is the quadratic form J which is our prime interest. Throughout the paper we will use the following constants γ c := 4π(π + 4 π4 + 4π 6/(π 4, γ B c := /(π/+/π, and d γ := γ 4 (3+ γ. Our goal is to show Theorem. For all nonnegative masses m the following holds:. If γ [,γ c ] then J is bounded from bellow, i.e., there exist a constant c R such that for all u C (R 3 C J (u cm u.. If γ > γ c, then J (u is unbounded from below. 3. If γ [,γ B c ] then J (u d γ m u. Note that γ c Because γ = αz where α is the Sommerfeld fine structure constant which has the physical value of about /37 and Z is the atomic number, this allows for the treatment of all known elements. Documenta Mathematica 7 ( 67 8

4 7 R. Brummelhuis, H. Siedentop, and Edgardo Stockmeyer It also means that the method is applicable for all αz where the Coulomb- Dirac operator can be defined in a natural way through form methods (Nenciu [5]. Note, in particular, that the energy is bounded from below, even if γ c > γ > although the perturbative derivation of the symmetric operator is questionable in this case. We would like to remark that the lower bound can most likely be improved for positive masses. In fact, we conjecture that the energy is positive for all sub-critical γ. However, this is outside the scope of this work. According to Friedrichs our theorem has the following immediate consequence: H ext D Corollary. The symmetric operator J γ has a unique self-adjoint extension whose form domain contains C (R 3 C for γ [,γ c ]. In fact for γ < γ c, since the potential turns out to be form bounded with relative bound less than one, the self-adjoint operator defined has form domain H / (R 3 C. The structure of the paper is as follow: in Section using spherical symmetry we decompose the operator in angular momentum channels. In Section 3 we prove the positivity of the massless operators. Since these operators are homogeneous under dilation an obvious tool to use is the Mellin transform, a method that previously has been used with success to obtain tight estimates on critical coupling constant (see, e.g., [3]. In Section 4 we find that the difference between the massless and the massive operator is bounded. Finally, some useful identities are given in the Appendix. Partial Wave Analysis of the Energy To obtain a sharp estimate for the potential energy we decompose the operator as direct sum on invariant subspaces. Because of the rotational symmetry of the problem one might suspect that the angular momenta are conserved quantities. Indeed, as a somewhat lengthy calculation shows, the total angular momentum J = (x p + σ commutes with Hext. In fact we can largely follow a strategy carried out by Hardekopf and Sucher [9] and Evans et al. [5] in somewhat simpler contexts. We begin by observing that those of the spherical spinors l+s+m (l+s Y l,m (ω l+s m (l+s Y s = l,m+ (ω Ω l,m,s (ω := (7 l+s m+ (l+s+ Y l,m (ω l+s+m+ (l+s+ Y s = l,m+ (ω with l =,,,... and m = l,...,l +, that do not vanish, form an orthonormal basis of L (S C. Here Y l,k are normalized spherical harmonics Documenta Mathematica 7 ( 67 8

5 Relativistic One-Electron Atoms 7 on the unit sphere S (see, e.g., [4], p. 4 with the convention that Y l,k =, if k > l. We denote the corresponding index set by I, i.e., I := {(l,m,s l N,m = l,...,l +,s = ±,Ω l,m,s }. Thus any u L (R 3 C can be written as u(p = p f l,m,s (pω l,m,s (ω p (8 where p = p, ω p = p/p, and (l,m,s I (l,m,s I f l,m,s (p dp = u(p dp. R 3 We now remind the reader that the expansion of the Coulomb potential in spherical harmonics is given by p p = π pp l l= m= l q l (p/p Y l,m (ω p Y l,m (ω p (9 where q l (x := Q l ((x + /x/; Q l are Legendre functions of the second kind, i.e., Q l (z = P l (t z t dt ( where the P l are Legendre polynomials. [See Stegun [6] for the notation and some properties of these special functions.] Inserting the expansion (8 and (9 into (5 yields J (u = J l,s (f l,m,s with and J l,s (f := e(p f(p dp γ π (l,m,s I + γ dp f(pk l,s (p,p f(pdpdp dp f(p k l,s (p,p f(p ( k l,s (p,p = (e(p + mq l ( p p (e(p + m + p q l+s ( p p p n(p n(p ( and k l,s (p,p = ( π dp e(p + e(p + e(p + e(p [ q l+s ( p p q l+s( p p b(pa(p b(p q l+s ( p p q l( p p b(pb(p a(p a(p +q l ( p p q l( p p a(pb(p a(p q l ( p ] p q l+s( p p a(pb(p a(p b(p. (3 Documenta Mathematica 7 ( 67 8

6 7 R. Brummelhuis, H. Siedentop, and Edgardo Stockmeyer The Legendre functions of the second kind appear here for exactly the same reasons as in the treatment of the Schrödinger equation for the hydrogen atom in momentum space (Flügge [6], Problem 77.] To obtain (, we also use that (ω p σω l,m,s (ω p = Ω l+s,m, s (ω p (see, e.g., Greiner [8], p. 7, (. The operators h l,s defined by the sesquilinear form ( via the equation (f,h l,s f = J l,s (f are reducing the operator H ext on the corresponding angular momentum subspaces. 3 The Massless Operators and Their Positivity To proceed, we will first consider the massless operators. The lower bound in the massive case will be a corollary of the positivity of the massless one. The energy in angular momentum channel (l,m,s in the massless case can be read of from (4 and is given by with B l,s (f = and J l,s (f := B l,s (f + γ dp p f(p dp γ π k l,s (p,p = 8π dp dp f(p k l,s (p,p f(p (4 dp f(p (q l ( pp + q l+s( pp f(p (5 ( dp p + p + p + p [ q l+s ( p p q l+s( p p q l+s( p p q l( p p +q l ( p p q l( p p q l( p p q l+s( p p Using the simplifications of Appendix A, Formulae (57 and (59 we get ]. (6 k l,s (p,p = dp ( 8π p q l ( p p q l( p p q l+s( p p q l( p p q l ( p p q l+s( p p + q l+s( p p q l+s( p p. (7 Since the operator in question is homogeneous of degree minus one we Mellin transform (see Appendix B the quadratic form ε l,s. If we write this form as a functional J # l,s of the Mellin transformed radial functions f#, we get J # ( l,s f # = B # ( l,s f # + ( γ dt f # (t + i/ F # (t (8 π Documenta Mathematica 7 ( 67 8

7 Relativistic One-Electron Atoms 73 where B # l,s is the Brown-Ravenhall energy in angular momentum channel (l,s in Mellin space, i.e., B # l,s (g := [ dt g (t + i/ γ ] (V l(t + V l+s (t (9 with V l (t = π q# l (t i/ = Γ ( l+ it Γ ( l+ it (3 (see Tix [9] [note also the factor /π which is different from Tix s original formula] and F # (t = ( π q # l (t i/ q# l+s (t i/. (3 Formulae (9, (3, and (3 are obtained from (4, (5, and (7 using the fact that the occurring integrals can be read as a Mellin convolution which is turned by the Mellin transform into a product (see Appendix B, Formulae (6 and (63. Note that V l is the Coulomb potential after Fourier transform, partial wave analysis, and Mellin transform. 3. Positivity of the Brown-Ravenhall Energy To warm up for the minimization of J # l,s we start with B# l,s we first note only. To this end Lemma. We have V l+ (t V l+ ( V l (. (3 Note, that this is similar to Lemma in [5]. Proof. First note that q q q... which follows from the integral representation in [], Chapter XV, Section 3, p This implies q # l+ (t i/ = π π which implies the lemma. q l+ (pp q l (p dp p, it dp p π q l+ (p dp p (33 Theorem. For all u C (R 3 C and m = we have B if and only if γ γ B c. Documenta Mathematica 7 ( 67 8

8 74 R. Brummelhuis, H. Siedentop, and Edgardo Stockmeyer Proof. Note that V l (t + V l+s (t V ( + V ( = π + π. (34 Thus B # l,s (g ( dt g (t + i/ γ ( π + π which implies that the energy is nonnegative if γ /(π/ + /π. (35 We remark that Theorem was proved by Evans et al. [5]. However, since g can be localized at t =, our method shows that Inequality (35 is sharp, i.e., the present proof shows also the sharpness of γ B c, a result of Hundertmark et al. [] obtained by different means. Since according to Tix [9] the difference of the massive and massless Brown-Ravenhall operators is bounded, Theorem shows also that the energy in the massive case is bounded from below under the same condition on γ as in the massless case. 3. The Jansen-Hess Energy We now wish to treat the full relativistic energy according to Jansen and Heß as given in (8 through (3. From these equations it is obvious that the energy is positive, if the coupling constant γ does not exceed γ B c, since the additional energy term is non-negative. However, as can be expected, the critical coupling constant is in fact bigger, i.e. we want to prove Theorem in the massless case. Lemma. For all u C (R 3 C, m =, and γ γ c we have (u, J u. Moreover, if γ > γ c, then J is not bounded from bellow. Proof. We write the energy density in Mellin space as given in Equations (8 through (3 as j l,s (t := γ (V l(t + V l+s (t + γ 8 (V l(t V l+s (t. (36 As in the case of the Brown-Ravenhall energy we want to show that j l,s attains its minimum for l = and t =. First we note, that j l,s (t = j l+s, s (t which means that we can restrict the following to s = /, i.e., to j l,/. Next we show that it is monotone decreasing in l. For γ 4/π we have γ V ( γ V l(t γ 4 V l(t γ 4 V l+(t + γ V l+(t γ 4 V l(t γ 4 V l+(t (37 Documenta Mathematica 7 ( 67 8

9 Relativistic One-Electron Atoms 75 where use successively (64, (3, Lemma 6 in Appendix C, and the positivity of the V l. Inequality (37 is after multiplication by γ((v l (t V l+ (t/ identical with the desired monotonicity inequality j l+,/ (t j l,/ (t. (38 For later purposes we note that functions j l,/ are symmetric about the origin. Next we will show that the energy density has its absolute minimum at the origin: to this end we simply show that the derivative of j,/ is nonnegative on the positive axis, if γ /(π/ + /π which is bigger than 4/π. Since we have V (t V (t and obviously we have Thus the derivative of the energy j,/ is (q (x q (x dx x = V ( V ( = π π + γ (V (t V (t (39 γ (V (t V (t. (4 j,/ (t = γ [ V (t V (t + γ (V (t V (t(v (t V (t] = γ {V (t[ + γ (V (t V (t] + V (t[ γ (V (t V (t]}, (4 since V and V are symmetrically decreasing about the origin (see Appendix C. Finally, the polynomial j,/ ( = γ ( π + ( + γ π π 8 π is nonnegative for γ γ c as defined in the hypothesis. Thus, we have j l,s (t j,/ (. 4 Lower Bound on the Energy According to Jansen and Heß To distinguish the massive and the massless expressions we will indicate in this section the dependence their on the mass m by a superscript m, if it seems appropriate. The goal of this section is to show Theorem for the massive case. We proceed by enunciating the following lemmata. Documenta Mathematica 7 ( 67 8

10 76 R. Brummelhuis, H. Siedentop, and Edgardo Stockmeyer Lemma 3 (Tix [8, ]. For all u C (R 3 C,m, and γ γ B c then B(u m( γ. Lemma 4 (Tix [9]. The expression B m (u B (u is bounded for u C (R 3 C. Lemma 5. For all m and for all u C (R 3 C we have where d := ( + 5/. K m (u K (u md u (4 We note that the first part of Theorem follows from Lemmata, 4, and 5. The third part is a consequence of Lemmata 3 and 5. Proof. First we remark that sup{ J m (u J (u u = } = m sup{ J (u J (u u = }. Then it is enough to start bounding (u, K u (u, K u : By the mean value theorem we have K (p,p K (p,p λ D(µ,p,p (43 for some µ (,λ where λ (, is a deformation parameter and D(µ,p,p is the derivative of K µ (p,p with respect to µ. Computing the derivative yields D(µ,p,p = dp F(µ,p,p,p (44 with F(µ,p,p,p := (π ( C λ (T T 4 + C (T T 4 (µ,p,p,p (45 λ where C and T,...,T 4 are defined right below (4. Note that a(p and b(p /, i.e., by the definition T,...,T 4 /. Furthermore we note that C λ = λ E(p p p p p ( (E(p + E(p E(p + (E(p + E(p E(p. (46 First we treat C λ (T +...+T 4. We get using the above estimates on T through T 4 and (46 C λ (T T 4 (µ,p,p,p ( p p p p p p + p + p + p (47 Documenta Mathematica 7 ( 67 8

11 Relativistic One-Electron Atoms 77 Next we treat C (T+...+T4 λ a λ (p. To this end we note = p 4E(p 3 E(p E(p + λ 4p (48 and Thus C (T T 4 (µ,p,p,p λ 3 3/ p p p p b λ (p = p E(p + λ 8E(p 5 p. (49 ( ( p + p + p + p p + p + p. (5 We now bound the integral operator K K by a multiplication operator: First pick α R. Then we have using the symmetry of F(µ,p,p,p in p and p for fixed p (u,( K K u = dp dp dp u(p F(µ,p,p,p u(p α dp dp u(p dp p F(µ,p,p,p (5 where we used the Schwarz inequality in the measure dpdp in the last step for fixed p. Now using the estimates (47 and (5 and collecting similar terms yields (u,( K K α u dp u(p dp dp p 5/ (π p p p p ( p + p + p + p p p ( 3 p p + 3 p (5 where we claim the last line to be bounded by 3( + 5/ π 4, i.e., (u,( K K u dp u(p ( + 5/. (53 To show the above bound we break the integral into three parts ( α ( p I := dp dp p p p p p p + p + p + p p, ( α ( p I := dp dp p p p p p p + p + p + p p, ( α ( p I := dp dp p p p p p p + p + p + p p. (54 Documenta Mathematica 7 ( 67 8

12 78 R. Brummelhuis, H. Siedentop, and Edgardo Stockmeyer We will also use the following integral (see [3], p.4 β β Γ( Γ( Υ(β := dp R e p = π pβ 3 Γ( β Γ( β, (55 where e is an (arbitrary unit vector in R 3 and β (,3. We observe that each of the integrals in (54 do not depend on the value of p (what becomes evident after substitution of p pp and p pp. So picking p = and doing p p p in each integral in (54 we find I = I = ( α { dp dp p p p u p u p + p + } + p Υ(α, ( α { } dp dp p p u p u p + p + p ( + p Υ(α + Υ(αΥ(α +, ( α I = dp dp p p u p u p Υ(α +, { } p ( + p + p p ( + p (56 We choose α = 3/ and using (55 we obtain the same bound for each integral, namely 3π 4. Equation (53 proves Lemma 5 and follows by using the latter bound in (5. A Some Useful Integral Identities Suppose f(x = f(/x and suppose f(x/(+x is integrable on (,. Then f(x + x dx = f(x x dx = To show (57 we split the first integral f(x dx (57 x dx x f(x x + x + dx x f(x x + x = = dx x f(x dx x f(x = dx x f(x dx x f(x dx f(x. (58 x where we used the invariance under inversion of f for the first and third equality. Next we wish to simplify the kernel j l,s. To this end we use again the abbrevi- Documenta Mathematica 7 ( 67 8

13 ation q l (x := Q l ( Relativistic One-Electron Atoms 79 ( x + x as in (6. We claim I(p,p := = dp p dp p ( q l ( p p q m( p p + q m( p p q l( p ( q l ( p p q m( p p + q m( p p q l( p p ( p p p + p + p p + p (59 To prove this we take the integral with the complete first factor times the first summand of the second factor we name I and the integral over the complete first factor times the second summand of the second factor, I. In I we substitute p pp whereas in I we substitute p p p. This yields using (57 I(p,p = I + I = dp p + q m (p q l ( p p p + q l ( p p [ ( p q l (p p q m p p q m (p +q m (p Undoing the substitutions yields the desired result. p p ] q l (p. (6 B The Mellin Transform The Mellin transform is a unitary map from L (R + to L (R given by the formula f # (s := f(pp is dp. π The Mellin convolution of two function f and g is defined as (f g(p = f ( p g(q dq q q. (6 If f C (R +, then f # extends to an entire function, and we have (p α f # (s = f # (s + iα. (6 We also have (f g # (s = πf # (sg # (s. (63 Both, (6 and (63, can be verified by direct computation. Documenta Mathematica 7 ( 67 8

14 8 R. Brummelhuis, H. Siedentop, and Edgardo Stockmeyer C Some Properties Related to the Partial Wave Analysis of the Coulomb potential in Mellin Space We first remark the follow property on the difference of V l and V l+. Lemma 6. For l =,,,... and t R we have V l+ (t < V l (t. Proof. From the definition of V l in (3 we see that the claim is equivalent to Γ ( l+ it Γ ( Γ ( l+3 it l+ it > Γ ( l+4 it. This, however, can be easily verified using the functional equation Γ(x + = xγ(x of the Gamma function in the numerator and denominator of the right hand side with x = (l + it/ and x = (l + it/. From the definition of the V l and from Formulae and in [7] one finds V and V in terms of the hyperbolic tangent and cotangent: V (t = Tg(πt/ (64 t t V (t = Ctg(πt/. (65 + t Moreover, both of these functions are decreasing symmetricly about the origin. Acknowledgment: We thank Dr. Doris Jakubaßa-Amundsen for careful reading of the manuscript and pointing out several mistakes. The work has been partially supported by the European Union through its Training, Research, and Mobility program, grant FMRX-CT 96-96, the Volkswagen Foundation through a cooperation grant, and the Deutsche Forschungsgemeinschaft (Schwerpunktprogramm 464 Theorie relativistischer Effekte in der Chemie und Physik schwerer Elemente. E. S. acknowledges partial support of the project by FONDECYT (Chile, project 4, CONICYT (Chile, and Fundación Andes for support through a doctoral fellowship. References [] Hans A. Bethe and Edwin E. Salpeter. Quantum mechanics of one- and two-electron atoms. In S. Flügge, editor, Handbuch der Physik, XXXV, pages Springer, Berlin, edition, 957. [] G. E. Brown and D. G. Ravenhall. On the interaction of two electrons. Proc. Roy. Soc. London Ser. A., 8:55 559, 95. [3] Raymond Brummelhuis, Norbert Röhrl, and Heinz Siedentop. Stability of the relativistic electron-positron field of atoms in Hartree-Fock approximation: Heavy elements. Doc. Math., J. DMV, 6: 8,. Documenta Mathematica 7 ( 67 8

15 Relativistic One-Electron Atoms 8 [4] Marvin Douglas and Norman M. Kroll. Quantum electrodynamical corrections to the fine structure of helium. Annals of Physics, 8:89 55, 974. [5] William Desmond Evans, Peter Perry, and Heinz Siedentop. The spectrum of relativistic one-electron atoms according to Bethe and Salpeter. Commun. Math. Phys., 78(3: , July 996. [6] Siegfried Flügge. Practical Quantum Mechanics I, volume 77 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, edition, 98. [7] I.S. Gradshteyn and I.M. Ryzhik. Table of Integrals, Series, and Products. Academic Press, 4th edition, 98. [8] Walter Greiner. Relativistic Quantum Mechanics, volume 3 of Theoretical Physics Text and Excercise Books. Springer, Berlin, edition, 99. [9] G. Hardekopf and J. Sucher. Relativistic wave equations in momentum space. Phys. Rev. A, 3(:73 7, August 984. [] Dirk Hundertmark, Norbert Röhrl, and Heinz Siedentop. The sharp bound on the stability of the relativistic electron-positron field in Hartree-Fock approximation. Commun. Math. Phys., (3:69 64, May. [] Georg Jansen and Bernd A. Heß. Revision of the Douglas-Kroll transformation. Physical Review A, 39(:66 67, June 989. [] V. Kellö, A. J. Sadlej, and B. A. Hess. Relativistic effects on electric properties of many-electron systems in spin-averaged Douglas-Kroll and Pauli approximations. Journal of Chemical Physics, 5(5:995 3, August 996. [3] Elliott H. Lieb and Michael Loss. Analysis. Number 4 in Graduate Studies in Mathematics. American Mathematical Society, Providence, edition, 996. [4] Albert Messiah. Mécanique Quantique, volume. Dunod, Paris, edition, 969. [5] G. Nenciu. Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms. Commun. Math. Phys., 48:35 47, 976. [6] Irene A. Stegun. Legendre functions. In Milton Abramowitz and Irene A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, chapter 8, pages Dover Publications, New York, 965. Documenta Mathematica 7 ( 67 8

16 8 R. Brummelhuis, H. Siedentop, and Edgardo Stockmeyer [7] J. Sucher. Foundations of the relativistic theory of many-electron atoms. Phys. Rev. A, (:348 36, August 98. [8] C. Tix. Lower bound for the ground state energy of the no-pair Hamiltonian. Phys. Lett. B, 45(3-4:93 96, 997. [9] C. Tix. Self-adjointness and spectral properties of a pseudo-relativistic Hamiltonian due to Brown and Ravenhall. Preprint, mp-arc: 97-44, 997. [] C. Tix. Strict positivity of a relativistic Hamiltonian due to Brown and Ravenhall. Bull. London Math. Soc., 3(3:83 9, 998. [] E. T. Whittaker and G. N. Watson. A Course of Modern Analysis; An Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions. Cambridge University Press, Cambridge, 4 edition, 97. Raymond Brummelhuis Birkbeck College University of London School of Economics, Mathematics and Statistics Gresse Street London WT LL United Kingdom r.brummelhuis@statistics.bbk.ac.uk Heinz Siedentop Mathematik Theresienstr München Germany h.s@lmu.de Edgardo Stockmeyer Pontificia Universidad Católica de Chile Departamento de Física Casilla 36 Santiago Chile estockme@maxwell.fis.puc.cl Documenta Mathematica 7 ( 67 8