A MINIMAX PRINCIPLE FOR THE EIGENVALUES IN SPECTRAL GAPS

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1 A MINIMAX PRINCIPLE FOR THE EIGENVALUES IN SPECTRAL GAPS MARCEL GRIESEMER AND HEINZ SIEDENTOP ABSTRACT A minimax principle is derived for the eigenvalues in the spectral gap of a possibly non-semibounded selfadjoint operator. It allows the nth eigenvalue of the Dirac operator with Coulomb potential from below to be bound by the nth eigenvalue of a semibounded Hamiltonian which is of interest in the context of stability of matter. As a second application it is shown that the Dirac operator with suitable non-positive potential has at least as many discrete eigenvalues as the Schro dinger operator with the same potential. 1. Introduction The minimax principle provides a variational characterization of all eigenvalues below (or above the essential spectrum of a self-adjoint operator that is bounded below (above (see, for example, Courant and Hilbert [2, Chapter VI, 1.4] or Reed and Simon [14, Chapter XIII.1]. It allows one to estimate, say, the nth eigenvalue without a priori knowledge on the spectrum or eigenfunctions, which might be the main reason why it is one of the most used and most powerful tools in the investigation of the spectrum. Clearly it would be desirable to have a similar variational characterization of eigenvalues in gaps of the essential spectrum and for operators which are not semibounded. There are important systems to which such a characterization would be applicable. We mention periodic Schro dinger operators with localized perturbation (they occur in the description of crystals with impurities and Dirac operators. In this paper we are mainly concerned with Dirac operators. Minimax principles for the lowest eigenvalue of the Dirac operator are in fact discussed in the quantum chemistry literature (see, for example, Rosenberg and Spruch [15], Drake and Goldman [5], Kutzelnigg [9], Talman [18] and Datta and Deviah [3]. According to Talman, and Datta and Deviah, min g 9 max (ψ, Hψ f (ψ, ψ : (1 should be the first positive eigenvalue of the Dirac operator H. (As a matter of fact Datta and Deviah not only do not refer to Talman, but also those articles that refer to these two papers according to the Science Citation Index of January 1997 always ignore the corresponding other author(s. Here ψ is a Dirac spinor whose upper and lower two components are denoted by g and f respectively. The arguments given for (1 are not stringent but they were still convincing enough to be the motivation for the present work on such minimax principles. Received 4 December Mathematics Subject Classification 34L15, 34L40. J. London Math. Soc. (2 60 (

2 A MINIMAX PRINCIPLE FOR THE EIGENVALUES IN SPECTRAL GAPS 491 Our main result is a minimax principle for discrete eigenvalues in a gap of the essential spectrum of a possibly non-semibounded self-adjoint operator. It generalizes the standard minimax principle and allows us to prove that (1, with the correction g 0, is indeed the lowest discrete eigenvalue of the Dirac operator for suitable potentials. As a second application of this new minimax principle we show that the nth eigenvalue of the Dirac operator with Coulomb potential is an upper bound for the nth eigenvalue of an operator due to Brown and Ravenhall [1]. Hamiltonians of this kind (also called no-pair Hamiltonians are of interest for the description of heavy atoms and molecules (Sucher [17] and are used in quantum chemistry (Ishikawa and Koc [7]. They have also been discussed in the context of stability of relativistic matter (Lieb et al.[12, 13]. Thirdly, we show that the Dirac operator with suitable non-positive potential has at least as many discrete eigenvalues as the Schro dinger operator, more precisely the Pauli operator, with the same potential. This generalizes to operators including a magnetic field. All these results are new to our knowledge. Moreover, it seems hard to prove them without the minimax principle. To conclude we would like to mention that a minimax principle for the eigenvalues of Dirac operators has also been announced by Dolbeault et al. [4]. This work is organized as follows. In Section 2 we formulate and prove the new minimax principle in an abstract form. Section 3 contains two applications of the main result, the second being the comparison of the discrete eigenvalues of Dirac and Brown Ravenhall operators. In Section 4 we re-cover the minimax principle of Talman and Datta and Deviah (Subsection 4.1 and finally prove the abovementioned lower bound on the number of eigenvalues of the Dirac operator (Subsection The minimax principle In this section we state and prove our main result, Theorem 3, a minimax principle for discrete eigenvalues of a possibly non-semibounded self-adjoint operator A. Although bounded operators are covered by this result, we state and prove it for bounded A separately (Theorem 1, since in this case the statement is less technical and the proof is short and simple. We begin with a few notations. Suppose that A is a self-adjoint operator in a Hilbert space. Then (A denotes the domain and (A the form-domain of A. The operator P (a,b (A is the spectral projection of A for the interval (a, b. If A is bounded from below then (A inf sup φ, Aφ M (A φ M dim(m=n φ = is the nth eigenvalue (counted from below and counting multiplicity of A or, if A has less then n eigenvalues below the essential spectrum, the bottom of the essential spectrum [14]. As usual σ(a denotes the spectrum of A and ρ(a σ(a. Finally B( stands for the set of bounded linear operators on the Hilbert space. THEOREM 1. and Let A A* B( and + where +. Let P (, (A (A inf + dim( =n sup φ M = (i If φ, Aφ 0 for all φ, then (A φ, Aφ, n dim +.

3 492 MARCEL GRIESEMER AND HEINZ SIEDENTOP (ii If φ, Aφ 0 for all non-zero φ +, then (A REMARK 2. (i For + one can drop in the definition of (A, that is, the theorem is trivial in this case. (ii The theorem says that need only be approximate spectral subspaces in the above sense in order that (A Proof of Theorem 1. Set (A and, and let and Λ be the orthogonal projection onto + and respectively. (i First note that : + is one-to-one. If not, there would exist a φ with φ 0 so that 0 φ, Aφ φ, A φ 0. Now pick ε 0 and let M P (, + ε (A. Then dim(m n and hence dim( M n. Therefore sup φ, Aφ sup φ, Aφ, φ M φ φ M+ φ = = where M M was used. To estimate this from above we first decompose φ M as φ φ φ, where φ M and φ M (M, and then φ as φ φ φ where φ M and φ. Since Aφ M and φ φ M we have Aφ, φ Aφ, φ. Using this, Aφ, φ 0, and φ, Aφ 0 we find φ, Aφ φ, Aφ φ, Aφ φ, Aφ φ, Aφ φ, Aφ φ, Aφ (µ n ε φ, φ. (ii Now is dense in + because otherwise there exists a φ + ( + P with φ 0 and thus 0 φ, Aφ P φ, AP φ 0. Since A is bounded it follows that (A inf sup φ dim( =n φ = φ, Aφ inf sup φ, Aφ M φ M dim(m=n φ = To prove the inequality use each is of the form M for some M with dim(m n, and M M. Notice that in the proof of Theorem 1(ii the assumption is only used to show that is dense in +, while in the proof of Theorem 1(i the assumption is used in the estimate. This is the reason for the asymmetry in the set of states on which these assumptions are imposed in the following unbounded case. THEOREM 3. Suppose that A is a self-adjoint operator in a Hilbert space + where +. Let Λ be the orthogonal projections onto and let be a subspace with (A (A and Λ (A. Let P (, (A, P P (,] (A,, and (A inf sup φ, Aφ. + φ dim( =n φ = (i If φ, Aφ 0 for all φ, then (A

4 A MINIMAX PRINCIPLE FOR THE EIGENVALUES IN SPECTRAL GAPS 493 (ii If φ, Aφ 0 for all φ (A + and (A1/ P B(, then (A REMARK 4. Notice that (A1/ P is bounded, if A is bounded from below. Proof of Theorem 3. (i The proof of (i is essentially the same as that for bounded A. (ii Pick a, set f(x minx, a, let A f(a, and let us assume that inf sup φ, A φ inf sup φ, A φ, (2 + (A dim( =n dim( =n φ φ = φ φ = where + on the left-hand side is replaced by (A on the right-hand side. Since A A the left-hand side is bounded from above by (A while the right-hand side is bounded from below by (A The latter is proved in the same way as in the bounded case. Since A and A have the same spectrum in (, a it follows that (A (A (3 To prove (2 we first show that (A is dense in +. If this were wrong, then there would be a + ( P with 0, so that (A1 / (A1/ P (A and we would arrive at the contradiction 0, A P, AP 0. Now, pick +, dim( n, and ε 0. Since (A is dense in +,we can find a subspace M + ε (A with dim(mε + n such that for each φε + Mε +, φ ε + 1, there is a with φ ε + ε and φε. (4 + (To prove the existence of M + ε approximate the vectors of a given basis of with vectors of (A. Now, equation (2 follows if sup φ φ = φ, A φ lim inf ε sup φ ε M+ ε φ ε = φ ε, A φ ε. (5 Since, by (3, the supremum on the right-hand side is not smaller than which is non-negative, we can restrict it to vectors φ ε with φ ε, A φ ε 1. To prove (5 it therefore suffices, if for any given φ ε M ε + with φε 1 and φ ε, A φ ε 1 there exists a φ (which will depend on φ ε with φ 1 such that φ ε, A φ ε φ, A φ0 (ε 0 (6 uniformly in φ ε. Pick such a φ ε and let φ ε Λ φ ε. We define φ φ ε where obeys (4, so that φ φ ε, and we define a semi-norm on (A by ψ ψ,(aa ψ with a as in the definition of A. Then φε, A φ ε φ, A φ φε φ φ ε φ (2φ ε φ ε φ Here φ ε a1 because φ ε, A φ ε 1, and φ ε φ φ ε + (φ ε +, AP (φε + aφε + constφ ε + φ const ε, + because (A1/ P is a bounded operator. This proves (6 and thus the theorem.

5 494 MARCEL GRIESEMER AND HEINZ SIEDENTOP 3. Applications In this section we give two applications of Theorem 3. The first exhibits its perturbative character and is of abstract nature. It assumes that A A A where A has a gap (a, b in the spectrum while A may have discrete eigenvalues in this gap. These eigenvalues are provided by the minimax principle for the decomposition P (a, (A P (,a] (A. The operator A could be, for example, a periodic Schro dinger operator or the free Dirac operator. As a second application we show that the nth eigenvalue of the Coulomb Dirac operator is an upper bound for the nth eigenvalue of the so-called Brown Ravenhall operator. (See the introduction The case of relatiely compact perturbations of self-adjoint operators with spectral gap THEOREM 5. Let A A A be the sum of the self-adjoint operators A and A B( in the Hilbert space. Suppose there are real numbers a b such that (i σ(a (a, b ; (ii 0 A ba; (iii A (A i is compact; and let P (a, (A and Λ 1. Then (AP (a, (A inf (A dim( =n sup φ Λ (A φ = φ, Aφ. Proof. We apply Theorem 3 to Aa, Λ and (A (A. Then Λ (A so that the right-hand side is exactly λ n (A. By definition of Λ and by hypothesis (ii we have φ,(aa φ 0 for φ Λ (A and φ,(aa φ 0 for φ (A. Thus (AP (a, (A λ (A a n by Theorem 3 and the definition of (A. If eigenvalues of A accumulate from above at a then (AP (a, (A a for all n and the theorem is proved. It remains to show that (A1/ P is bounded in the case where (a, aε contains no eigenvalues for some ε 0. From condition (iii on A it follows that σ ess (A σ ess (A (a, b and hence (a, aε (ρ(aρ(a. In particular the path γ: with γ(t aε2it is contained in ρ(aρ(a. Therefore 1 2 2πi& 1 dz(za γ and similarly for and A (Kato [8, p. 359] which leads us to (A1/ ( 2πi& 1 dz(a1/ (za (A (za. γ Since A is bounded and (A1/ (za A (za const(1t / the integral converges absolutely which implies that Ran( (A and that (A1/ ( is bounded. Hence (A1/ P P (A1/ ( is bounded as well.

6 A MINIMAX PRINCIPLE FOR THE EIGENVALUES IN SPECTRAL GAPS Relation between the spectra of Dirac and no-pair Hamiltonians Let L(; and let α, α, α and β denote the usual Dirac matrices α i 0 0 σ i σ 01 i 0, β , σ i being the Pauli matrices. Let H iαβm with (H H(, and let H denote the closure of the operator H V where 0 V(x κ x, κ 3 2. Then H is self-adjoint [16] and (H (H (Landgren and Rejto [10] and Landgren et al. [11]. Let P (, (H, and consider the quadratic form φ φ, Hφ, φ (H. It is bounded from below (in fact even positive [20] and closable as recently shown by Evans et al.[6]. The closure defines a unique self-adjoint operator which we denote by B. THEOREM 6. With the aboe notations we hae for all n (B (HP ( m, (H Proof. Apply Theorem 3 to A Hm, Λ and (H (H. Then Λ (H, and for all φ Λ (H we have φ,(hm φ φ,(h m φ 0. Hence µ n (HP ( m, (H λ (H by Theorem 3(i. On the other n hand we get by dropping Λ (H in the definition of λ n (H (H inf sup φ, Hφ (B. (H φ dim( =n φ = 4. Perturbed supersymmetric Dirac operators In this section we derive the minimax principle for the Dirac operator and the decomposition of a Dirac spinor into upper and lower (large and small components. (Note that Theorem 3 cannot be applied to this case, because the boundedness assumption is not satisfied. In particular we prove formula (1 of Talman [18], and Datta and Deviah [3], for certain bounded potentials. Moreover, we show that the Dirac operator with suitable non-positive potential has at least as many discrete eigenvalues as the Schro dinger operator with the same potential. These results generalize to systems including a magnetic field The minimax principle of Talman and Datta and Deiah To compensate for the boundedness of (A1/ P in the proof of our minimax principle it suffices that the considered operator has the form QβmV where Qβm is a so-called Dirac operator with supersymmetry and V is a bounded

7 496 MARCEL GRIESEMER AND HEINZ SIEDENTOP perturbation. It is therefore natural to work with the class of such generalized Dirac operators, which in particular includes Dirac operators with magnetic and (bounded electric potential. DEFINITION 7. Let Q be a self-adjoint operator in a Hilbert space and let m +. Furthermore let β β* B( with σ(β 1. Ifβ(Q (Q and βqqβ 0 on (Q then Qβm is called a Dirac operator with supersymmetry. REMARK 8. A consequence of the anticommutativity βqqβ 0 is that (Qβm Qm m which means that the spectrum of Qβm has the gap (m, m. A typical example of a Dirac operator with supersymmetry is the Dirac operator α(i Aβm with magnetic field A(x. For many other examples we refer to Thaller [19]. In the following we will use the notations β 12(1β, β and V β Vβ if V B( THEOREM 9. Let H Qβm be a Dirac operator with supersymmetry in the Hilbert space and let V B( with V* V. Let H H V, P + P (, (H and (H inf β + (H dim( =n sup φ β (H φ = φ, Hφ. (i If V m, then (H (H (ii If V ++ m and (Va H is compact for some a with m a m, then (H (H The theorem applies in particular to the usual Dirac operator where Q iα and V is multiplication with a real-valued function. COROLLARY 10. Let H iαβmv where V L ( Suppose that V(x 0 as x and 2m V(x 0 almost eerywhere. Then for all n (HP ( m, (H inf sup φ, Hφ. (7 β + (H φ β (H dim( =n φ = REMARK 11. Specializing our assertion to n 1 proves that (1 with the correction g 0 is indeed the lowest discrete eigenvalue of H. Note, however, that the requirement g 0 cannot be dropped. Already in the case of a diagonal matrix with eigenvalues 1 and 1 one can easily prove the corresponding assertion without this additional condition false. Proof of Theorem 9. Assertion (i follows from Theorem 3(i for (A, Λ (H, β because φ, Hφ φ,(mv φ 0 for all φ. (ii Since (Va H is compact, σ ess (H(ma, ma, and in particular σ ess (H(0, ε for some ε 0 by assumption on a. If the eigenvalues of H accumulate from above at 0, then the assertion is trivial, because then (H

8 A MINIMAX PRINCIPLE FOR THE EIGENVALUES IN SPECTRAL GAPS for all n, while (H 0 as a consequence of φ, Hφ φ,(mv + φ 0on β + (Q. We may therefore assume that (0, ε ρ(h for some ε 0(ε m. It now suffices to show that (H inf sup φ, Hφ (8 M β + + (H φ M β + (H dim( =n φ = because (ii then follows in the same way as Theorem 3(ii followed from (2. To prove (8 we show that β + (H is dense in β + (H with respect to the form-norm of H. We do this in three steps, step (1 and step (2 being prerequisites for step (3. Step 1. Let P + P (, (H. Then H / ( P is a bounded operator. + For the proof of this assertion see the proof of Theorem 5 and recall that (0, ε ρ(h. Step 2. Let P 1. Then P + (H. Let P 1P + and pick φ P +. Then φ P φ (P P φp φ and β φ 0. Therefore β P φ β ( P + φ. Since β commutes with H / the right-hand side belongs to (H by step (1. Hence (H β P φ β P β + φ. Using P 12(1H H and Qβ + (Q we get QH φ (H which implies that QH / φmh / φ φ,(qm H φ φ, H φ. This proves step (2. Step 3. The set β + (Q β + (Q is dense with respect to the norm φ H / φ. To see this claim, note that it is equivalent to β + H / P + (Q + because H / commutes with β +. Let us assume that this is wrong. Then there exists a + with 0 and, H / P + φ 0 φ (Q. (9 Writing now P + P + and φ H / ψ with ψ (H one finds H / B*φ + P + 0 (10 where B H / (P + P + B( by step 1. Thus P + (H and hence (H 2β + P + β + (1H H φ + (1mH φ +, that is, (H. Now (9 and step 2 mean that ψ + H / φ + P + (H, which leads to the contradiction 0 ψ +,(mv ψ + ψ +, Hψ + P ψ +, HP ψ + 0. This proves the theorem Comparing discrete eigenalues of Dirac and Schro dinger operators As an application of Theorem 9 we now compare the number of eigenvalues in (m, m of the Dirac operator with the number of negative eigenvalues of the Schro dinger operator with the same potential.

9 498 MARCEL GRIESEMER AND HEINZ SIEDENTOP THEOREM 12. Let H Qβm be a Dirac operator with supersymmetry in the Hilbert space and suppose that V V* B( commutes with β and VH is compact. Let H H V. If 0 V and V ++ 2m, then dim P ( m,m (H dim P (, (Q(2mV +. REMARK 13. By Weyl s theorem the spectrum of H V in (m, m is discrete because σ(h (m, m and VH is compact. Similarly the spectrum of Q2mV in (, 0 is discrete because σ(q2m [0, and V(Qm VH H is compact. Let us again specialize to the case where Q iα and V is multiplication with a real-valued function. COROLLARY 14. Suppose that V L (, V(x 0 as x and 2m V(x 0 almost eerywhere. Then, counting multiplicity, the operator iαβmv has at least as many discrete eigenalues as (2mV onl(r, This can be understood as an expression of the fact that the non-relativistic kinetic energy of an electron is at least as large as the relativistic one: p(2mm pm. The proof of Theorem 12 is based on the minimax principle Theorem 9 and the following lemma, where D Qβ + (Q. Note that Q has the form Q 0 0 D* D 0 1 (11 with respect to the decomposition + (Thaller [19]. LEMMA 15. Assume the hypotheses of Theorem 12. Define φ, Hφ E( sup φ β φ, φ (Q φ = + φ for β + (Q0. IfE( 0 then E(,[D*(mE( V DmV ++ ],. Proof. By the special form of Q given in (11 and because V B(, the map φ φ, Hφ is continuous. Therefore φ, Hφ E( sup φ φ, φ. (12 φ = + φ If there exists a maximizer φ of (12 then it obeys the corresponding Euler Lagrange equation φ (me( V D. Substitute this for φ on the right-hand side of E(, φ, HφE( φ, φ to get the claimed equation. It remains to prove the existence of a maximizer. Choose a sequence φn φ φn + for which lim n φn, Hφn φn, φn E(.

10 A MINIMAX PRINCIPLE FOR THE EIGENVALUES IN SPECTRAL GAPS 499 Since E( 0 also φn, Hφn 0 for large n and 0 φn, Hφn,(mV ++ 2Re D, φnφn,(mv φn const2dφ φnmφn +, φn. This implies that sup φn n. Hence there exists a φ and a weakly convergent subsequence, call it φn again, with φn φ as n. Since mv 0 the map φ φ, Hφ is weakly upper semicontinuous. The map φ φ, φ is weakly lower semicontinuous. Thus φn, Hφn 1 E( lim lim sup φn, Hφn lim sup n φn, φn n n φn, φn φ*, Hφ* φ*, φ* E( where φ* φ. Therefore φ is a maximizer. Proof of Theorem 12. Since P (, ((12m QV + P ((12m D*D (, V ++ +, it suffices to show that (HP ( m, n0 (H m µ 1 2m ++1 D*DV 0. (13 By Theorem 9(ii applied to Vm and a m (H (HP ( m, (H m. (14 Here we can add the condition β + φ 0 in the definition of (H, because φ,hφ 0, if β + φ 0. It then follows from (14 and Lemma 15 that for every subspace β + (Q with dim( n m sup φ β (H φ = φ, Hφ sup = E( sup =,[D*(mE( V DmV ++ ],. (15 Obviously the second and third supremum may be restricted to with E( m(12ε and ε 0 small. Then (me( V (2m(1ε and thus (15 implies that m(1ε ++1 D*DV 0 ε 1. To see that this proves (13 multiply both sides by (1ε and use that V ++ is bounded. Acknowledgements. The authors thank Dirk Hundertmark and Christian Tix for useful discussions. They also acknowledge the support of the European Union through its Training, Research, and Mobility program, grant FMRX-CT The first author also thanks the Swiss national science foundation for support. Note added in proof. Theorem 3 has been strengthened to make it applicable to Dirac operators with Coulomb potentials (see Griesemer, M., Lewis and Siedentop, H., Doc. Math. J. DMV 4 (

11 500 A MINIMAX PRINCIPLE FOR THE EIGENVALUES IN SPECTRAL GAPS References 1. G. E. BROWN and D. G. RAVENHALL, On the interaction of two electrons, Proc. Roy. Soc. London Ser. A 208 (A 1095 ( R. COURANT and D. HILBERT, Methoden der Mathematischen Physik I, 3rd edn, Heidelberger Taschenbu cher 30 (Springer, Berlin, S. N. DATTA and G. DEVIAH, The minimax technique in relativistic Hartree Fock calculations, Pramana 30 ( J. DOLBEAULT, M.J.ESTEBAN and E. SE RE, International Conference on Differential Equations and Mathematical Physics, Atlanta, Georgia, March G. W. F. DRAKE and S. P. GOLDMAN, Relativistic Sturmian and finite basis set methods in atomic physics, Ad. Atomic Molecular Phys. 25 ( W. D. EVANS,P. PERRY and H. SIEDENTOP, The spectrum of relativistic one-electron atoms according to Bethe and Salpeter, Comm. Math. Phys. 178 ( Y. ISHIKAWA and K. KOC, Relativistic many-body perturbation theory based on the no-pair Dirac Coulomb Breit Hamiltonian: relativistic correlation energies for the noble-gas sequence through Rn (Z 86, the group-iib atoms through Hg, and the ions of Ne isoelectronic sequence, Phys. Re. A 50 ( T. KATO, Perturbation theory for linear operators, Grundlehren der Mathematischen Wissenschaften 132 (Springer, Berlin, W. KUTZELNIGG, Relativistic one-electron Hamiltonians for electrons only and the variational treatment of the Dirac equation, Chemical Physics 225 ( J. J. LANDGREN and P. A. REJTO, An application of the maximum principle to the study of essential selfadjointness of Dirac operators. I, J. Math. Phys. 20 ( J. J. LANDGREN, P. A. REJTO and M. KLAUS, An application of the maximum principle to the study of essential selfadjointness of Dirac operators. II, J. Math. Phys. 21 ( E. H. LIEB, H. SIEDENTOP and J. P. SOLOVEJ, Stability and instability of relativistic electrons in classical electromagnetic fields, J. Statist. Phys. 89 ( E. H. LIEB, H. SIEDENTOP and J. P. SOLOVEJ, Stability of relativistic matter with magnetic fields, Phys. Re. Lett. 79 ( M. REED and B. SIMON, Analysis of operators, Methods of Modern Mathematical Physics 4 (Academic Press, New York, L. ROSENBERG and L. SPRUCH, Extremum principles for the determination of relativistic bound-state energies, Phys. Re. A 34 ( U.-W. SCHMINCKE, Distinguished selfadjoint extensions of Dirac operators, Math. Z. 129 ( J. SUCHER, Foundations of the relativistic theory of many-electron atoms, Phys. Re. A 22 ( J. D. TALMAN, Minimax principle for the Dirac equation, Phys. Re. Lett. 57 ( B. THALLER, The Dirac equation, Texts and Monographs in Physics (Springer, Berlin, C. TIX, Strict positivity of a relativistic Hamiltonian due to Brown and Ravenhall, Bull. London Math. Soc. 30 ( Mathematik Uniersita t Regensburg D Regensburg Germany Marcel.Griesemermathematik.uni-regensburg.de Heinz.Siedentopmathematik.uni-regensburg.de

76 Griesemer, Lewis, Siedentop the Dirac operator in the gap with the number of negative eigenvalues of a corresponding Schrodinger operator (see [5])

76 Griesemer, Lewis, Siedentop the Dirac operator in the gap with the number of negative eigenvalues of a corresponding Schrodinger operator (see [5]) Doc. Math. J. DMV 75 A Minimax Principle for Eigenvalues in Spectral Gaps: Dirac Operators with Coulomb Potentials Marcel Griesemer, Roger T. Lewis, Heinz Siedentop Received: March 9, 999 Communicated