A Minimax Principle

for Eigenvalues in Spectral Gaps:

Dirac Operators with Coulomb Potentials^{1}

Marcel Griesemer, Roger T. Lewis, Heinz Siedentop

Received: March 29, 1999

Communicated by Bernold Fiedler

Abstract. We prove the minimax principle for eigenvalues in spectral gaps introduced in [5] based on an alternative set of hypotheses. In the case of the Dirac operator these new assumptions allow for potentials with Coulomb singularites.

1991 Mathematics Subject Classification: 47A75, 81Q10

Keywords and Phrases: Minimax principle, Dirac operator, Coulomb singularity

1 Introduction

Recently Dolbeault, Esteban, and Séré [4, 3, 2] have found a minimax principle for Dirac operators with Coulomb potentials. Independently, Griesemer and Siedentop [5] have found a minimax principle characterizing the eigenvalues of self-adjoint operators in their spectral gaps, which is flexible enough to adapt to various situations. In particular it can also be applied to Dirac operators. Such a minimax principle is of particular interest for applications, e.g., in solid state physics and relativistic quantum chemistry where differential operators having gaps in their spectra naturally arise. Apart from the computational point of view (see, e.g., Kutzelnigg [7]) it can serve as a tool to obtain nonasymptotic eigenvalue estimates, e.g., comparing the number of eigenvalues of the Dirac operator in the gap with the number of negative eigenvalues of a corresponding Schrödinger operator (see [5]). Comparing [3, 2] and [5] shows, that although the hypotheses for the validity of the minimax principle overlap, the methods of proof are quite different. On the other hand, with these different hypotheses different classes of operators can be treated: Dolbeault, Esteban, and Séré’s result allows for Dirac operators with singular potentials of Coulomb type. Griesemer and Siedentop’s result allows for a flexible formulation of the minimax principle adaptable to various situations, e.g., an earlier minimax principle for the first positive eigenvalue of the Dirac operator considered by Talman [9] and Datta and Deviah [1] can be proved. This difference in hypotheses indicates that the optimal assumption for the abstract minimax principle is yet to be found. The present paper is a step in this direction. In Section 2 we prove the abstract minimax principle under assumptions alternative to those in [5]. In Section 3 we show that these hypotheses allow for Dirac operators with Coulomb potentials. Applications to other self-adjoint operators with eigenvalues in spectral gaps like perturbed periodic Schrödinger operators are also conceivable.

1 This work has been partially supported by the European Union through the TMR network FMRX-CT 96-0001.

Documenta Mathematica 4 (1999) 275–283

2 The Minimax Principle

In this section we formulate and prove the abstract minimax principle. Suppose A and A_{0} are self-adjoint operators in a Hilbert space ℌ and assume that their form domains are equal

𝔔(A) = 𝔔(A_{0}) = 𝔔.  (1)

Let 𝔇(A) and 𝔇(A_{0}) denote the domains of A and A_{0} respectively and let P_{I}(A) be the spectral projection of A corresponding to the interval I ⊂ ℝ. Define

\begin{aligned}Λ_{+} & = P_{(0,∞)}(A_{0}),  Λ_{−} = 1 − Λ_{+}, \\ P_{+} & = P_{(0,∞)}(A),  P_{−} = 1 − P_{+}. \end{aligned}  (2)

We set ℌ_{±} := Λ_{±}ℌ and 𝔔_{±} := Λ_{±}𝔔. Then ℌ = ℌ_{+} ⊕ ℌ_{−} and, by assumption (1), 𝔔_{±} ⊂ 𝔔. The minimax values in which we are interested are given by

λ_{n}(A) := inf_{\begin{matrix}𝔐_{+}⊂𝔔_{+} \\ dim(𝔐_{+})=n \end{matrix}} sup_{\begin{matrix}ψ∈𝔐_{+}⊕𝔔_{−} \\ ∥ψ∥=1 \end{matrix}}(ψ, Aψ),  (3)

and have been introduced in [5]. These minimax values are to be compared with the standard (Courant) minimax values

μ_{n}(B) := inf_{\begin{matrix}𝔐⊂𝔔(B) \\ dim(𝔐)=n \end{matrix}} sup_{\begin{matrix}ψ∈𝔐 \\ ∥ψ∥=1 \end{matrix}}(ψ, Bψ)

for the eigenvalues of a self-adjoint operator B which is bounded from below. The value μ_{n}(B) is the n-th eigenvalue of B counting from below (see, e.g., Reed and Simon [8]).

Theorem 1. Suppose A and A_{0} are self-adjoint operators in ℌ with the same form domain 𝔔 and define Λ_{±}, P_{±}, 𝔔_{±}, λ_{n}(A) and μ_{n}(·) as above. If (ψ, Aψ) ≤ 0 for all ψ ∈ 𝔔_{−} and if

∥(|A_{0}| + 1)^{1/2}Λ_{+}P_{−}(|A_{0}| + 1)^{−1/2}∥ < 1  (4)

then λ_{n}(A) = μ_{n}(A|` P_{+}ℌ) for all n ≤ dim ℌ_{+}.

We remark that |A_{0}| + 1 can be replaced by |A_{0}| in (4), if we assume that 0 is in the resolvent set of A_{0}. This will be obvious from the proof.

Proof. We prove the theorem in two steps. Although these are partly contained in [5] we do not omit the similar parts in order to be self-contained: First, we show that it suffices to prove that Λ_{+} : P_{+}𝔔 → 𝔔_{+} is a bijection. Secondly, we verify this property using assumption (4) and the negativity of (ψ, Aψ) on 𝔔_{−}. Step 1. If Λ_{+}P_{+}𝔔 = 𝔔_{+}, then we have

λ_{n}(A) = inf_{\begin{matrix}𝔐_{+}⊂Λ_{+}P_{+}𝔔 \\ dim(𝔐_{+})=n \end{matrix}} sup_{\begin{matrix}ψ∈𝔐_{+}⊕𝔔_{−} \\ ∥ψ∥=1 \end{matrix}}(ψ, Aψ)  (5)

using the defining Equation (3). Since for each 𝔐_{+} ⊂ Λ_{+}P_{+}𝔔 with dim(𝔐_{+}) = n, we can find a subspace 𝔐 ⊂ P_{+}𝔔 with dim(𝔐) = n such that 𝔐_{+} = Λ_{+}𝔐 and since Λ_{+}𝔐 ⊕ 𝔔_{−} ⊃ 𝔐, we get from (5)

λ_{n}(A) = inf_{\begin{matrix}𝔐_{+}⊂Λ_{+}P_{+}𝔔 \\ dim(𝔐_{+})=n \end{matrix}} sup_{\begin{matrix}ψ∈𝔐_{+}⊕𝔔_{−} \\ ∥ψ∥=1 \end{matrix}}(ψ, Aψ)

≥ inf_{\begin{matrix}𝔐⊂P_{+}𝔔 \\ dim(𝔐)=n \end{matrix}} sup_{\begin{matrix}ψ∈𝔐 \\ ∥ψ∥=1 \end{matrix}}(ψ, Aψ) = μ_{n}(A|` P_{+}ℌ).

To prove the converse inequality we proceed as in [5]: pick ϵ > 0 and let 𝔐 := P_{(0,μ_{n}+ϵ)}(A)𝔔. Then dim(𝔐) ≥ n and hence dim(Λ_{+}𝔐) ≥ n by the remark above. Therefore

λ_{n} ≤ sup_{\begin{matrix}ψ∈Λ_{+}𝔐⊕𝔔_{−} \\ ∥ψ∥=1 \end{matrix}}(ψ, Aψ) = sup_{\begin{matrix}ψ∈𝔐+𝔔_{−} \\ ∥ψ∥=1 \end{matrix}}(ψ, Aψ),

where Λ_{+}𝔐 ⊕ 𝔔_{−} = 𝔐 + 𝔔_{−} was used. To estimate this from above we first decompose ψ ∈ 𝔐 + 𝔔_{−} as ψ = ψ_{1} + ψ_{2}, where ψ_{1} ∈ 𝔐 and ψ_{2} ∈ 𝔐^{⊥} ∩ (𝔐 + 𝔔_{−}), and then ψ_{2} as ψ_{2} = ψ_{3} + ψ_{−} where ψ_{3} ∈ 𝔐 and ψ_{−} ∈ 𝔔_{−}.

Since Aψ_{3} ∈ 𝔐 and ψ_{3} + ψ_{−} ∈ 𝔐^{⊥} we have (Aψ_{3}, ψ_{−}) = −(Aψ_{3}, ψ_{3}). Using this, (Aψ_{3}, ψ_{3}) ≥ 0, and (ψ_{−}, Aψ_{−}) ≤ 0 we find

(ψ, Aψ) = (ψ_{1}, Aψ_{1}) + (ψ_{2}, Aψ_{2})

= (ψ_{1}, Aψ_{1}) − (ψ_{3}, Aψ_{3}) + (ψ_{−}, Aψ_{−}) ≤ (ψ_{1}, Aψ_{1}) ≤ (μ_{n} + ϵ)(ψ, ψ)

which implies λ_{n} ≤ μ_{n}. Step 2. Surjectivity: Since Λ_{+}P_{+}𝔔 ⊂ 𝔔_{+} it suffices that Λ_{+}P_{+}𝔔_{+} = 𝔔_{+}, which is equivalent to (|A_{0}|+1)^{1/2}Λ_{+}P_{+}(|A_{0}|+1)^{−1/2}ℌ_{+} = ℌ_{+}. Now Λ_{+}P_{+} = 1 − Λ_{+}P_{−} on ℌ_{+} so that

(|A_{0}| + 1)^{1/2}Λ_{+}P_{+}(|A_{0}| + 1)^{−1/2} = 1 − (|A_{0}| + 1)^{1/2}Λ_{+}P_{−}(|A_{0}| + 1)^{−1/2}

on ℌ_{+}. By assumption (4) the latter is an isomorphism from ℌ_{+} to ℌ_{+}. Injectivity: Suppose Λ_{+} : P_{+}𝔔 → 𝔔_{+} would not be one-to-one. Then there would exist a non-zero ψ ∈ ℌ_{−} ∩ P_{+}𝔔 such that

0 ≥ (ψ, Aψ) = (P_{+}ψ, AP_{+}ψ) > 0.

3 Application to the Dirac Operator

The hypothesis (4) of Theorem 1 contains the a priori unknown operator P_{−}, i.e., it is not straightforward to check. In this section we will show how to verify it for given operators nevertheless. To be specific we restrict ourselves to the Dirac operator D_{γ} with a screened Coulomb potential, i.e., D_{γ} := (1/i)∇ · α + mβ − γφ in ℌ := L^{2}(ℝ^{3})^{4}, where φ(x) = y(x)/|x| with measurable y and y(ℝ^{3}) ⊂ [0, 1]. By Hardy’s inequality we have that D_{γ} is an operator perturbation of D_{0} for γ ∈ (−1/2, 1/2). We will assume this restriction on γ henceforth. In particular, perturbation theory for |D_{0}| = (−Δ+m^{2})^{1/2} implies by Hardy’s and Kato’s inequality

∀_{γ∈[0,1/2)} 𝔇(D_{γ}) = H^{1}(ℝ^{3}) ⊗ ℂ^{4} =: 𝔇,  (6)
∀_{γ∈[0,2/π)} 𝔔(D_{γ}) = H^{1/2}(ℝ^{3}) ⊗ ℂ^{4} =: 𝔔  (7)

for the operator and form domain of D_{γ}, respectively. To make connections with Section 2 we pick A_{0} := D_{0} and A := D_{γ}. The notation (2) is used correspondingly here. By γ_{0} we denote the real solution of 2γ_{0}^{3} − 3γ_{0}^{2} + 4γ_{0} = 1. Note that 0.305 < γ_{0} < 0.306 holds.

Theorem 2. For γ ∈ [0, γ_{0})

inf_{\begin{matrix}𝔐_{+}⊂𝔔_{+} \\ dim 𝔐_{+}=n \end{matrix}} sup_{\begin{matrix}ψ∈𝔐_{+}⊕𝔔_{−} \\ ∥ψ∥=1 \end{matrix}}(ψ, D_{γ}ψ)  (8)

is equal to the n-th positive eigenvalue – counting multiplicity – of the Dirac operator D_{γ} or equals the mass m.

Our strategy is to roll the proof back to a verification of the hypotheses of Theorem 1. The main step is the verification of (4) which we break up into several steps:

Lemma 1. For all f ∈ ℌ

\begin{matrix}Λ_{+}P_{−}f = & −\frac{γ}{2π} Λ_{+} ∫_{−∞}^{∞}(D_{0} − iz)^{−1}φ(D_{γ} − iz)^{−1}dzf \\ = & −\frac{γ}{π} Λ_{+} ∫_{0}^{∞} [(D_{0}^{2} + z^{2})^{−1}(D_{0}φD_{γ} − z^{2}φ)(D_{γ}^{2} + z^{2})^{−1}] dzf. \end{matrix} (9)

Proof. Since for γ ∈ [0, 2/π), zero is in the resolvent set of D_{γ}, we have that

P_{±} = \frac{1}{2} ± \frac{1}{2π} ∫_{−∞}^{∞}(D_{γ} − iz)^{−1}dz = \frac{1}{2} ± \frac{1}{π} ∫_{0}^{∞} D_{γ}(D_{γ}^{2} + z^{2})^{−1}dz (10)

(Kato [6], Chapter VI.5, Lemma 5.6); Λ_{±} is obtained from (10) by setting γ = 0. Therefore, by (10), and the second resolvent identity

P_{−} = Λ_{−} − \frac{γ}{2π} ∫_{−∞}^{∞}(D_{0} − iz)^{−1}φ(D_{γ} − iz)^{−1}dz

from which we may conclude that the first part of (9) holds. We can simplify

∫_{−∞}^{∞}(D_{0} − iz)^{−1}φ(D_{γ} − iz)^{−1}dzf
= ∫_{0}^{∞} [(D_{0} − iz)^{−1}φ(D_{γ} − iz)^{−1} + (D_{0} + iz)^{−1}φ(D_{γ} + iz)^{−1}] dzf
= ∫_{0}^{∞} [\frac{D_{0} + iz}{D_{0}^{2} + z^{2}} φ\frac{D_{γ} + iz}{D_{γ}^{2} + z^{2}} + \frac{D_{0} − iz}{D_{0}^{2} + z^{2}} φ\frac{D_{γ} − iz}{D_{γ}^{2} + z^{2}} ] dzf
= 2 ∫_{0}^{∞} [(D_{0}^{2} + z^{2})^{−1}(D_{0}φD_{γ} − z^{2}φ)(D_{γ}^{2} + z^{2})^{−1}] dzf

which implies that the second part of (9) holds.

Lemma 2. For γ ∈ ℝ_{+} we have (1/2 − γ)^{2}φ^{2} ≤ |D_{γ}|^{2} ≤ (1 + 2γ)^{2}|D_{0}|^{2}.

Proof. For all ψ ∈ 𝔇(D_{0}) we have ∥D_{γ}ψ∥ ≥ ∥D_{0}ψ∥ − γ∥φψ∥ ≥ (1/2 − γ)∥φψ∥, where we first use the triangle inequality and then Hardy’s inequality. This implies the first stated operator inequality. The second one follows from ∥D_{γ}ψ∥ ≤ ∥D_{0}ψ∥ + γ∥φψ∥ ≤ (1 + 2γ)∥D_{0}ψ∥.

Lemma 3. For all γ ∈ (0, \frac{1}{2}) and f ∈ ℌ we have

∥|D_{0}|^{1/2} ∫_{0}^{∞}(D_{0}^{2} + z^{2})^{−1}(D_{0}φD_{γ} − z^{2}φ)(D_{γ}^{2} + z^{2})^{−1}dz|D_{0}|^{−1/2}f∥

≤ π \frac{\sqrt{1 + 2γ}}{1 − 2γ} ∥f∥. (11)

Proof. Using the fact that

∥h∥ = sup_{∥g∥=1} |(g, h)|, h ∈ ℌ

and setting f′ := |D_{0}|^{−1/2}f we see that the norm on the left hand side of (11) can be approximated by finding an upper bound for

|(g, |D_{0}|^{1/2} ∫_{0}^{∞} [(D_{0}^{2} + z^{2})^{−1}(D_{0}φD_{γ} − z^{2}φ)(D_{γ}^{2} + z^{2})^{−1}] dzf′)|, ∥g∥ = 1. (12)

First, consider the term

|(g, |D_{0}|^{1/2} ∫_{0}^{∞} [(D_{0}^{2} + z^{2})^{−1}(D_{0}φD_{γ})(D_{γ}^{2} + z^{2})^{−1}] dzf′)|

≤ [∫_{0}^{∞} ∥D_{0}(D_{0}^{2} + z^{2})^{−1}|D_{0}|^{1/2}g∥^{2}dz]^{\frac{1}{2}} [∫_{0}^{∞} ∥φD_{γ}(D_{γ}^{2} + z^{2})^{−1}f′∥^{2}dz]^{\frac{1}{2}} . (13)

Note that

∫_{0}^{∞} \frac{dz}{(1 + z^{2})^{2}} = ∫_{0}^{∞} \frac{z^{2}dz}{(1 + z^{2})^{2}} = \frac{π}{4} .  (14)

Thus, the first factor yields

∫_{0}^{∞} ∥D_{0}(D_{0}^{2} + z^{2})^{−1}|D_{0}|^{1/2}g∥^{2}dz = ∫_{0}^{∞}(g, \frac{|D_{0}|^{3}}{(D_{0}^{2} + z^{2})^{2}} g)dz = \frac{π}{4} (g, g). (15)

In a similar manner we show for γ ∈ (0, 1/2)

∫_{0}^{∞} ∥φD_{γ}(D_{γ}^{2} + z^{2})^{−1}f′∥^{2}dz  (16)
= ∫_{0}^{∞}(f′, (D_{γ}^{2} + z^{2})^{−1}D_{γ}φ^{2}D_{γ}(D_{γ}^{2} + z^{2})^{−1}f′)dz  (17)
≤ \frac{1}{(1/2 − γ)^{2}} ∫_{0}^{∞}(f′, (D_{γ}^{2} + z^{2})^{−1}|D_{γ}|^{4}(D_{γ}^{2} + z^{2})^{−1}f′)dz  (18)
= \frac{π}{(1 − 2γ)^{2}} (f′, |D_{γ}|f′) ≤ \frac{π(1 + 2γ)}{(1 − 2γ)^{2}}(f′, |D_{0}|f′) ≤ \frac{π(1 + 2γ)}{(1 − 2γ)^{2}}(f, f)(19)

where we have used the first inequality of Lemma 2 to go from (17) to (18) and the second inequality of that Lemma in (19). Thus we have for the product

|(g, |D_{0}|^{1/2} ∫_{0}^{∞} [(D_{0}^{2} + z^{2})^{−1}(D_{0}φD_{γ})(D_{γ}^{2} + z^{2})^{−1}] dzf′)| ≤ \frac{π}{2} \frac{\sqrt{1 + 2γ}}{1 − 2γ} ∥f∥.

Likewise, we estimate the second term in (12)

|(g, |D_{0}|^{1/2} ∫_{0}^{∞}(D_{0}^{2} + z^{2})^{−1}z^{2}φ(D_{γ}^{2} + z^{2})^{−1}dz|D_{0}|^{−1/2}f)|

= | ∫_{0}^{∞}(z(D_{0}^{2} + z^{2})^{−1}|D_{0}|^{1/2}g, zφ(D_{γ}^{2} + z^{2})^{−1}f′)dz|

≤ [∫_{0}^{∞} ∥z(D_{0}^{2} + z^{2})^{−1}|D_{0}|^{1/2}g∥^{2}dz]^{\frac{1}{2}} [∫_{0}^{∞} ∥zφ(D_{γ}^{2} + z^{2})^{−1}f′∥^{2}dz]^{\frac{1}{2}} . (20)

By scaling and (14) we get for the first factor

∫_{0}^{∞} ∥z|D_{0}|^{1/2}(D_{0}^{2} + z^{2})^{−1}g∥^{2}dz = \frac{π}{4} .  (21)

The second factor yields using Lemma 2 twice

∫_{0}^{∞} ∥zφ(D_{γ}^{2} + z^{2})^{−1}f′∥^{2}dz = (f′, ∫_{0}^{∞}(D_{γ}^{2} + z^{2})^{−1}φ^{2}z^{2}(D_{γ}^{2} + z^{2})^{−1}dzf′)

≤\frac{1}{(1/2 − γ)^{2}} (f′, ∫_{0}^{∞}(D_{γ}^{2} + z^{2})^{−1}|D_{γ}|^{2}z^{2}(D_{γ}^{2} + z^{2})^{−1}dzf′)
=\frac{π}{4(1/2 − γ)^{2}} (f′, D_{γ}f′) ≤ π \frac{1 + 2γ}{(1 − 2γ)^{2}} (f′, D_{0}f′).

Thus we get

|(g, |D_{0}|^{1/2} ∫_{0}^{∞}(D_{0}^{2} + z^{2})^{−1}z^{2}φ(D_{γ}^{2} + z^{2})^{−1}dzf′)| ≤ \frac{π}{2} \frac{\sqrt{1 + 2γ}}{1 − 2γ} ∥f∥, (22)

i.e., the same upper bound as for the first term. By (11), (12), and the calculations above, we have the upper bound

∥|D_{0}|^{1/2} ∫_{0}^{∞} [(D_{0}^{2} + z^{2})^{−1}(D_{0}φD_{γ} − z^{2}φ)(D_{γ}^{2} + z^{2})^{−1}] dz|D_{0}|^{−1/2}f∥

≤ π \frac{\sqrt{1 + 2γ}}{1 − 2γ} ∥f∥

for γ ∈ [0, 1/2) which we claimed.

From Lemmata 1 and 3 we have the immediate

Corollary 1. For all γ ∈ (0, \frac{1}{2})

∥|D_{0}|^{1/2}Λ_{+}P_{−}|D_{0}|^{−1/2}∥ ≤ γ \frac{\sqrt{1 + 2γ}}{1 − 2γ} .

We remark that an argument similar to the proofs of Lemmata 1 and 3 shows that ∥Λ_{+}P_{−}∥ = O(γ) as γ → 0 which implies that Λ_{+}P_{+}ℌ = ℌ_{+} and ℌ_{+} ∩ P_{−}ℌ = {0} for small enough positive γ. We turn now to the proof of Theorem 2.

Proof. First, we reiterate our remark (7) that for γ ∈ [0, 2/π) the form domain of 𝔔 := 𝔔(D_{γ}) = H^{1/2}(ℝ^{3}) ⊗ ℂ^{4}. In particular, it is independent of γ. This also means that P_{±} and Λ_{±} leave 𝔔 invariant. Moreover, Λ_{−}D_{γ}Λ_{−} is certainly non-positive. Finally, Corollary 1 implies that (4) holds true for γ ∈ [0, γ_{0}) which completes the proof.

Finally, we remark, that the construction of this Section is easily generalized to other types of potentials, as long as one can prove an analogue of Lemma 3.

Acknowledgment. This work has been partially supported by the European Union through its Training, Research, and Mobility program, grant FMRX-CT 96-0001.

References

[1] S. N. Datta and G. Deviah. The minimax technique in relativistic Hartree-Fock calculations. Pramana, 30(5):387–405, May 1988.

[2] J. Dolbeault, M. J. Esteban, and E. Séré. Variational characterization for eigenvalues of Dirac operators. Preprint, mp-arc: 98-177, 1998.

[3] Jean Dolbeault, Maria J. Esteban, and Eric Séré. International Conference on Differential Equations and Mathematical Physics, Atlanta, Georgia, March 23–29, 1997.

[4] Maria J. Esteban and Eric Séré. Existence and multiplicity of solutions for linear and nonlinear Dirac operators. In Paritial Differential Equations and their Applications (Toronto, ON, 1995), pages 107–118. Amer. Math. Soc., Providence, RI, 1997.

[5] Marcel Griesemer and Heinz Siedentop. A minimax principle for the eigenvalues in spectral gaps. J. London Math. Soc., Accepted for publication. Preprint, mp-arc 97-492, 1997.

[6] Tosio Kato. Perturbation Theory for Linear Operators, volume 132 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, 1 edition, 1966.

[7] Werner Kutzelnigg. Relativistic one-electron Hamiltonians ‘for electrons only’ and the variational treatment of the Dirac equation. Chemical Physics, 1997.

[8] Michael Reed and Barry Simon. Methods of Modern Mathematical Physics, volume 4: Analysis of Operators. Academic Press, New York, 1 edition, 1978.

[9] James D. Talman. Minimax principle for the Dirac equation. Phys. Rev. Lett., 57(9):1091–1094, September 1986.

Marcel Griesemer Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170 USA marcel@math.uab.edu Roger T. Lewis Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170 USA lewis@math.uab.edu

Heinz Siedentop Mathematik I Universität Regensburg D-93040 Regensburg Germany Heinz.Siedentop@mathematik.uniregensburg.de 284

Documenta Mathematica 4 (1999)