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

Size: px

Start display at page:

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

Ezra Harrison
6 years ago
Views:

1 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 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. 99 Mathematics Subject Classication: 47A75, 8Q Keywords and Phrases: Minimax principle, Dirac operator, Coulomb singularity Introduction Recently Dolbeault, Esteban, and Sere [4, 3, ] 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 exible 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 dierential 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 This work has been partially supported by the European Union through the TMR network FMRX-CT 96-. Documenta Mathematica 4 (999) 75{83

2 76 Griesemer, Lewis, Siedentop the Dirac operator in the gap with the number of negative eigenvalues of a corresponding Schrodinger operator (see [5]). Comparing [3, ] and [5] shows, that although the hypotheses for the validity of the minimax principle overlap, the methods of proof are quite dierent. On the other hand, with these dierent hypotheses dierent classes of operators can be treated: Dolbeault, Esteban, and Sere's result allows for Dirac operators with singular potentials of Coulomb type. Griesemer and Siedentop's result allows for a exible formulation of the minimax principle adaptable to various situations, e.g., an earlier minimax principle for the rst positive eigenvalue of the Dirac operator considered by Talman [9] and Datta and Deviah [] can be proved. This dierence 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 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 Schrodinger operators are also conceivable. The Minimax Principle In this section we formulate and prove the abstract minimax principle. Suppose A and A are self-adjoint operators in a Hilbert space H and assume that their form domains are equal Q(A) = Q(A ) = Q: () Let D(A) and D(A ) denote the domains of A and A respectively and let P I (A) be the spectral projection of A corresponding to the interval I R. Dene + = P (;) (A ); = + ; P + = P (;) (A); P = P + : () We set H := H and Q := Q. Then H = H + H and, by assumption (), Q Q. The minimax values in which we are interested are given by n (A) := inf M +Q + dim(m +)=n sup ( ; A ); (3) M +Q k k= and have been introduced in [5]. These minimax values are to be compared with the standard (Courant) minimax values n (B) := inf MQ(B) M dim(m)=n k k= sup ( ; B ) Documenta Mathematica 4 (999) 75{83

3 Minimax for Dirac Operators 77 for the eigenvalues of a self-adjoint operator B which is bounded from below. The value n (B) is the n-th eigenvalue of B counting from below (see, e.g., Reed and Simon [8]). Theorem. Suppose A and A are self-adjoint operators in H with the same form domain Q and dene ; P ; Q ; n (A) and n () as above. If ( ; A ) for all Q and if k(ja j + ) = + P (ja j + ) = k < (4) then n (A) = n (AjP + H) for all n dim H +. We remark that ja j + can be replaced by ja j in (4), if we assume that is in the resolvent set of A. 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 suces to prove that + : P + Q! Q + is a bijection. Secondly, we verify this property using assumption (4) and the negativity of ( ; A ) on Q. Step. If + P + Q = Q +, then we have n (A) = inf M + +P +Q dim(m +)=n sup ( ; A ) (5) M +Q k k= using the dening Equation (3). Since for each M + + P + Q with dim(m + ) = n, we can nd a subspace M P + Q with dim(m) = n such that M + = + M and since + M Q M, we get from (5) n (A) = inf M + +P +Q dim(m +)=n sup ( ; A ) M +Q k k= inf sup ( ; A ) = n (AjP + H): MP +Q M dim(m)=n k k= To prove the converse inequality we proceed as in [5]: pick > and let M := P (;n+) (A)Q. Then dim(m) n and hence dim( + M) n by the remark above. Therefore n sup ( ; A ) = sup ( ; A ); +MQ M+Q k k= k k= where + M Q = M + Q was used. To estimate this from above we rst decompose M + Q as = +, where M and M? \ (M + Q ), and then as = 3 + where 3 M and Q. Documenta Mathematica 4 (999) 75{83

4 78 Griesemer, Lewis, Siedentop Since A 3 M and 3 + M? we have (A 3 ; ) = (A 3 ; 3 ). Using this, (A 3 ; 3 ), and ( ; A ) we nd ( ; A ) = ( ; A ) + ( ; A ) = ( ; A ) ( 3 ; A 3 ) + ( ; A ) ( ; A ) ( n + )( ; ) which implies n n. Step. Surjectivity: Since + P + Q Q + it suces that + P + Q + = Q +, which is equivalent to (ja j+) = + P + (ja j+) = H + = H +. Now + P + = + P on H + so that (ja j + ) = + P + (ja j + ) = = (ja j + ) = + P (ja j + ) = on H +. By assumption (4) the latter is an isomorphism from H + to H +. Injectivity: Suppose + : P + Q! Q + would not be one-to-one. Then there would exist a non-zero H \ P + Q such that ( ; A ) = (P + ; AP + ) > : 3 Application to the Dirac Operator The hypothesis (4) of Theorem 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 specic we restrict ourselves to the Dirac operator D with a screened Coulomb potential, i.e., D := (=i)r + m ' in H := L (R 3 ) 4, where '(x) = y(x)=jxj with measurable y and y(r 3 ) [; ]. By Hardy's inequality we have that D is an operator perturbation of D for ( =; =). We will assume this restriction on henceforth. In particular, perturbation theory for jd j = ( +m ) = implies by Hardy's and Kato's inequality 8 [;=) D ) = H (R 3 ) C 4 =: D; (6) 8 [;=) Q ) = H = (R 3 ) C 4 =: Q (7) for the operator and form domain of D, respectively. To make connections with Section we pick A := D and A := D. The notation () is used correspondingly here. By we denote the real solution of =. Note that :35 < < :36 holds. Theorem. For [; ) inf M +Q + dim M +=n sup ( ; D ) (8) M +Q k k= is equal to the n-th positive eigenvalue { counting multiplicity { of the Dirac operator D or equals the mass m. Documenta Mathematica 4 (999) 75{83

5 Minimax for Dirac Operators 79 Our strategy is to roll the proof back to a verication of the hypotheses of Theorem. The main step is the verication of (4) which we break up into several steps: Lemma. For all f H R + R + P f = = + iz) ' iz) dzf + z ) 'D z ') + z ) dzf: Proof. Since for [; =), zero is in the resolvent set of D, we have that P = iz) dz = D + z ) dz () (Kato [6], Chapter VI.5, Lemma 5.6); is obtained from () by setting =. Therefore, by (), and the second resolvent identity P = iz) ' iz) dz from which we may conclude that the rst part of (9) holds. We can simplify = = = iz) ' iz) dzf iz) ' iz) + + iz) ' + iz) dzf D + iz D + ' D + iz z D + z + D iz D + ' D iz z D + z dzf + z ) 'D z ') + z ) dzf which implies that the second part of (9) holds. Lemma. For R + we have (= ) ' jd j ( + ) jd j. Proof. For all D ) we have kd k kd k k' k (= )k' k, where we rst use the triangle inequality and then Hardy's inequality. This implies the rst stated operator inequality. The second one follows from kd k kd k + k' k ( + )kd k. Lemma 3. For all (; ) and f H we have kjd j = + z ) 'D z ') + z ) dzjd j = fk p + kfk: () (9) Documenta Mathematica 4 (999) 75{83

6 8 Griesemer, Lewis, Siedentop Proof. Using the fact that khk = sup j(g; h)j; kgk= h H and setting f := jd j = f we see that the norm on the left hand side of () can be approximated by nding an upper bound for j(g; jd j = First, consider the term j(g; jd j = Note that + z ) 'D z ') + z ) dzf )j; kgk = : + z ) 'D ) + z ) dzf )j kd + z ) jd j = gk dz Thus, the rst factor yields kd + z ) jd j = gk dz = k'd + z ) f k dz : () (3) dz ( + z ) = z dz ( + z ) = 4 : (4) In a similar manner we show for (; =) jd j 3 (g; + z ) g)dz = (g; g): (5) 4 = = k'd + z ) f k dz (6) (f ; + z ) D ' D + z ) f )dz (7) (= ) ( ) (f ; jd jf ) (f ; + z ) jd j 4 + z ) f )dz (8) ( + ) ( ) (f ; jd jf ) ( + ) (f; f)(9) ( ) where we have used the rst inequality of Lemma to go from (7) to (8) and the second inequality of that Lemma in (9). Thus we have for the product j(g; jd j = + z ) 'D ) + z ) dzf )j p + kfk: Documenta Mathematica 4 (999) 75{83

7 Minimax for Dirac Operators 8 Likewise, we estimate the second term in () j(g; jd j = = j + z ) z ' + z ) dzjd j = f)j (z + z ) jd j = g; z' + z ) f )dzj kz + z ) jd j = gk dz By scaling and (4) we get for the rst factor The second factor yields using Lemma twice kz' + z ) f k dz = (f ; kz' + z ) f k dz : () kzjd j = + z ) gk dz = 4 : () + z ) ' z + z ) dzf ) (= ) (f ; + z ) jd j z + z ) dzf ) = 4(= ) (f ; D f ) + ( ) (f ; D f ): Thus we get j(g; jd j = + z ) z ' + z ) dzf )j p + kfk; () i.e., the same upper bound as for the rst term. By (), (), and the calculations above, we have the upper bound kjd j = + z ) 'D z ') + z ) dzjd j = fk p + kfk for [; =) which we claimed. From Lemmata and 3 we have the immediate Corollary. For all (; ) kjd j = + P jd j = k p + : We remark that an argument similar to the proofs of Lemmata and 3 shows that k + P k = O() as! which implies that + P + H = H + and H + \ P H = fg for small enough positive. We turn now to the proof of Theorem. Documenta Mathematica 4 (999) 75{83

8 8 Griesemer, Lewis, Siedentop Proof. First, we reiterate our remark (7) that for [; =) the form domain of Q := Q ) = H = (R 3 ) C 4. In particular, it is independent of. This also means that P and leave Q invariant. Moreover, D is certainly non-positive. Finally, Corollary implies that (4) holds true for [; ) 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-. References [] S. N. Datta and G. Deviah. The minimax technique in relativistic Hartree- Fock calculations. Pramana, 3(5):387{45, May 988. [] J. Dolbeault, M. J. Esteban, and E. Sere. Variational characterization for eigenvalues of Dirac operators. Preprint, mp-arc: 98-77, 998. [3] Jean Dolbeault, Maria J. Esteban, and Eric Sere. International Conference on Dierential Equations and Mathematical Physics, Atlanta, Georgia, March 3{9, 997. [4] Maria J. Esteban and Eric Sere. Existence and multiplicity of solutions for linear and nonlinear Dirac operators. In Paritial Dierential Equations and their Applications (Toronto, ON, 995), pages 7{8. Amer. Math. Soc., Providence, RI, 997. [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 97-49, 997. [6] Tosio Kato. Perturbation Theory for Linear Operators, volume 3 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, edition, 966. [7] Werner Kutzelnigg. Relativistic one-electron Hamiltonians `for electrons only' and the variational treatment of the Dirac equation. Chemical Physics, 997. [8] Michael Reed and Barry Simon. Methods of Modern Mathematical Physics, volume 4: Analysis of Operators. Academic Press, New York, edition, 978. [9] James D. Talman. Minimax principle for the Dirac equation. Phys. Rev. Lett., 57(9):9{94, September 986. Documenta Mathematica 4 (999) 75{83

9 Minimax for Dirac Operators 83 Marcel Griesemer Department of Mathematics University of Alabama at Birmingham Birmingham, AL USA Roger T. Lewis Department of Mathematics University of Alabama at Birmingham Birmingham, AL USA Heinz Siedentop Mathematik I Universitat Regensburg D-934 Regensburg Germany Heinz.Siedentop@mathematik.uniregensburg.de Documenta Mathematica 4 (999) 75{83

10 84 Documenta Mathematica 4 (999)

A MINIMAX PRINCIPLE FOR THE EIGENVALUES IN SPECTRAL GAPS

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