math-ph/ Aug 1999

Transcription

1 Measures of Fermi surfaces and absence of singular continuous spectrum for magnetic Schrodinger operators math-ph/ Aug 1999 Michael J. Gruber y August 25, 1999 Abstract Fermi surfaces are basic objects in solid state physics and in the spectral theory of periodic operators. We dene several measures connected to Fermi surfaces and study their measure theoretic properties. From this we get absence of singular continuous spectrum and of singular continuous components in the density of states for symmetric periodic elliptic dierential operators acting on vector bundles. This includes Schrodinger operators with periodic magnetic eld and rational ux, as well as the corresponding Pauli and irac-type operators. Introduction Fermi surfaces are basic objects in solid state physics of metals. They dene the region in phase space which is relevant at room temperatures or below: the only electrons which take part in transport are those diering by almost an energy k B T from the electrons on the Fermi surface which have energy k B T F. Here, k B is Boltzmann's constant, T F is the Fermi temperature which is of the order of K for ordinary metals compared to room temperatures of 300 K. The curvature of the Fermi surface endows electrons with a dynamic mass, singularities of the Fermi surface can show up in measurements. In the spectral theory of periodic elliptic operators, Bloch theory (also known as Floquet theory in the one-dimensional case) is the preferred tool. First of all it gives the band structure of the spectrum in a topological sense (locally nite union of closed intervals). Many elements from solid state physics show up, although named dierently. In this paper we focus on the notion of Fermi surface and its importance for spectral theory. Since it is dened by one real Institut fur Mathematik, Humboldt-Universitat zu Berlin, Berlin, Germany. mailto:gruber@mathematik.hu-berlin.de y epartment of Mathematics, MIT 2-167, 77 Massachusetts Avenue, Cambridge, MA , USA. mailto:mjg@math.mit.edu 1

2 equation one expects it to be of codimension one in general (in a suitable sense) and therefore would have zero measure as a subset of the Brillouin zone. On the other hand, for the 2-dimensional Schrodinger operator with constant magnetic eld and zero potential (the Landau operator) the spectrum consists of a discrete set of innitely degenerate eigenvalues, the Fermi 'surface' is either empty or given by the whole Brillouin zone (which has full measure, of course), depending on the Fermi energy. This is denitely not considered as band structure by physicists, but it is band structure in the topological sense mentioned above. One mathematical notion which distinguishes these situations is the measure theoretic type of the spectrum: absolutely continuous, pure point, or singular continuous. Physically this corresponds to band structure, break-down of band structure, and physically not occuring structure (in our context of smooth periodic potentials and absence of disorder). We dene a (quasi-) measure which associates to each energy interval the measure of the corresponding Fermi shell in momentum space. Its measure theoretic properties determine those of the spectrum. The aim of this paper is to show that the physically unexpected (singular continuity) does not occur mathematically for symmetric periodic elliptic dierential operators acting on sections of vector bundles over manifolds. This class of operators contains the Schrodinger operator with periodic magnetic eld and rational magnetic ux, the Pauli and the irac operator under the same conditions. The spectral history of these operators is of course long. Related to our work is: Thomas (1973) was the rst to show absence of eigenvalues for periodic Schrodinger operators in R n. Gerard (1990) discussed the extension properties of resolvents for these operators, using analytic families of Fredholm determinants (we will use real-analytic families of -regularized determinants). Hempel & Herbst (1995) attacked the periodic Schodinger operator with constant magnetic eld in R n and showed absence of singular continuous spectrum in the case of zero ux (we will follow their ideas). They were able to prove absence of eigenvalues for small enough magnetic eld also. This was improved by Birman & Suslina (1998) who proved absence of eigenvalues in the 2-dimensional case with minimal regularity assumptions (operators dened by quadratic forms). Sobolev (1999) proved the same in higher dimensions for more regular potentials. In this paper we will not deal with singular potentials but assume all coef- cients to be smooth. This allows to work in a more geometric context which pays o already in the Euclidean case as soon as the magnetic ux is non-zero. Outline In section 1 we recall the basic notions of Bloch theory for symmetric periodic elliptic operators acting on sections of vector bundles. In section 2 we dene Fermi surfaces and construct several measures (Fermi measure, integrated density of states). We analyze their mutual relations and their relations to the spectrum. 2

3 In section 3 we investigate the properties of the Fermi measure and draw our conclusions for the spectrum. Acknowledgements This work is a (commutative) part of my Ph.. thesis \Nichtkommutative Blochtheorie" (non-commutative Bloch theory; Gruber, 1998). I gratefully appreciate the advice and supervision given by Jochen Bruning at Humboldt- University at Berlin. This work was supported by eutsche Forschungsgemeinschaft as project 6 at the Sonderforschungsbereich 288 (dierential geometry and quantum physics). 1 Bloch theory on vector bundles In this section we recall the basic elements of Bloch theory for periodic operators in the geometric context of vector bundles, since even in the scalar case of a magnetic Schrodinger operator one is lead to consider possibly non-trivial complex line bundles. The standard reference for the theory of direct integrals is (ixmier, 1957, chapter II), for Bloch theory in Euclidean space see (Reed & Simon, 1978, chapter XIII.16). Our general assumptions are: X is an oriented smooth Riemannian manifold without boundary, a discrete abelian group acting on X freely, isometrically, and properly discontinuously. Furthermore, we assume the action to be cocompact in the sense that the quotient M : X is compact. Next, let E be a smooth Hermitian vector bundle over X. Example 1 (solid crystals). The main motivating example for our setting comes from solid state physics. Here, X R n is the conguration space of a single electron (n 2; 3). It is supposed to move in a crystal whose translational symmetries are described by a lattice n ' R n, which acts on X by translations, of course. Note that this does not take into account the point symmetries. could be extended by them but the action would not be free any more. Considering just the translations is enough to achieve the compactness of the quotient M ' T n : Wave functions of electrons are just complex-valued functions on R n, so we can set E R n C. One may also include the spin of the electrons into the picture by choosing the appropriate trivial spinor bundle E R n C 2. enition 1 (periodic operator). Assume there is an isometric lift of the action of fom X to E in the following sense: : E x! E x for x 2 X; 2 : (1) This denes an action T on the sections: For s 2 C 1 c (E) we dene (T s)(x) : s( 1 x) for x 2 X; 2 : (2) 3

4 (T ) 2 induces a unitary representation of in L 2 (E) since acts isometrically and T (T ) 1. A dierential operator on () : C 1 c (E) is called periodic if, on (), we have: 8 2 : [T ; ] 0 (3) Example 2 (periodic Schrodinger operator). Given a manifold as described above, we may lift the action to any trivial vector bundle E : X C k canonically. If is a periodic operator on X (for example any geometric operator, i.e. dened by the metric on X) and V 2 C 1 (X; M(k; C)) a periodic eld of endomorphisms, then + V is a periodic operator on E. In the case of a crystal, we choose the Laplacian (which describes the kinetic energy quantum mechanically) and a periodic potential V 2 C 1 (R n ; R) (which describes the electric eld of the ions at the lattice sites) to get the periodic Schrodinger operator + V. Example 3 (Schrodinger operator with exact periodic magnetic eld). Let b 2 2 (X) be a magnetic eld 2-form. In dimension 3 this corresponds (by the Hodge star) to a vector eld B, in dimension 2 to a scalar function which may be thought of as the length and orientation of a normal vector B. From physical reasons one has div B 0, i.e. db 0. For simplicity we assume that b is not only closed but exact, so there is a 2 1 (X) with b da (B rot A for the corresponding vector elds). This denes a magnetic Hamiltonian operator a : (d {a) (d {a) (4) (the minimally coupled Hamiltonian), where d is the ordinary dierential (corresponding to the gradient) and the adjoint of an operator between the Hilbert spaces of L 2 -functions L 2 (X) and of L 2-1-forms L 2 (X; T X). For later convenience we set, for 2 and! 2 (X),! : 1!, considering 1 as a map X! X and using the usual pull-back of forms. This puts the action on forms in a notation compatible with the action on sections (2) from the preceding denition. Now, if b is periodic, a does not need to be so: If b 2 2 (R n ) is constant then a is ane linear. So the translations are no symmetries for the magnetic Hamiltonian. ak (1968) was the rst to dene the so-called magnetic translations: Since d(a a) da da b b 0, one can (at least if H 1 (X) 0) nd a function with d a a. One may dene such a function explicitly by (x) : x x 0 (a a) which is well-dened if H 1 (X) 0. If we now dene a gauge function s : e { 4

5 then (d {a)(s f) s df + {(a a)s f {as f s df { a f s df {s (af) s ((d {a)f) : So we have found symmetries of the magnetic Hamiltonian operator, the gauged translations coming from the lifted action T : C 1 (X)! C 1 (X); (T s)(x) s (x)( s)(x) : X C! X C ; (x; c) (x; s (x)c) : The commutation relation for the magnetic translations is (T 1 T 2 s)(x) s 1 (x)s 2 ( 1 1 x)s(1 2 1x) 1 exp { exp { exp exp { { x x x 0 a 1 a + 1x 0 x 0 a 1 a + x x 0 1x 0 x 0 : ( 1 ; 2 )s 1 2 (x)s( x) ( 1 ; 2 )(T 1 2 s)(x)!! 1 1 x a a 2 x 0 1x 0 1 a ( 1 2 ) a s( x) x ( 1 2 ) a a + 1 a ( 1 2 ) a x 0 ( 1 2 ) a a 1 s 1 2 (x)s( 1 s( x) s( x) x) (5) with ( 1 ; 2 ) 2 S 1. In general this is just a projective representation of. If a itself is periodic, then 0 for 2, i.e. there is no gauge, and we have just ordinary translations forming a proper representation. But even if a is not periodic it can happen that the magnetic translations commute with each other. This is called the case of integral ux since the term occuring in the exponential in line (5) is just the magnetic ux through one lattice face. A periodic a obviously gives rise to zero magnetic ux. Furthermore, if V 2 C 1 (X; R) is -periodic it commutes with the magnetic translations as well, so a + V is a (symmetric elliptic) periodic operator. Finally, the very same magnetic translations can be used for the Pauli Hamiltonian and the magnetic irac operator. 5

6 Remark 1 (non-integral ux). If the magnetic ux is rational one can nd a superlattice of, i.e. a subgroup of nite index, such that the ux is integral. The quotient will still be compact, of course, so that the rational case can be completely reduced to the integral. If the magnetic ux is irrational there is no such superlattice. Still, one may try to make use of the projective representation dened above. There are several approaches, similar in the objects which are used, dierent in the objectives that are aimed at and accordingly in the results. For references and one approach which mimicks Bloch theory non-commutatively we refer to (Gruber, 1999a). Remark 2 (non-exact magnetic eld). If b is closed but not exact one rst has to agree upon the quantization procedure used. (4) may be identied as a Bochner Laplacian for a connection with curvature b, and such a connection exists if and only if b denes an integral cohomology class, i.e. [b] 2 H 2 (X; ). There may exist dierent quantizations for the same magnetic eld. This is connected to the Bloch decomposition again, For this and the construction of the magnetic translations in this case see (Gruber, 1999b). Lemma 1 (associated bundle). E is the lift E 0 of a Hermitian vector bundle E 0 over M by the projection : X! M. E and X are -principal ber bundles over E 0 resp. M. To every -principal ber bundle and every character 2 we associate a line bundle. This gives the relations depicted in the following diagram (\ " denotes association of line bundles.): y,! E,! X y y y C N C N C N C N! E 0 C,!E! E 0 y y E E y y 0! M C,! F! M principal ber bundles and associated line bundles In this situation we have E ' E 0 F. Proof. E is a -principal ber bundle, so we can use the lifted -action to dene E 0 : E. Since this action is a lift of the -action on X, E 0 has a natural structure of a vector bundle over M. If E0 : E 0! M is the bundle projection of E 0, then the pull back by is dened as E 0 X E 0 f(x; e) 2 X E 0 j (x) E0 (e)g: If E : E! X is the bundle projection of E and : E! E 0 is the quotient map, then we get a bundle isomorphism E! E 0 by E 3 e 7! ( E (e); (e)) 2 E 0 : 6

7 Therefore, in this representation the lift of acts on (x; e) 2 E 0 as (x; e) (x; e). Sections into an associated bundle P V are just those sections of the bundle P V which have the appropriate transormation property. By construction, E is a complex line bundle over E 0, but from E it inherits the vector bundle structure, so its sections fulll: C 1 (E ) ' C 1 (E) ; fs 2 C 1 (E) j 8 2 : s ()sg (6) An analogous equation holds for the line bundle F over M. Finally, (6) shows E E C ( E 0 ) C (X E 0 ) C ' E 0 (X C ) E 0 F : Here, all equalities are immediate from the denitions, besides the last but one, which may be seen as follows: with the -action whereas with the -action (X E 0 ) C (X E 0 C ) (x; e; z) (x; e; ()z); E 0 (X C ) E 0 ((X C )) (x; z) (x; ()z): So, both bundles are quotients of isomorphic bundles with respect to the same -action. Example 4 (magnetic bundles). Consider again the case of the magnetic translations for a periodic magnetic 2-form b 2 2 (X), E being a complex line bundle with curvature b (b 2 H 2 (X; )). Hence we have c 1 (E) [b] for the Chern class (up to factors of 2, depending on the convention). Since b is periodic we may restrict it to a form b M 2 2 (M) on the quotient. The existence of the lifted action, i.e. the fact that E can be written as a pull-back E E 0, corresponds to the integrality of b M from c 1 (E 0 ) [b M ] 2 H 2 (M; ). Tensoring E 0 with the at line bundle F does not change the Chern class (up to torsion). In particular, in dimension 2 the integrality of b M is equivalent to the integrality of the ux, and E 0 is trivial only for zero ux. 7

8 Next we want to decompose the Hilbert space L 2 (E) of square-integrable sections of E into a direct integral over the character space. On we use the Haar measure. From the theory of representations of locally compact groups we need the following character relations for abelian discrete, i.e. for abelian, compact (see e.g. Rudin, 1962, x1.5): Lemma 2 (character relations). For 2 1; e; () d 0; 6 e: For ; 0 2 X 2 in distributional sense, i.e. for f 2 C() X 2 (7) () 0 () ( 0 ) (8) () 0 ()f() d f( 0 ): We dene for every character 2 a mapping : C 1 c (E) 3 s 7! ~s 2 C 1 (E) by ~s (x) : X 2 () s( 1 x): (9) Since ~s ( 0 x) X 2 () s( 1 0 x) X 2 ( 0 01 )( 0 01 ) s ( 01 ) 1 x we have ( 0 ) 0 ~s (x) ~s 2 C 1 (E) ; fr 2 C 1 (E) j 8 2 T r ()rg which denes a section s 2 C 1 (E ). Let be S a fundamental domain for the -action, i.e. an open subset of X such that X up to a set of measure 0 and \ ; for 6 e. 2 Then ks k 2 L 2 (E ) d j~s (x)j 2 dx d X 2 ksk 2 L 2 (E) : X 1; 22 ( 1 1 2)h 1 s( 1 1 x) j 2s( 1 2 x)i Ed dx js( 1 x)j 2 dx 8

9 On Q the one hand, this shows that we can dene a measurable structure on 2 L2 (E ) by choosing a sequence in Cc 1(E) which R is total in L2 (E). On the other hand, we can see that the direct integral L2 (E ) d is isomorphic to L 2 (E) via the isometry, whose inverse is given by : (s ) 2 7! ~s (x) d; as is easily seen from the character relations (7) and (8). This shows Lemma 3 (direct integral). The mapping dened by (9) can be extented continuously to a unitary : L 2 (E)! L 2 (E ) d: (10) For the direct integral of Hilbert spaces H R H d the set of decomposable bounded operators L 1 (; L(H)) is given by the commutant (L 1 (; C)) 0 in L(H). Since commutants are weakly closed and C(; C ) is weakly dense in L 1 (; C) one has (L 1 (; C )) 0 (C(; C)) 0. Therefore, in order to determine the decomposable operators one has to determine the action of C() on L 2 (E). This is easily done using the explicit form of : Proposition 1 (C()-action). f 2 C() acts on s 2 C 1 c (E) by and one has M f s : fs; (11) (M f s)(x) X 2 ^f() : ^f( 1 )T s(x); where (12) f() () d (13) is the Fourier transform of f. M f is a bounded operator with norm kfk 1. Proof. For x 2 X one has: (M f s)(x) ( fs)(x) (fs) (x) d f() X 2 () s( 1 x) d X 2 ^f( 1 ) s( 1 x) Since f is a multiplication operator in each ber it has berwise norm kfk 1, and so have f and M f f. 9

10 Corollary 1 (decomposable operators). Conjugation R by denes an isomorphism between decomposable bounded operators on L2 (E ) d and - periodic bounded operators on L 2 (E). Proof. \)" A decomposable operator commutes with the C()-action, especially with f 2 C() which is dened by ^f ( 0 ) : ( 1; if 0 ; 0 else. By (12) commuting with f is equivalent to commuting with. \(" To commute with the -action means to commute with all f for 2. Because of f () () the f are just the characters b of the compact group, and by the Peter-Weyl theorem (or simpler: by the Stone-Weierstra theorem) they are dense in C(). Since the operator norm of M f and the supremum norm of f coincide the commutation relation follows for all f 2 C() by continuity. An unbounded operator is decomposable if and only if its (bounded) resolvent is decomposable. For a periodic symmetric elliptic operator we have a domain of denition () C 1 c (X) on which is essentially self-adjoint. This domain is invariant for as well as for the -action, and one has [; ] 0 for all 2. Thus all bounded functions of commute with the -action, and one has: Theorem 1 (decomposition of periodic operators). The closure of every periodic symmetric R elliptic operator is decomposable with respect to the direct integral L2 (E ) d. A core for the domain of is given by C 1 (E ), and the action of on C 1 (E ) ' C 1 (E) ; is just the action of as differential operator on C 1 (E) ;. We have, where : j C 1 (E); (14) and the closures are to be taken as operators in L 2 (E ). Proof. Given the remark above we have shown the decomposability already. C 1 c (X) is a core for, its image under is contained in C 1 (E) ; and is a core for, sinece is an isometry. On this domain (9) gives the action of as asserted in the theorem. Since is a symmetric elliptic operator on the compact manifold M it is essentially self-adjoint. is a ber of (which is self-adjoint by, e.g., Atiyah, 1976) and therefore self-adjoint, thus both dene the same unique self-adjoint extension of. 10

11 In passing we harvest a corollary which we will not use in the sequel, but which is well known in the Euclidean setting: Corollary 2 (reverse Bloch property). Every symmetric elliptic abelian periodic operator has the reverse Bloch property, i.e. to every 2 spec there is a bounded generalized eigensection s 2 C 1 (E) with s s: Proof. If 2 spec then, by the general theory for direct integrals, f 2 j ( "; + ") \ spec 6 ;g has positive measure for every " > 0. The bers are elliptic operators on a compact manifold and thus have discrete spectrum; the eigenvalues depend continuously on (even piecewise real-analytically; see below). We choose a sequence ( n ) n2nwith ( 1n; + 1n) \ spec n 6 ;, so that there is an accumulation point 1 ( is compact), and 2 spec 1 due to continuity. Since spec 1 is discrete is an eigenvalue of 1. The lift of an eigensection (which is smooth due to ellipticity) lies in C 1 (E) ; and therefore is bounded. Furthermore the lift satises the same eigenvalue equation because of (14). 2 Measures of Fermi surfaces In the previous section we noticed that the spectrum of an abelian-periodic operator can be computed form the spectra of its Bloch bers which have discrete, nitely degenerate spectra. In this section we will assume that is bounded from below since certain tools which we use can only be dened in this case. In our main theorem about the spectrum (section 3, theorem 2) we will show how to reduce the general case to the semi-bounded. enition 2 (Fermi surface). Let be an abelian-periodic symmetric operator, its Bloch bers, and E 0 () E 1 () E 2 () : : : the corresponding eigenvalues repeated according to multiplicity. E n :! R; 7! E n () (15) denes the n-th energy band (in the \ reduced zone scheme"). The Fermi surface of the n-th band at energy E is dened to be the Fermi surface at energy E is F (E) : [ F n (E) : f 2 j E n () Eg; (16) n2n0 F n (E) (17) f 2 j E 2 spec g: 11

12 \Usually" the Fermi surface is of codimension 1 (if this notion makes sense) so that its measure vanishes, whereas E is an eigenvalue of if and only if meas F (E) > 0 (from the direct integral). enition and proposition 1 (Fermi measure). For every Borel set B R of energies we dene the measure of [ its Fermi shell by F () (B) : meas F (E) (18) E2B measf 2 j spec \ B 6 ;g This is a quasi-measure (i.e. it is a measure up to additivity which is replaced by sub-additivity). E is an atom of F () if and only if F () (feg) > 0 if and only if E is an eigenvalue of. Proof. First, the energy band functions are measurable since they are the eigenvalue family of a decomposable operator in a direct integral; so the Fermi surface and Fermi shell are measurable. Since the band functions are continuous and is bounded, the measure is nite on bounded sets. The statement about the atoms follows from the remark above, referring to the basic properties of direct integrals, since by denition d (B) meas F (E). If E 1 and E 2 are dierent it may happen that F (E 1 ) and F (E 2 ) intersect, but still meas F (E 1 ) [ F (E 2 ) meas F (E 1 ) + meas F (E 2 ) and analogous for disjoint Borel sets, so that F () is sub-additive. enition 3 (integrated density of states). Let P (E) be the spectral projections of. The integrated density of states (IS) of is dened by N (E) : tr P (E) (19) : tr P (E) d tr P (E) d The IS denes a Lebesgue{Stieltjes measure N such that N ((1; E]) N (E). For f 2 L 2 (E) we have the spectral measure at f given by f (B) 1 B d f P (E)f (20) f P B f (21) f d (22) for Borel sets B R. This is the usual denition from spectral theory. 12

13 enition 4 (determinant measure). ( ) 2 is a family of semi-bounded elliptic dierential operators. Then we have the family (d (; )) 2 of - regularized determinants d (; ) det( ). The associated quasi-measure is dened by for every Borel set B R. d (B) : measf 2 j 9 2 B : d (; ) 0g (23) The fact that d is a quasi-measure can be seen as for the Fermi measure. We will show that they coincide anyway. To nish this section we investigate the relations between the various measures: Lemma 4 (relations between measures). For the measures dened above we have the following relations: 1. d F () 2. N and F () are equivalent, i.e. they have the same null-sets (and atoms). 3. f is continuous with respect to N : f (B) kfk2 N (B) Proof. 1. d F () just by denition, since d (E; ) 0 if and only if E 2 spec. 2. For every Borel set B R we have: and tr P B N (B) tr P B d 6 0, PB 6 0, B \ spec 6 ;, 9E 2 B : d (E; ) f f P B E f kf k 2 tr P B 13

14 3 Spectral consequences So far the operators act on dierent bundles, therefore a priori on dierent domains. We want to construct an isomorphism S : F 0! F of Hermitian line bundles such that 0 : (1 E 0 S) (1 E 0 S ) becomes a real-analytic family of operators on E 0. We get the real-analytic structure on by choosing an isomorphism ' T n, where is the (discrete) torsion part. Then the real-analytic standard structure of the n-torus. (As already mentioned, is nitely generated, so that indeed n < 1.) T n induces a real-analytic structure on. Note that in our situation the sequence of coverings induces the exact sequences ~X k ~M 1 (X)! X! M k 1 (M )! M 0! 1 (X)! 1 (M)!! 0 0 \ 1 (X) \ 1 (M) 0 of groups. Here, ^G denes the character group of a locally compact topological group G. So we can embed the characters from (injectively) into \ 1 (M). This allows to consider the bundle F X C as a bundle M ~ C associated to M: ~ Proposition 2 (real-analytic family). Let 0 2. Then for every in the connected component of 0 there exists an isomorphism S : F 0! F of Hermitian line bundles. If we dene 0 : (1 E 0 S ) (1 E 0 S ) then ( 0 ) 2 becomes a real-analytic family with constant domain of denition (family of type A). Proof. First assume 0 1. Let ~x 0 2 M ~ be an arbitraily choosen base point. For! 2 1 (M); d! 0; denote by ~! 2 1 ( M) ~ the pull back to the covering space. Then ~x! ~s! (~x) : exp { ~! ~x 0 is well-dened since d~! 0 and H 1 ( ~ M; ) 0. For 2 1 (M) one has ~s! (~x) exp { ~s! exp ~x ~x 0 ~! + { ~s! (~x)! () c()!! ~x ~x ~!! 14

15 with the character! 2 \ 1 (M), dened by! () : exp 2{ for some choice of a closed path c() representing the class 2 1 (M). This is well-dened because! is closed. On the other hand it is easily seen (see e.g. Katsuda & Sunada, 1987; Gruber, 1999b) that every character in \ 1 (M) 0 may be represented in the form! for some!. Especially this is possible for every character in () 0. Thus ~s! denes a section of s! 2 C 1 ( M ~! C ) with pointwise norm 1 and hence an Hermitian line bundle isomorphism so that c()!! S! : M C! ~ M! C ; f 7! fs! 1 E 0 S : E 0! E! : Finally d~s! (~x) {~s! ~!. Therefore we have on C 1 0 (M) 0! : (1 E 0 S! )! (1 E 0 S! ) + T! ; where T! is a dierential operator of order ord() 1 with smooth bounded coecients; these are real-analytic in because they contain! only polynomially. So the domain of 0 is the Sobolev space W ord() (E 0 ), independent of. If is in an arbitrary connected component of we can use E 0 ' E 0 (M C ) and get an Hermitian line bundle isomorphism 1 E0 S! : E 0! E 0! with the same properties as above. So we have the desired real-analytic family over every connected component. Lemma 5 (-regularized determinant). If is bounded-below then every ber 0 has a -regularized determinant d (; ) det ( 0 ) such that d (; ) 0, 2 spec 0. d is real-analytic in and analytic in. 15

16 Proof. 0 is bounded below ( C ) C because of (14)), selfadjoint, elliptic, and dened on a compact manifold. Therefore it has a - regularized determinant d (; ) det ( 0 ), which is analytic in and fullls d (; ) 0, 2 spec 0 (see, e.g., Gilkey, 1995). Since the family ( 0 ) 2 depends real-analytically on the coecients in the heat trace asymptotics do so. The asymptotics is uniform in, so the -function as well as the -regularized determinant depend real-analytically on. From Hempel & Herbst (1995) we cite Lemma 6 (real analyticity and measure). 1. Let O R n be open and connected, h : O! C real-analytic. If h 1 (f0g) has zero Lebesgue measure then h Let O R n be open and connected, I Ran open intervall, f : I O! C real-analytic, f 6 0. For Borel sets B R f (B) measfk 2 O j 9 2 B : f(; k) 0g (24) denes a quasi-measure. Then we have (a) f has no singular continuous component (i.e. there is no decomposition containing a singular continuous component). (b) The atoms of f are discrete in R. Theorem 2 (spectral nature of abelian-periodic operators). Let be a symmetric elliptic abelian-periodic dierential operator. Then we have for the spectrum of : 1. The singular continuous spectrum is empty. 2. The density of states measure N has no singular continuous component. 3. The point spectrum is discrete as subset of R. 4. If is an eigenvalue of then there exists a connected component of such that for 2 : 2 spec. Proof. First we show that we may assume 0 without loss of generality. For this we note that the spectral projections P and P 2 of and 2 are related by: P 2 () 1 (1;] 2 1 (1;] () 2 1 [ sgn p ; sgnp ] P (P p P p [ sgnp ; sgnp ] ; if 0; 0 else, (25) 16

17 where we set sgnp : sgn p jj. For the point spectrum as well as the whole spectrum as a set we can use the spectral mapping theorem: spec p 2 spec p 2 ; spec 2 spec 2 Because of the continuity of () 2 and p this implies spec p:p: 2 spec p:p: 2, da spec p:p: spec p. For 3 and 4 it therefore suces to consider 2. If 2 has no singular continuous spectrum then, given a set B with zero Lebesgue measure and B Rn spec p:p: 2 and an arbitrary s 2 2 n 0 n 0, the spectral measure s;s(b) 2 at s vanishes on B. If we put q B : ~B : n o 2 R j 2 2 B ~ o n 2 R j jj 2 ~ B q j ~ Bj [ j ~ Bj B 2 [ B 2 ~ B [ ; B ~ B ~ for a Lebesgue zero-set ~ B Rn spec p:p: then B has Lebesgue measure zero and contains, by the spectral mapping theorem, no elements of spec p:p: 2. From (25) we get 0 2 s;s(b) E sp 2 ()s 1 B () d sp p E s 1 B 2 d s P ()s R 1 B() d R+ R+ R 1 B s;s s;s 2 d s P ()s ~B ~B : R+ R+ 1 B () d sp p 1 B ( 2 ) d s P ()s Since 2 is dense one has spec s:c: ;. So, even for 1 it is sucient to consider 2. Finally, by assumption 2 is a symmetric elliptic abelian-periodic dierential operator. Given because of the essential self-adjointness we can assume 0 in the following. E s Let d be the real-analytic family of -determinants of which exists by lemma 6. 17

18 4. If 2 spec p ( ) then, by the general theory for direct integrals, 0 < meas(f 2 j 2 spec p ( g) meas(f 2 j d (; ) 0g) meas(h 1 (f0g)) with h d (; ): Since has only nitely many connected components there must be a component with meas(h 1 (0) \ ) > 0. Then hj fullls the assumptions of lemma 6.1 so that hj 0 and thus 2 spec for As above 2 spec p, meas(f 2 j 2 spec p ( g) > 0, meas(f 2 j d (; ) 0g) > 0, d (fg) > 0, is an atom of d : But by lemma 6.2b the atoms are discrete. 1. Consider a Borel set B R with B \ spec p ; such that B contains no atoms of d. If meas(b) 0 then d (B) 0 since d has, by lemma 6, 2a, no singular continuous component with respect to the Lebesgue measure meas. Since N and d are equivalent (by lemma 4, 1. and 2.) we have N (B) 0. Since f is continuous with respect to N (by lemma 4, 3.) we conclude f (B) 0. Thus f has no singular continuous component with respect to meas. 2. From 1. we get the assertion about the density of states, since N and F () are equivalent by lemma 4.2. Remark 3. For the periodic Schrodinger operator one has absolutely continuous spectrum only. On the other hand, the Landau operator (constant magnetic eld, dimension 2) has only pure point spectrum. This shows that the theorem is optimal with respect to possible measure types under the given conditions. Remark 4. As is well known, any possibly occuring eigenvalue automatically has innite degeneracy (from periodicity under the innite group). So the spectrum is purely essential. References Atiyah, M. F. (1976). Elliptic operators, discrete groups and von Neumann algebras. Asterisque 32{33, 43{72 18

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