Essential spectra of difference operators on $\sZ^n$-periodic graphs
aa r X i v : . [ m a t h - ph ] A p r Essential spectra of difference operators on Z n -periodic graphs V. S. Rabinovich, S. RochVladimir S. Rabinovich, Instituto Polit´ecnico Nacional,ESIME-Zacatenco, Av. IPN, edif.1, M´exico D.F.,07738, M ´EXICO, e-mail: [email protected]ffen Roch, Technische Universit¨at Darmstadt,Schlossgartenstrasse 7, 64289 Darmstadt, Germany,e-mail: [email protected]
Abstract
Let ( X , ρ ) be a discrete metric space. We suppose that the group Z n acts freely on X and that the number of orbits of X with respect to thisaction is finite. Then we call X a Z n -periodic discrete metric space. Weexamine the Fredholm property and essential spectra of band-dominatedoperators on l p ( X ) where X is a Z n -periodic discrete metric space. Ourapproach is based on the theory of band-dominated operators on Z n andtheir limit operators.In case X is the set of vertices of a combinatorial graph, the graphstructure defines a Schr¨odinger operator on l p ( X ) in a natural way. Weillustrate our approach by determining the essential spectra of Schr¨odingeroperators with slowly oscillating potential both on zig-zag and on hexag-onal graphs, the latter being related to nano-structures. In the last years, spectral properties of Schr¨odinger operators on quantumgraphs have attracted a lot of attention due to their interesting mathemati-cal properties and due to existing and expected applications in nano-structuresas well (see, for instance, [4, 9, 38]). Quantum graph models also occur in chem-istry [26, 37] and physics [4, 15] (see also the references therein). The spectralproperties of Schr¨odinger operators on quantum graphs considered by P. Kuch-ment and collaborators in a series of papers [14, 15, 16, 17, 18, 19]. Direct andinverse spectral problems for Schr¨odinger operators on graphs connected withzig-zag carbon nano-tubes was considered in [12, 13].It was shown in [16, 17] that the spectral analysis of quantum Hamiltonianon periodic graphs splits into two parts: the spectral analysis of a Hamiltonianon a single edge, and the spectral analysis on a combinatorial graph. This1bservation makes difference operators on combinatorial graphs to an essentialtool in the theory of differential operators on quantum graphs.The main theme of this paper is the essential spectrum of difference oper-ators (with the Schr¨odinger operators as a prominent example) acting on thespaces l p ( X ) where X is the set of the vertices of a combinatorial graph Γ. Weexclusively consider discrete graphs Γ on which the group Z n acts freely andwhich have a finite fundamental domain with respect to this action.We introduce a Banach algebra A p ( X ) of so-called band-dominated differ-ence operators l p ( X ) for 1 < p < ∞ . Following [31, 32] and [33], we introducefor each operator A ∈ A p ( X ) a family op p ( A ) of limit operators of A , and weshow that an operator A ∈ A p ( X ) is Fredholm on l p ( X ) if and only if all oper-ators in op p ( A ) are invertible and if the norms of their inverses are uniformlybounded. In general, the limit operators of an operator A are simpler objectsthan the operator A itself. Thus, the limit operators method often provides aneffective tool to study the Fredholmness of operators in A p ( X ).For operators in the so-called Wiener algebra W ( X ) (which is a non-closedsubalgebra of every algebra A p ( X ), the uniform boundedness of norms of inverseoperators to limit operators follows already from their invertibility. This basicfact implies the useful identitysp ess A = [ A h ∈ op A sp A h (1)where the set of the limit operators of A , the spectra sp A h of the limit operatorsof A and, hence, also the essential spectrum sp ess A of A are independent of p .In case X = Z n , formula (1) was obtained in [31], see also [33]. In [29],we applied this formula to study electromagnetic Schr¨odinger operators on thelattice Z n . In particular, we determined the essential spectra of the Hamiltonianof the 3-particle problem on Z n .In [27], one of the authors obtained an identity similar to (1) for perturbedpseudodifferential operators on R n . He applied this result to study the locationof the essential spectra of electromagnetic Schr¨odinger operators, square-rootKlein-Gordon, and Dirac operators under general assumptions with respect tothe behavior of magnetic and electric potentials at infinity. By means of thismethod, also a very simple and transparent proof of the well known Hunziker,van Winter, Zjislin theorem (HWZ-Theorem) on the location of essential spectraof multi-particle Hamiltonians was obtained.It should be noted that formulas similar to (1) have been obtained indepen-dently (but later) in [21] by means of admissible geometric methods. We alsomention the papers [8, 7, 23, 3] and the references therein where C ∗ -algebratechniques have been applied to study essential spectra of Schr¨odinger opera-tors.The present paper is organized as follows. In Section 2 we collect some aux-iliary material from [33] on matrix band-dominated operators on the lattice Z n .In Section 3 we introduce the Banach algebra A p ( X ) of band-dominated oper-ators acting on l p ( X ) where X is a periodic discrete metric space on which the2roup Z n acts freely. We construct an isomorphism between the Banach alge-bra A p ( X ) and the Banach algebra A p ( Z n , C N ) of all (block) band-dominatedoperators on l p ( Z n , C N ) where N is the number of points in the fundamen-tal domain of X with respect to the action of Z n . Applying this isomorphismand the results of Section 2, we derive necessary and sufficient conditions for A ∈ A p ( X ) to be a Fredholm operator. We also introduce a Wiener algebra W ( X ) and derive formula (1) for operators in W ( X ).In Section 4 we introduce the class of periodic band-dominated operators.We say that A ∈ A p ( X ) is a periodic operator if it commutes with each op-erator L h of left shift by h ∈ Z n on l p ( X ). Note that, for periodic operators,sp ess A = sp A . With each periodic operator A ∈ W ( X ), we associate a contin-uous function σ A : T n → C N × N , called the symbol of A . In the terminologyof [15, 16], σ A ( t ) is just the Floquet transform of A . We prefer to follow thetheory of discrete convolutions and use the discrete Fourier transform to define σ A .Let λ j ( t ), j = 1 , . . . , N be the eigenvalues of σ A ( t ). Thensp A = N [ j =1 C j ( A )where C j ( A ) := { λ ∈ C : λ = λ j ( t ) , t ∈ T n } . If A is a self-adjoint operator on l ( X ), then the C j ( A ) can be identified with segments.In Section 5 we consider operators in the Wiener algebra W ( X ) with slowlyoscillating coefficients. These operators are distinguished by two remarkableproperties: their limit operators are periodic operators, and all limit operatorsbelong to the Wiener algebra again. Via formula (1) we thus obtain a com-plete description of the essential spectra of operators with slowly oscillatingcoefficients.In Section 6 we apply these results to Schr¨odinger operators with slowlyoscillating electrical potentials. As already mentioned, every Z n -periodic graphinduces a related Schr¨odinger operator in a natural way (it is only this placewhere the graph structure becomes important). As illustrations we calculate theessential spectra of Schr¨odinger operators with slowly oscillating potentials onthe zig-zag graph and on the hexagonal graph. Some other spectral problems onsuch graphs which are connected with carbon nano-structures were consideredin [12, 13, 18].In Section 7 we examine the essential spectrum of the Hamiltonian of themotion of two particles on a periodic graph Γ around a heavy nucleus. Forthe lattice Γ = Z n we considered this problem in [29]. See also the papers[2, 1, 20, 24, 25] and the references therein which are devoted to discrete multi-particle problems.The limit operators approach does also apply to study the essential spectrumof pseudodifferential operators on periodic quantum graphs. We plan to developthese ideas in a forthcoming paper.The authors are grateful for the support by CONACYT (Project 43432) andby the German Research Foundation (Grant 444 MEX-112/2/05).3 Band-dominated operators on Z n In this section we fix some notations and recall some facts concerning the Fred-holm property of band-dominated operators on l p ( Z n ). The Fredholm propertiesof these operators are fairly well understood. All details can be found in [31];see also the monograph [33] for a comprehensive account.We will use the following notations. Given a Banach space X , let L ( X )refer to the Banach algebra of all bounded linear operators on X and K ( X )to the closed ideal of the compact operators. An operator A ∈ L ( X ) is calleda Fredholm operator if its kernel ker A := { x ∈ X : Ax = 0 } and its cokernelcoker A := X/A ( X ) are finite dimensional linear spaces. Equivalently, A isFredholm if the coset A + K ( X ) is invertible in the Calkin algebra L ( X ) / K ( X ).The essential spectrum of A is the set of all complex numbers λ for whichthe operator A − λI is not Fredholm on X , whereas the discrete spectrumof A consists of all isolated eigenvalues of finite multiplicity. We denote theessential spectrum of A by sp ess A , the discrete spectrum by sp dis A , and theusual spectrum by sp A . Sometimes we also write sp ( A : X → X ) instead ofsp A in order to emphasize the underlying space X (with obvious modificationsfor the essential and the discrete spectrum). Clearly,sp dis ( A ) ⊆ sp ( A ) \ sp ess ( A )for every operator A ∈ L ( X ). If A is a self-adjoint operator, then equality holdsin this inclusion.Let p ≥ n a positive integer. As usual, we write l p ( Z n ) for the Banach space of all functions u : Z n → C for which k u k pl p ( Z n ) := X x ∈ Z n | u ( x ) | p < ∞ and l ∞ ( Z N ) for the Banach space of all bounded functions u : Z n → C withnorm k u k l ∞ ( Z n ) := sup x ∈ Z n | u ( x ) | . For every positive integer N , let l p ( Z n ) N stand for the Banach space of allvectors u = ( u , . . . , u N ) of functions u i ∈ l p ( Z n ) with norm k u k pl p ( Z n ) N := N X i =1 k u i k pl p ( Z n ) Likewise, one can identify l p ( Z ) N with the Banach space l p ( Z n , C N ) of all func-tions u : Z n → C N for which k u k pl p ( Z n , C N ) := X x ∈ Z n N X i =1 | u j ( x ) | p < ∞ . l p ( Z n ) N and l p ( Z n , C N ) are isometric to each other.We also consider the Banach spaces l ∞ ( Z n ) N and l ∞ ( Z n , C N ) with norms k u k l ∞ ( Z n ) N := sup ≤ i ≤ N k u i k l ∞ ( Z n ) and k u k l ∞ ( Z n , C N ) := sup x ∈ Z n sup ≤ i ≤ N | u i ( x ) | . Again, these spaces are isometric to each other in a natural way. Note also that l ∞ ( Z n , C N × N ) can be made to a C ∗ -algebra by providing the matrix algebra C N × N with a C ∗ -norm.We consider operators on l p ( Z n , C N ) which are constituted by shift operatorsand by operators of multiplication by bounded functions. The latter are definedas follows: For α ∈ Z n , the shift operator V α is the isometry acting on l p ( Z n , C N )by ( V α u )( x ) := u ( x − α ). Further, each function a in l ∞ ( Z n , C N × N ) induces amultiplication operator aI on l p ( Z n , C N ) via ( au )( x ) := a ( x ) u ( x ). Clearly, k aI k L ( l p ( Z n , C N )) = k a k l ∞ ( Z n , C N × N ) . A band operator on l p ( Z n , C N ) is an operator of the form A = X | α |≤ m a α V α (2)with coefficients a α ∈ l ∞ ( Z n , C N × N ). The closure in L ( l p ( Z n , C N )) of the set ofall band operators is a subalgebra of L ( l p ( Z n , C N )). We denote this algebra by A ( l p ( Z n , C N )) and call its elements band-dominated operators (BDO for short).In a completely analogous way, band-dominated operators on l ∞ ( Z n , C N ) aredefined.Our main tool to study Fredholm properties of band-dominated operatorsare the associated limit operators. Definition 1
Let A ∈ L ( l p ( Z n , C N )) , and let h : N → Z n be a sequence tendingto infinity. A linear operator A h is called the limit operator of A with respect tothe sequence h if V − h ( m ) AV h ( m ) → A h and V − h ( m ) A ∗ V h ( m ) → A ∗ h strongly as m → ∞ . We let op p A denote the set of all limit operators of A . Here and in what follows, convergence of a sequence in Z n to infinity meansconvergence of this sequence to infinity in the one-point compactification of Z n (which makes sense since Z n is a locally compact metric space).There are operators on l p ( Z n , C N ) which do not possess limit operators atall. But if A is a band-dominated operator then one can show via a Cantordiagonal argument that every sequence h tending to infinity has a subsequence g for which the limit operator A g exists. Moreover, the operator spectrum of5 stores the complete information on the Fredholmness of A , as the followingtheorem states. (In case n = 1 there is also a sufficiently nice formula for theFredholm index of A which expresses this index in terms of local indices of thelimit operators of A , see [30].) Theorem 2
An operator A ∈ A ( l p ( Z n , C N )) is Fredholm if and only if all limitoperators of A are invertible and if sup A h ∈ op p ( A ) k A − h k < ∞ . (3)The uniform boundedness condition (3) is often difficult to check: It is onething to verify the invertibility of an operator and another one to provide agood estimate for the norm of its inverse. It is therefore of vital importanceto single out classes of band-dominated operators for which this condition isautomatically satisfied. One of these classes is defined by imposing conditionsof the decay of the norms of the coefficients. More precisely, we consider band-dominated operators of the form A := X α ∈ Z n a α V α where X α ∈ Z n k a α k l ∞ ( Z n , C N × N ) < ∞ . (4)One can show that the set W ( Z n , C N ) of all operators with property (4) formsan algebra and that the term on the left-hand side of (4) defines a norm whichmakes W ( Z n , C N ) to a Banach algebra. We refer to this algebra as the Wieneralgebra and write k A k W ( Z n , C N ) for the norm of an operator in W ( Z n , C N ).Clearly, operators in the Wiener algebra act boundedly on each of the spaces l p ( Z n , C N ) (including p = ∞ ) and k A k L ( l p ( Z n , C N )) ≤ k A k W ( Z n , C N ) . Hence, W ( Z n , C N ) ⊆ A ( l p ( Z n , C N )) for every p .One important property of the Wiener algebra is its inverse closedness ineach of the algebras L ( l p ( Z n , C N )), i.e., if A ∈ W ( Z n , C N ) has an inverse in L ( l p ( Z n , C N )), then A − belongs to W ( Z n , C N ) again. This fact implies thatthe spectrum of an operator A ∈ W ( Z n , C N ) considered as acting on l p ( Z n , C N )does not depend on p ∈ (1 , ∞ ). Also the operator spectrum op p ( A ) proves tobe independent of p , which justifies to write op A instead. Note finally that alllimit operators of operators in the Wiener algebra belong to the Wiener algebraagain.For operators in the Wiener algebra, the Fredholm criterion in Theorem 2reduces to the following much simpler assertion. Theorem 3
Let A ∈ W ( Z n , C N ) . The operator A is Fredholm on l p ( Z n , C N ) if and only if there exists a p ∈ [1 , ∞ ] such that all limit operators of A areinvertible on l p ( Z n , C N ) . Theorem 4
For A ∈ W ( Z n , C N ) , the essential spectra of A : l p ( Z n , C N ) → l p ( Z n , C N ) do not depend on p ∈ (1 , ∞ ) , and sp ess A = [ A h ∈ op A sp A h . (5) By a discrete metric space we mean a countable set X together with a metric ρ such that every ball B r ( x ) := { x ∈ X : ρ ( x, x ) ≤ r } is a finite set. For each discrete metric space X , we introduce some standardBanach spaces over X . For p ∈ (1 , ∞ ), let l p ( X ) denote the Banach space ofall complex-valued functions u on X with norm k u k pl p ( X ) := X x ∈ X | u ( x ) | p , and write l ∞ ( X ) for the Banach space of all bounded functions u of X withnorm k u k l ∞ ( X ) := sup x ∈ X | u ( x ) | A periodic discrete metric space is a discrete metric space provided with the freeaction of the group Z n . More precisely, let X be a discrete metric space, andlet there be a mapping Z n × X → X, ( α, x ) → α · x satisfying 0 · x = x and ( α + β ) · x = α · ( β · x )for arbitrary elements α, β ∈ Z n and x ∈ X , which leaves the metric invariant, ρ ( α · x, α · y ) = ρ ( x, y ) (6)for all elements α ∈ Z n and x, y ∈ X . Recall also that the group Z n acts freely on X if whenever the equality x = α · x holds for elements x ∈ X and α ∈ Z n then, necessarily, α = 0.For each element x ∈ X , consider its orbit { α · x ∈ X : α ∈ Z n } with respectto the action of Z n . Any two orbits are either disjoint or identical. Hence, thereis a binary equivalence relation on X , by calling two points equivalent if theybelong to the same orbit. The set of all orbits of X with respect to the action of7 n is denoted by X/ Z n . A basic assumption throughout what follows is that theorbit space X/ Z n is finite . Thus, there is a finite subset M := { x , x , . . . , x N } of X such that the orbits X j := { α · x j ∈ X : α ∈ Z n } satisfy X i ∩ X j = ∅ if x i = x j and ∪ Ni =1 X i = X . If all these conditions aresatisfied then we call X is a periodic discrete metric space with respect to Z n orsimply Z n -periodic.The free action of Z n on X guarantees that the mapping U j : Z n → X j , α α · x j is a bijection for every j = 1 , . . . , N . For each complex-valued function f on X , let U f : Z n → C N be the function( U f )( α ) := (( U f )( α ) , . . . , ( U N f )( α )) . Clearly, the mapping U is a linear isometry from l p ( X ) onto l p ( Z n , C N ), andthe mapping A U AU − is an isometric isomorphism from L ( l p ( X )) onto L ( l p ( Z n , C N )) for every p ∈ [1 , ∞ ].Another consequence of our assumptions is thatlim Z n ∋ α →∞ ρ ( α · x, y ) = ∞ . (7)for all points x, y ∈ X . Indeed, suppose that (7) is wrong. Then there arepoints x, y ∈ X , a positive constant M , and a sequence α of pairwise differentpoints in Z n such that ρ ( α ( n ) · x, y ) ≤ M for all n ∈ N . (8)The free action of Z n on X implies that ( α ( n ) · x ) n ∈ N is a sequence of pairwisedifferent points in X . Hence, (8) implies that the ball with center y and radius M contains infinitely many points, a contradiction. X Let X be a periodic discrete metric space and p ∈ [1 , ∞ ). We consider linearoperators A on l p ( X ) for which there exists a function k A ∈ l ∞ ( X × X ) suchthat ( Au )( x ) = X y ∈ X k A ( x, y ) u ( y ) for all x ∈ X (9)and for all finitely supported functions u on X (note that the latter form adense subspace of l p ( X )). The function k A is called the generating function of the operator A . It is easily seen that every bounded operator A on l p ( X )is of this form and is, thus, generated by a bounded function. The converse is8ertainly not true. It is also clear that every operator A determines its generatingfunction uniquely, since ( Aδ y )( x ) = k A ( x, y )where δ y is the function on X which is 1 at y and 0 at all other points.An operator A of the form (9) is called a band operator if there exists an R > k A ( x, y ) = 0 whenever ρ ( x, y ) > R . Example 5
Every operator aI of multiplication by a function a ∈ l ∞ ( X ) is aband operator. Example 6
For α ∈ Z n , let T α be the operator of shift by α on l p ( X ), i.e.,( T α u )( x ) := u (( − α ) · x ). Clearly, T α is a band operator which acts as an isometryon l p ( X ). Hence, every operator of the form X | α |≤ m a α T α (10)with a α ∈ l ∞ ( X ) is a band operator (but there are band operators which cannot be represented of this form). Proposition 7 If A is a band operator on l p ( X ) , then U AU − is a band oper-ator on l p ( Z n , C N ) . Proof.
The operator
U AU − has the matrix representation( U AU − f ) i ( α ) = N X j =1 X β ∈ Z n r ijA ( α, β ) f j ( β ) (11)where α ∈ Z n , i = 1 , . . . , N and r ijA ( α, β ) := k A ( α · x i , β · x j ) . (12)From (7) we conclude that ρ ( α · x i , β · x j ) = ρ ( x i , ( β − α ) · x j ) → ∞ as | α − β | → ∞ . Thus, there is an R > r ijA ( α, β ) = 0 if | α − β | > R .In other words, every r ijA is the generating function of a band operator on l p ( Z n ),implying that U AU − is a band operator on l p ( Z n , C N ).The preceding proposition implies in particular that every band operator isbounded on l p ( X ) for p ∈ [1 , ∞ ].For p ∈ [1 , ∞ ], let A p ( X ) stand for the closure in L ( l p ( X )) of the set of allband operators. The operators in A p ( X ) are called band-dominated operatorson X . Note that the class A p ( X ) depends heavily on p (whereas the class ofthe band operators is independent of p ). One can show easily (for example, byemploying the preceding proposition and the well properties of band-dominatedoperators on Z n ) that A p ( X ) is a Banach algebra and even a C ∗ -algebra if p = 2. 9 roposition 8 Let X be a periodic discrete metric space and p ∈ [1 , ∞ ] . Themapping A U AU − is an isomorphism between the Banach algebras A p ( X ) and A p ( Z n , C N ) . Proof.
Note that an operator A is a band operator on l p ( X ) if and only if U AU − is a band operator on l p ( Z n , C N ). The assertion follows since the map-ping A U AU − is a continuous isomorphism between the Banach algebras L ( l p ( X )) and L ( l p ( Z n , C N )). Let X be a Z n -periodic discrete metric space. The goal of this section is acriterion for the Fredholmness of band-dominated operators on l p ( X ). Thiscriterion makes use of the limit operators of A which, in a sense, reflect thebehaviour of A at infinity. Here is the definition. Definition 9
Let < p < ∞ , and h : N → Z n be a sequence tending to infinity.We say that A h is a limit operator of A ∈ L ( l p ( X )) defined by the sequence h if T − h ( m ) AT h ( m ) → A h and T − h ( m ) A ∗ T h ( m ) → A ∗ h as m → ∞ strongly on l p ( X ) and l p ( X ) ∗ = l q ( X ) with /p + 1 /q = 1 , respectively. Wedenote the set of all limit operators of A ∈ L ( l p ( X )) by op p ( A ) and call this setthe operator spectrum of A . Note that the generating function of the shifted operator T − α AT α is relatedwith that of A by k T − α AT α ( x, y ) = k A (( − α ) · x, ( − α ) · y ) (13)and that the generating functions of T − h ( m ) AT h ( m ) converge point-wise on X × X to the generating function of the limit operator A h if the latter exists.It is an important property of band-dominated operators that their operatorspectrum is not empty. More general, one has the following result which can beproved by an obvious Cantor diagonal argument (see [31, 32, 33]). Proposition 10
Let p ∈ (1 , ∞ ) and A ∈ A p ( X ) . Then every sequence h : N → G which tends to infinity possesses a subsequence g such that the limit operator A g of A with respect to g exists. The following theorem settles the basic relation between the Fredholmness ofa band-dominated operator A and the invertibility of its limit operators. Itfollows easily from Theorem 2 if one takes into account that the mapping A p ( X ) → A p ( Z n , C N ) , A U AU − is an isomorphism of Banach algebras and that the relation( U AU − ) h = U A h U − between the limit operators of A and U AU − holds.10 heorem 11 Let p ∈ (1 , ∞ ) and A ∈ A p ( X ) . Then A is a Fredholm operatoron l p ( X ) if and only if all limit operators of A are invertible and if the normsof their inverses are uniformly bounded, sup A h ∈ op ( A ) k A − h k < ∞ . (14) X The goal of this section is to single out a class of band-dominated operators forwhich the uniform boundedness condition (14) is redundant.
Definition 12
Let X be a Z n -periodic discrete metric space. The set W ( X ) consists of all linear operators A for which there is a function h A in l ( Z n ) suchthat max j ∈{ , ..., N } N X i =1 | r ijA ( α, β ) | ≤ h A ( α − β ) (15) for all α, β ∈ Z n . We introduce a norm in W ( X ) by k A k W ( X ) := inf k h k l ( Z n ) (16)where the infimum is taken over all sequences h ∈ l ( Z n ) for which inequality(15) holds in place of h A . Proposition 13
The set W ( X ) with the norm (16) is a Banach algebra, andthe mapping A U AU − is an isometrical isomorphism between the Banachalgebras W ( X ) and W ( Z n , C N ) . The proof is straightforward. We refer to the algebra W ( X ) as the Wieneralgebra . Proposition 14
Let p ∈ [1 , ∞ ] .(i) Every operator A ∈ W ( X ) is bounded on each of the spaces l p ( X ) .(ii) The algebra W ( X ) is inverse closed in each of the algebras L ( l p ( X )) . Proposition 14 follows from Proposition 13 and the related results for the specialcase X = Z n presented in [31, 32] and [33].The following result highlights the importance of the Wiener algebra in ourcontext. Theorem 15
Let A ∈ W ( X ) . Then A is a Fredholm operator on l p ( X ) with p ∈ (1 , ∞ ) if and only if there is a p ∈ [1 , ∞ ] such that all limit operators of A are invertible on l p ( X ) . Moreover sp ess A does not depend on p ∈ (1 , ∞ ) , and sp ess A = [ A h ∈ op ( A ) sp A h . (17)11heorem 15 follows immediately from Proposition 13 and Theorems 3 and 4.The following result states a sufficient condition for the absence of the dis-crete spectrum of an operator A ∈ A p ( X ). Proposition 16
Let A ∈ A p ( X ) and suppose there is a sequence h : N → Z n for which the limit operator A h exists in the sense of norm convergence, lim m →∞ k T − h m AT h m − A h k = 0 . (18) Then sp ess A = sp A . Proof.
Let λ / ∈ sp ess A . Then, by Theorem 11, λ / ∈ sp A h . It follows from (18)that λ / ∈ sp A . Hence, sp A ⊆ sp ess A , which implies the assertion. Let X be a Z n -periodic discrete metric space. An operator A ∈ L ( l p ( X )) is saidto be Z n -periodic if it is invariant with respect to left shifts by elements of Z n ,that is if T α A = AT α for every α ∈ Z n . The following is a straightforward consequence of Proposition 16.
Proposition 17
Let A ∈ A p ( X ) be a Z n -periodic operator. Then sp ess A = sp A. The explicit description of the spectrum (= the essential spectrum) of Z n -periodic operators is possible by means of the Fourier transform. One easilychecks that A ∈ W ( X ) is Z n -periodic on X if and only if the generating func-tion k A of A satisfies the following periodicity condition: For all group elements γ ∈ Z n and all points x, y ∈ X , k A ( γ · x, γ · y ) = k A ( x, y ) . This equality implies that the functions r ijA ( α, β ) := k A ( α · x i , β · x j ) satisfy r ijA ( α, β ) = k A (( α − γ ) · x i , ( β − γ ) · x j )for all γ ∈ Z n , whence r ijA ( α, β ) = r ijA ( α − β, i = 1 , . . . , N ,( U AU − f ) i ( α ) = N X j =1 X β ∈ Z n r ijA ( α, β ) ( U j f )( β )= N X j =1 X β ∈ Z n r ijA ( α − β,
0) ( U j f )( β )= N X j =1 X β ∈ Z n r ijA ( β,
0) ( V β U j f )( α )12here | r ijA ( β, | ≤ h ( β )for a some non-negative function h ∈ l ( Z n ). Thus, we arrived at the followingproposition. Proposition 18
Every Z n -periodic operator A ∈ W ( X ) is isometrically equiv-alent to the shift invariant matrix operator U AU − ∈ W ( Z n , C N ) . Under the conditions of the previous proposition, we associate with A a function σ A : T n → C N × N via σ A ( t ) := X β ∈ Z n r A ( β ) t β where T is the torus { z ∈ C : | z | = 1 } , r A ( β ) is the matrix ( r ijA ( β, Ni, j =1 ,and t β := t β . . . t β n n for t = ( t , . . . , t n ) ∈ T n and β = ( β , . . . , β n ) ∈ Z n . Thefunction σ A is referred to as the symbol of A . It is well known that the operator( ˜ Au )( α ) := X β ∈ Z n r A ( α − β, u ( β )is invertible on l p ( Z n , C N ) with p ∈ [1 , ∞ ] if and only if det σ A = 0 on T n .For t ∈ T n , let λ jA ( t ) with j = 1 , . . . , N denote the eigenvalues of the matrix σ A ( t ). The enumeration of the eigenvalues can be chosen in such a way that λ jA ( t ) depends continuously on t for every j . Thus, the sets C j ( A ) := { λ ∈ C : λ = λ jA ( t ) , t ∈ T n } , j = 1 , . . . , N (19)are compact and connected curves in the complex plane, called the spectral or dispersion curves of A . Proposition 19
Let A ∈ W ( X ) be a Z n -periodic operator. Then sp A = sp ess A = N [ j =1 C j ( A ) . (20)If, moreover, A ∈ W ( X ) is a self-adjoint Z n -periodic operator on l ( X ), then σ A is a Hermitian matrix-valued function. Hence, the λ jA are continuous real-valuedfunctions, and C j ( A ) = [ α j ( A ) , β j ( A )] for j = 1 , . . . , N where α j ( A ) := min t ∈ T n λ jA ( t ) and β j ( A ) := max t ∈ T n λ jA ( t ). Thus, the spec-trum of a self-adjoint Z n -periodic operator on a periodic metric space is theunion of at most N compact intervals (with N the number of orbits of X underthe action of Z n ). 13 Operators with slowly oscillating coefficientson periodic metric spaces
Let again X be a Z n -periodic discrete metric space. A function a ∈ l ∞ ( X ) iscalled slowly oscillating if, for every two points x, y ∈ X ,lim α →∞ ( a ( α · x ) − a ( α · y )) = 0 . (21)The set of all slowly oscillating functions on X forms a C ∗ -subalgebra of l ∞ ( X )which we denote by SO ( X ). Note that the class SO ( X ) does not only dependon X but also on the action of Z n on X .Let a ∈ SO ( X ) and h : N → G be a sequence tending to infinity. TheBolzano-Weierstrass Theorem and a Cantor diagonal argument imply that thereis a subsequence g of h such that the functions x a ( g ( m ) · x ) converge point-wise to a function a g ∈ l ∞ ( X ) as m → ∞ . The condition (21) ensures that thelimit function a g is Z n -periodic on X . Indeed, for every α ∈ Z n , a g ( x ) − a g ( α · x ) = lim m →∞ ( a ( g ( m ) · x ) − ( a ( g ( m ) · ( α · x ))) = 0 . We consider the operators of the form A = ∞ X k, l =1 b k A kl c l I (22)where the A kl are Z n -periodic operators in W ( X ) and the b k and c l are slowlyoscillating functions satisfying ∞ X k, l =1 k b k k l ∞ ( X ) k A kl k W ( X ) k c l k l ∞ ( X ) < ∞ . Let h : N → Z n be a sequence tending to infinity. Then T − h ( m ) AT h ( m ) = ∞ X k, l =1 ( T − h ( m ) b k ) A kl ( T − h ( m ) c l ) I. One can assume without loss that the point-wise limitslim m →∞ ( T − h ( m ) b k )( x ) =: b hk , lim m →∞ ( T − h ( m ) c l )( x ) =: c hl exist (otherwise we pass to a suitable subsequence of h ). As we have seenabove, the limit functions b hk and c hl are Z n -periodic on X . Consequently, thelimit operators A h of A are Z n -periodic operators of the form A h = ∞ X k, l =1 b hk A kl c hl I. Now, the following is an immediate consequence of Theorem 15.14 heorem 20
Let A be an operator with slowly oscillating coefficients of theform ( ) . Then A is a Fredholm operator on l p ( X ) if and only if, for everyoperator A h ∈ op A , det σ A h ( t ) = 0 for every t ∈ T n . Moreover, sp ess A = [ A h ∈ op ( A ) sp A h = [ A h ∈ op ( A ) N [ j =1 C j ( A h ) . By a discrete infinite graph we mean a countable set X together with a binaryrelation ∼ which is anti-reflexive (i.e., there is no x ∈ X such that x ∼ x )and symmetric and which has the property that for each x ∈ X there are onlyfinitely many y ∈ X such that x ∼ y . The points of X are called the vertices and the pairs ( x, y ) with x ∼ y the edges of the graph. Due to anti-reflexivity,the graphs under consideration do not possess loops. We write m ( x ) for thenumber of edges starting (or ending) at the vertex x of X . If x ∼ y , we say thatthe vertices x, y are adjacent .For technical reasons it will be convenient to assume that the graph ( X, ∼ )is connected, i.e., given distinct points x, y ∈ X , there are finitely many points x , x , . . . , x n ∈ X such that x = x , x n = y and x i ∼ x i +1 for i = 0 , . . . , n .The smallest number n with this property defines the graph distance ρ ( x, y ) of x and y . Together with ρ ( x, x ) := 0, this defines a metric ρ on X which makes X to discrete metric space.We call ( X, ∼ ) a Z n -periodic discrete graph if it is a connected discreteinfinite graph, if the group Z n operates freely from the left on X , and if thegroup action respects the graph structure, i.e., x ∼ y if and only if α · x ∼ α · y for arbitrary vertices x, y ∈ X and group elements α ∈ Z n . Clearly, every groupwith these properties leaves the graph distance invariant, that is, X becomes a Z n -periodic discrete metric space. If ( X, ∼ ) is a Z n -periodic graph, then thefunction m is Z n -periodic, too, that is, m ( α · x ) = m ( x ) for every x ∈ X and α ∈ Z n .Every Z n -periodic discrete graph Γ := ( X, ∼ ) induces a canonical differenceoperator ∆ Γ on l p ( X ), called the (discrete) Laplace operator or Laplacian of Γ,via (∆ Γ u )( x ) := 1 m ( x ) X y ∼ x u ( y ) , x ∈ X. (23)Evidently, ∆ Γ is a Z n -periodic band operator.Let v ∈ l ∞ ( X ). The operator H Γ := ∆ Γ + vI is referred to as the (discrete) Schr¨odinger operator with electric potential v on the graph X . Given a sequence15 : N → Z n tending to infinity, there exist a subsequence g of h and a function v g ∈ l ∞ ( X ) such that v ( g ( m ) · x ) → v g ( x ) as m → ∞ for every x ∈ X . It turnsout that the operator H g Γ := ∆ Γ + v g I is the limit operator of H Γ defined by the sequence g and that every limitoperator of H Γ is of this form. Thus, Theorem 15 implies the following. Theorem 21
The Schr¨odinger operator H Γ = ∆ Γ + vI with bounded potential v is a Fredholm operator on l p ( X ) with p ∈ (1 , ∞ ) if and only if there is a p ∈ [1 , ∞ ] such that all limit operators of H Γ are invertible on l p ( X ) . Theessential spectrum of H Γ does not depend on p ∈ (1 , ∞ ) , and sp ess H Γ = [ H h Γ ∈ op ( H Γ ) sp H h Γ . (24)For an explicit description of the essential spectrum of the Schr¨odinger operator H Γ we first assume that v is a periodic potential. Then the operator U vU − isthe operator of multiplication by the diagonal matrix diag ( v ( x ) , . . . , v ( x N )).Hence, U H Γ U − = X α ∈{− , , } n a α V α + diag ( v ( x ) , . . . , v ( x N )) , where the a α are certain constant N × N matrices which depend on the structureof the graph Γ. Consequently, σ H Γ ( t ) = X α ∈{− , , } n a α t α + diag ( v ( x ) , . . . , v ( x N )) , t ∈ T n . If the potential v is real-valued, then H Γ acts as a self-adjoint operator on l ( X ),and σ H Γ is a Hermitian matrix-valued function on T n . From Proposition 19 weconclude that sp H Γ = N [ j =1 C j ( H Γ )where C j ( H Γ ) is the real interval [ a j , b j ] with a j := min t ∈ T n λ j H Γ ( t ) and b j :=max t ∈ T n λ j H Γ ( t ).Next we consider Schr¨odinger operators H Γ = ∆ Γ + vI with slowly oscillatingpotential v . As we have seen in the previous section, all limit operators of H Γ are of the form H g Γ = ∆ Γ + v g I with periodic potentials v g . Theorem 21 together with Theorem 15 yield thefollowing. Theorem 22
Let H Γ = ∆ Γ + vI with v ∈ SO ( X ) . Then sp ess H Γ = [ H g Γ ∈ op ( H Γ ) N [ j =1 C j ( H g Γ )16 ith the spectral curves C j ( H g Γ ) defined as in ( ) . If the slowly oscillating potential v is real-valued, then the spectral curves C j ( H g Γ ) are (possibly overlapping) intervals on the real line.The following examples clarify the structure of the essential spectrum ofSchr¨odinger operators on some special periodic graphs. The graphs under con-sideration are embedded into R n for some n . This embedding allows one toconsider the vertices of the graph as vectors and to use the linear structure of R n in order to describe the group action. Example 23 (The Cayley graph of Z n ) As every finitely generated group,the group Z n induces a graph (called the Cayley graph of the group) the verticesof which are the points in Z n and with edges ( α, α ± e i ) where α ∈ Z n and where e i := (0 , . . . , , , , . . . ,
0) with the 1 at the i th position and i = 1 , . . . , n . TheLaplace operator ∆ Z n is of the form(∆ Z n u )( x ) = 12 n n X i =1 ( u ( x + e i ) + u ( x − e i )) , which leads to the symbol σ ∆ Z n ( t ) := 12 n n X i =1 ( t i + t − i ) , t ∈ T n . Hence, sp ∆ Z n = [ − , Example 24 (The zigzag graph)
Let Γ = ( X, ∼ ) be the zigzag graph in theplane R as shown in Figure 24. The graph Γ is periodic with respect to theaction g · x n := x n +2 g of the group Z , and the set M = { x , x } of verticesrepresents the fundamental domain. x x x x One should mention that, as a graph, the zigzag graph is isomorphic to theCayley graph of the group Z and, in both cases, it is the same group Z whichacts on the graph. The difference lies in the way in which Z acts. For the Cayleygraph, the group element α maps the vertex x to α + x , whereas α maps x to2 α + x for the zigzag graph. The latter action is visualized by the zigzag form.The operator U ∆ Γ U − has the matrix representation U ∆ Γ U − = 12 (cid:18) I + V (1 , I + V ( − , (cid:19)
17n the basis induced by M . Hence, σ ∆ Γ ( t ) = 12 (cid:18) t t − (cid:19) , t ∈ T , and a straightforward calculation shows that the spectral curves of ∆ Γ are { λ ∈ C : λ = ± cos ϕ/ , ϕ ∈ [0 , π ] } . Hence, the spectrum of the Laplacian ∆ Γ of the zigzag graph is the interval[ − , H Γ := ∆ Γ + vI with Z -periodicpotential v . Thus, v is completely determined by its values on M , and we write v := v ( x ) and v := v ( x ). Then σ H Γ − λI ( t ) = (cid:18) v − λ (1 + t ) / t − ) / v − λ (cid:19) , t ∈ T , which implies that the spectral curves of H Γ are ( λ ∈ C : λ = 12 ± p ( v − v ) + 4 cos ϕ/ v + v ) , ϕ ∈ [0 , π ] ) . If, for example, v and v are real numbers with v < v , then sp ess H Γ = sp H Γ is the union of the disjoint intervals " − p ( v − v ) + 42( v + v ) , v v + v v v + v ,
12 + p ( v − v ) + 42( v + v ) , (25)that is, one observes a gap ( v v + v , v v + v ) in the spectrum.Finally, let the potential v be slowly oscillating. Then the essential spectrumof H Γ is the union [ h " − p ( v h − v h ) + 42( v h + v h ) , min { v h , v h } v h + v h (26) [ h " max { v h , v h } v h + v h ,
12 + p ( v h − v h ) + 42( v h + v h ) where the unions are taken with respect to all sequences h for which the limits v hj := lim m →∞ v ( h ( m ) · x j ) , j = 1 , , (27)exist. Set a H Γ := lim sup Z ∋ α →∞ v ( α · x ) v ( α · x ) + v ( α · x ) ,b H Γ := lim inf Z ∋ α →∞ v ( α · x ) v ( α · x ) + v ( α · x ) . a H Γ < b H Γ (28)holds, then the operator H Γ has the gap ( a H Γ , b H Γ ) in its essential spectrum.Of course, this interval can contain points of the discrete spectrum of H Γ . Example 25 (The honeycomb graph)
Let Γ = ( X, ∼ ) be the hexagonalgraph shown in Figure 25. We consider this graph as embedded into R and let e and e be the vectors indicated in the figure. The group Z operates on Γvia ( α , α ) · x := x + α e + α e (where α , α ∈ Z and x ∈ X ). A fundamental domain M for this action isprovided by any two vertices x , x as marked in the figure. (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) x x ~e ~e x η ω ω η Hence, we have to identify l p ( X ) with l p ( Z , C ), and the Laplacian ∆ Γ has19he following matrix representation with respect to M U ∆ Γ U − = 13 (cid:18) I + V e + V e I + V − e + V − e (cid:19) . Consequently, σ ∆ Γ ( t ) = 13 (cid:18) t + t t − + t − (cid:19) , t = ( t , t ) ∈ T , and the spectral curves of the Laplacian ∆ Γ are C ± := { λ ∈ C : λ = ±| e iϕ + e iϕ | / , ϕ , ϕ ∈ [0 , π ] } . The curves C ± coincide with the intervals [0 ,
1] and [ − , Γ = [ − , v be a Z -periodic potential and set v j := v ( x j ) for j = 1 ,
2. A cal-culation similar to Example 24 yields that the spectral curves of the Schr¨odingeroperator H Γ := ∆ Γ + vI are ( λ ∈ C : λ = 12 ± p ( v − v ) + 4 µ ( ϕ , ϕ )2( v + v ) ) , where µ ( ϕ , ϕ ) := | e iϕ + e iϕ | / , ϕ , ϕ ∈ [0 , π ] . Hence, as in Example 24, sp ess H Γ = sp H Γ is given by the union (25). Letfinally v be a slowly oscillating potential on X . Since the image of the function µ is the interval [0 , a H Γ , b H Γ ) occurs in the essential spectrum of H Γ . Let Γ := ( X, ∼ ) be a Z n -periodic discrete graph. We consider the Schr¨odingeroperator H u := ∆ Γ ⊗ I X + I X ⊗ ∆ Γ + (29)+( W I X ) ⊗ I X + I X ⊗ ( W I X ) + W I on l ( X × X ). This operator describes the motion of two particles with coordi-nates x , x ∈ X with masses 1 on the graph Γ around a heavy nuclei located atthe point x ∈ X . Therefore, H is also called a 3-particle Schr¨odinger operator.In (29), ∆ Γ is again the Laplacian on the graph Γ, I X is the identity operatoron l ( X ), I = I X ⊗ I X is the identity operator on l ( X × X ), W and W arereal-valued functions on X defined by W j ( x j ) = w j ( ρ ( x j , x )) , j = 1 , , W is a real-valued function on X × X given by W ( x , x ) = w ( ρ ( x , x )) . Here ρ denotes the given metric on X , and w , w and w are functions on thereal interval [0 , ∞ ) which satisfylim z →∞ w ( z ) = lim z →∞ w ( z ) = lim z →∞ w ( z ) = 0 . Clearly, H is a band operator on l ( X × X ). We are going to describe its essentialspectrum via formula (24), for which we need the limit operators of H and theirspectra. Note that the spectrum of the Laplacian ∆ Γ depends on the structureof the graph Γ and that this spectrum has a band structure (= is the union ofclosed intervals). In Examples 23 – 25 we had sp ∆ Γ = [ − , E, F of R ,we let E + F := { z ∈ R : z = x + y, x ∈ E, y ∈ F } denote their algebraic sum, and we set 2 E := E + E .Let g = ( g , g ) : N → Z n × Z n be a sequence tending to infinity. We have todistinguish the following cases (all other possible cases can be reduced to thesecases by passing to suitable subsequences of g ): Case 1.
The sequence g tends to infinity, whereas g is constant. Then thelimit operator H g of H is unitarily equivalent to the operator H := ∆ Γ ⊗ I X + I X ⊗ (∆ Γ + W I X ) . (30) Case 2.
Here g tends to infinity and g is constant. Then the limit operator H g of H is unitarily equivalent to the operator H := (∆ Γ + W I X ) ⊗ I X + I X ⊗ ∆ Γ . (31) Case 3.
Both g and g tend to infinity. There are two subcases: Case 3a.
The sequence g − g tends to infinity. In this case the limit operatoris the free discrete Hamiltonian∆ Γ ⊗ I X + I X ⊗ ∆ Γ the spectrum of which is equal to 2 sp ∆ Γ . Case 3b.
The sequence g − g is constant. Then the limit operator H g of H is unitarily equivalent to the operator of interaction H := ∆ Γ ⊗ I X + I X ⊗ ∆ Γ + W I. (32)Note that the operators H , H and H are invariant with respect to shifts byelements of the form (0 , g ) , ( g,
0) and ( g, g ) of Z n × Z n , respectively. It follows21rom Proposition 16 that these operators do not possess discrete spectra. Fromformula (24) we further concludesp ess H = sp H ∪ sp H ∪ sp H . (33)The following proposition is well known. For a proof see [34], Theorem VIII.33and its corollary. Proposition 26
Let A ∈ L ( H ) and B ∈ L ( K ) be bounded self-adjoint opera-tors on Hilbert spaces H, K . Then sp ( A ⊗ I K + I H ⊗ B ) = sp A + sp B. This proposition implies in our setting thatsp H = sp ∆ Γ + sp (∆ Γ + W I X ) . Since the Schr¨odinger operator ∆ Γ + W I X is a compact perturbation of theLaplacian ∆ Γ , one hassp ess (∆ Γ + W I X ) = sp ∆ Γ ∪ { λ (2) k } ∞ k =1 where { λ (2) k } ∞ k =1 is the sequence of the eigenvalues of ∆ Γ + W I X which arelocated outside the spectrum of ∆ Γ . Thus,sp H = 2 sp ∆ Γ + ∪ ∞ k =1 ( λ (2) k + sp ∆ Γ ) . In the same way one findssp H = 2 sp ∆ Γ + ∪ ∞ k =1 ( λ (1) k + sp ∆ Γ )where the λ (1) k run through the points of the discrete spectrum of ∆ Γ + W I X which are located outside the spectrum of ∆ Γ .Recall that in Examples 23 – 25, sp ∆ Γ = [ − , H j = [ − , ∞ [ k =1 [ λ ( j ) k − , λ ( j ) k + 1] . One can also give a simple estimate for the location of the spectrum of H bymeans of the following well-known result (see, e.g., [22], p. 357). Proposition 27
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