Spectral theory of a class of Block Jacobi matrices and applications
aa r X i v : . [ m a t h . SP ] D ec Spectral theory of a class of Block Jacobi matrices and applications
Jaouad SahbaniInstitut de Math´ematiques de Jussieu-Paris Rive Gauche-UMR7586Universit´e Paris Diderot (Paris 7)Bˆatiment Sophie Germain–case 70125 rue Thomas Mann, 75205 Paris Cedex 13, [email protected]
Abstract
We develop a spectral analysis of a class of block Jacobi operators based on the conjugateoperator method of Mourre. We give several applications including scalar Jacobi operators withperiodic coefficients, a class of difference operators on cylindrical domains such as discrete wavepropagators, and certain fourth-order difference operators.
Keyword:
Essential spectrum, absolutely continuous spectrum, Mourre estimate
MSC (2000):
Primary 47A10, 47B36; Secondary 47B47, 39A70
Let H = l ( Z , C N ), for some integer N ≥
1, be the Hilbert space of square summable vector-valuedsequences ( ψ n ) n ∈ Z endowed with the scalar product h φ, ψ i = X n ∈ Z h φ n , ψ n i C N where h· , ·i C N is the usual scalar product of C N . Let A n and B n be two sequences of N by N matricessuch that B n = B ∗ n for all n ∈ Z . Here we denote by T ∗ the adjoint matrix of a given matrix T . Letus consider the difference operator H = H ( { A n } , { B n } ) acting in H by( Hψ ) n = A ∗ n − ψ n − + B n ψ n + A n ψ n +1 , for all n ∈ Z . (1.1)In the literature, H is usually called the block Jacobi operator realized by the sequences A n and B n . Operators of such a form are naturally related to matrix orthogonal polynomials theory, seefor example [2, 9, 10, 12, 14, 21, 22] and references therein. Of course, the case where N = 1corresponds to the usual scalar Jacobi operators that are well studied by different approaches, see e.g.[2, 6, 7, 8, 10, 11, 13, 15, 16, 17, 18, 20, 23, 25, 27, 28] and their references. (This list is very restrictiveand contains only some papers related directly to the present one). Applications given in the secondpart of this paper represent additional motivations of our interest to block Jacobi operators.Our main goal here is to study spectral properties of the operator H in terms of the asymptoticbehavior of the entries A n and B n . The key step of our analysis is the construction of a conjugateoperator for H in the sense of the Mourre estimate, see [24]. Recall that such an estimate has manyimportant consequences on the spectral measure of H as well as on the asymptotic behavior of theresolvent of ( H − z ) − when the complex parameter z approaches the real axis, see [1, 4, 5, 24, 26].We need the following standard notations. For a self-adjoint operator T we denote by E T ( · ) itsspectral measure, σ ( T ) its spectrum, σ ess ( T ) its essential spectrum, σ p ( T ) the set of its eigenvalues, σ sc ( T ) its singular continuous spectrum, and σ ac ( T ) its absolutely continuous spectrum.1n this paper we will focus on the so-called generalized Nevai class. More specifically, we assumethat there exist two matrices A and B such that,lim | n |→∞ ( k A n − A k + k B n − B k ) = 0 . (1.2)In particular, H is a bounded self-adjoint operator in H . Notice that in the literature, it is customaryto assume that the matrices A n are positive definite or at least non singular. Here we do not demandsuch an assumption, which turns out to be quite useful in many applications, see Section 10.It is more convenient to decompose H into the sum H = H + V where the operator H = H ( A, B )is defined on H by ( H ψ ) n = A ∗ ψ n − + Bψ n + Aψ n +1 , for all n ∈ Z (1.3)and V = H ( { A n − A } , { B n − B } ) is the Block Jacobi operator realized by the sequences A n − A, B n − B .The operator H is clearly bounded and self-adjoint in H while, according to (1.2), the perturbation V is compact. Hence, by Weyl theorem, H and H have the same essential spectra, σ ess ( H ) = σ ess ( H ) . The point here is that H is a block Toeplitz matrix whose spectrum can be determined explicitly.Indeed, by using Fourier transform we easily show that H is unitarily equivalent to the direct integralˆ H = Z ⊕ [ − π,π ] h ( p ) dp acting in ˆ H = Z ⊕ [ − π,π ] C N dp (1.4)where the reduced operators h ( p ) are given by, h ( p ) = e − ip A + e ip A ∗ + B. (1.5)The family h ( p ) , p ∈ [ − π, π ], is an analytic family of self-adjoint matrices. So, see [19], there existtwo analytic functions λ j and W j such that, for all p ∈ [ − π, π ], { λ j ( p ); 1 ≤ j ≤ N } are the repeatedeigenvalues of h ( p ) and { W j ( p ); 1 ≤ j ≤ N } is a corresponding orthonormal basis of eigenvectors. Inparticular, for any j = 1 , · · · , N , the critical set of λ j defined by κ ( λ j ) = { λ j ( p ) | λ ′ j ( p ) = 0 } is clearly finite. Therefore, the critical set of H defined by κ ( H ) = ∪ i = Ni =1 κ ( λ j )is finite too. We have Proposition 1.1
The spectrum of H is purely absolutely continuous outside the critical set κ ( H ) .More precisely, σ ( H ) = ∪ i = Ni =1 λ i ([ − π, π ]) , σ p ( H ) ⊂ κ ( H ) and σ sc ( H ) = ∅ . Let α j = min p ∈ [ − π,π ] λ j ( p ) and β j = max p ∈ [ − π,π ] λ j ( p ). Assume that the λ j ’s are arranged so that α ≤ α · · · ≤ α N . So each λ j gives arise to a spectral band Σ j = λ j ([ − π, π ]) = [ α j , β j ], j = 1 , · · · , N .Two successive bands Σ j , Σ j +1 may overlap if α j +1 ≤ β j ; or may be disjoint if β j < α j +1 . In thesecond case [ β j , α j +1 ] is called a non degenerate spectral gap. Notice also that, in some situations,the band Σ j can degenerate into a single point, i.e. α j = β j . This happens if, and only if, λ j ( p ) isconstant on [ − π, π ]. In which case, this constant value is an infinitely degenerate eigenvalue of H .To illustrate these considerations we will give in Section 2 some explicit examples and show necessaryand/or sufficient conditions to a spectral band to be non degenerate.Using the eigenvalues λ j ( p )’s of the reduced operators h ( p ), we will construct a conjugate operatorfor H . That is to say a self-adjoint operator A such that, for each compact real interval J includedin R \ κ ( H ), we have E H ( J )[ H , i A ] E H ( J ) ≥ c E H ( J ) (1.6)2or some c >
0. For the definition of the commutator [ H , i A ] see Section 3. In particular, we obtainseveral important resolvent estimates for H , see Theorem 5.1. We used a similar strategy in [3] toconstruct conjugate operators from dispersion curves for perturbations of fibered systems in cylinders.Next we prove that if A n − A and B n − B tend to zero fast enough as n tends to infinity, thenthe Mourre estimate (1.6) is preserved for H up to a compact operator. More precisely, we will showthat, if (1.8) below holds, then for the same A , J and c > E H ( J )[ H, i A ] E H ( J ) ≥ c E H ( J ) + K (1.7)for some compact operator K . In particular, we get Theorem 1.1
Suppose that lim | n |→∞ | n | ( k A n − A k + k B n − B k ) = 0 . (1.8) Then outside κ ( H ) the eigenvalues of H are all finitely degenerate and their possible accumulationpoints are contained in κ ( H ) . Put h x i = √ x for a real number x . We denote by N the multiplication operator defined by( N ψ ) n = nψ n for all ψ ∈ H , and by B ( H ) the Banach algebra of bounded operators in the Hilbertspace H . We also need the interpolation space K := ( D ( N ) , H ) / , which can be described, accordingto Theorem 3.6.2 of [1], by the norm, k ψ k / , = ∞ X j =0 j/ k θ (2 − j | N | ) ψ k where θ ∈ C ∞ ( R ) with θ ( x ) > − < x < θ ( x ) = 0 otherwise. Clearly, h N i − s H ⊂ K ⊂ H for any s > / Theorem 1.2
Suppose that the perturbation V satisfies Z ∞ k θ ( | N | /r ) V k dr < ∞ . (1.9) Then the limits ( H − x ∓ i − := lim µ → ( H − x ∓ iµ ) − exist locally uniformly on R \ [ κ ( H ) ∪ σ p ( H )] for the weak* topology of B ( K , K ∗ ) . In particular, the singular continuous spectrum σ sc ( H ) is empty. Mention that one may show that condition (1.9) is equivalent to Z ∞ sup r< | n | < r ( k A n − A k + k B n − B k ) dr < ∞ . (1.10)Hence these conditions are clearly satisfied if, for some θ >
0, one hassup n ∈ Z | n | θ ( k A n − A k + k B n − B k ) < ∞ . (1.11)Remark also that Theroem 1.2 implies that h N i − s ( H − x ∓ i − h N i − s ∈ B ( H ), for any s > . Inour next theorem we describe continuity properties of these boundary values of the resolvent of H asfunctions of x , as well as some of their propagation consequences. For we need the following definition.Let ( E, k · k ) be a Banach space and f : R → E be a bounded continuous function. For an integer m ≥ w m be the modulus of continuity of order m of f defined on (0 ,
1) by w m ( f, ε ) = sup x ∈ R (cid:13)(cid:13)(cid:13)(cid:13) m X j =0 ( − j (cid:18) mj (cid:19) f ( x + jε ) (cid:13)(cid:13)(cid:13)(cid:13) .
3e say that f ∈ Λ α , α >
0, if there is an integer m > α such thatsup <ε< ε − α w m ( ε ) < ∞ . Notice that if α ∈ (0 ,
1) then Λ α is nothing but the space of H˝older continuous functions of order α . In contrast, if α = 1 then Λ consists of smooth functions in Zygmund’s sense (they are notLipschitz in general, see [29]). Finally, if α > n α is the greatest integer strictly less than α then f belongs to Λ α if, and only if, f is n α − times continuously differentiable with bounded derivativesand its derivative f ( n α ) of order n α is of class Λ α − n α . For example, f belongs to Λ means that f is continuously differentiable with a bounded derivative and f ′ is of class Λ ( f ′ is not Lipschitz ingeneral). Theorem 1.3
Suppose that (1.11) holds. Then, for any < s ≤ + θ , the maps x N i − s ( H − x ∓ i − h N i − s ∈ B ( H ) are locally of class Λ s − on R \ [ κ ( H ) ∪ σ p ( H )] . Moreover, ||h N i − s e − iHt ϕ ( H ) h N i − s || ≤ C (1 + | t | ) − ( s − ) . for all ϕ ∈ C ∞ ( R \ [ κ ( H ) ∪ σ p ( H )]) and some C > . Notice that for H we have more than that, see Theorem 5.1. Moreover, Theorems 1.1-1.3 are actuallyvalid for operators of the form H = H + V where V is a symmetric compact perturbation such that h N i θ V is bounded with θ >
0. See section 6 where this remark is discussed more precisely.Finally, in the second part of the paper, we apply these abstract results to some concrete models.The first application concerns the case where A is positive definite. It is one of the most commonassumptions in the studies of block Jacobi matrices and their applications to matrix orthogonal poly-nomial theory. In this case, all the eigenvalues of h ( p ) are monotonic functions of p . This appliesdirectly to a class of difference operators on cylindrical domains such as discrete wave propagators.We devote the next applications to cases where the matrix A is singular. Then we show how Jacobioperators with periodic coefficients fit in this situation. Notice that while the spectral results of Jacobioperators with periodic coefficients are well known our Mourre estimate is a new ingredient. Finally,we study a model where A is a lower triangular matrix that we apply to symmetric difference operatorsof fourth-order.The paper is organized as follows. • In Section 2 we study in details the unperturbed operator H . • Section 3 contains a brief review on what we need from Mourre’s theory. • In Sections 4 and 5 we construct a conjugate operator for H . • In section 6 we extend our Mourre estimate to the operator H . • In section 7 we study the case where A is positive definite. • Section 8 is devoted to discrete wave propagators in stratified cylinders. • In Sections 9-11 we discuss cases where A is singular. • In section 12 we consider a case where A is a lower triangular. • Finally in Section 13 we explain how our method extends to difference operator of higher orderwith matrix coefficients.
Acknowledgements:
We would like to thank J. Janas and P. Cojuhari for their hospitality anduseful discussions on the subject during my visit to IPAN (Krakow).4
Basic properties of H Consider the unitary transform F : l ( Z , C N ) → L ([ − π, π ] , C N ) defined by F ( ψ )( p ) = b ψ ( p ) = X n ∈ Z e inp ψ n , for each ψ ∈ l ( Z , C N ) . Recall that the unperturbed operator H is defined in H by( H ψ ) n = A ∗ ψ n − + Bψ n + Aψ n +1 , for all n ∈ Z . Direct computation shows that, for each ψ ∈ l ( Z , C N ), d H ψ ( p ) = ( e − ip A + e ip A ∗ + B ) b ψ ( p ) = h ( p ) b ψ ( p ) , for all p ∈ [ − π, π ] . Hence (1.4)-(1.5) are proved. Recall that, for all p ∈ R , { λ j ( p ); 1 ≤ j ≤ N } are the repeatedeigenvalues of h ( p ) and { W j ( p ); 1 ≤ j ≤ N } is a corresponding orthonormal basis of eigenvectors.The Proposition 1.1 follows from: Proposition 2.1
The operator H is unitarily equivalent to M = ⊕ j = Nj =1 λ j ( p ) acting in the direct sum ⊕ j = Nj =1 L ([ − π, π ]) . Proof
Consider the operator U : l ( Z , C N ) → ⊕ j = Nj =1 L ([ − π, π ]) defined by U ψ = ( f j ) ≤ j ≤ N , where f j ( p ) = h W j ( p ) , b ψ ( p ) i C N = X n ∈ Z h W j ( p ) , ψ n i C N e inp . (2.1)Clearly, U is a unitary operator with the following inversion formula:( U − f ) n = N X j =1 π Z π − π f j ( p ) W j ( p ) e − inp dp, for all f = ( f , · · · , f N ) in ⊕ j = Nj =1 L ([ − π, π ]). Moreover, direct computation shows that, for each ψ ∈ l ( Z , C N ), ( U H ψ ) j ( p ) = h W j ( p ) , d H ψ ( p ) i C N = h W j ( p ) , h ( p ) b ψ ( p ) i C N = λ j ( p ) f j ( p ) . The proof is complete.In the sequel we denote the imaginary part of a complex number z by ℑ z . Proposition 2.2
The eigenvalue λ j ( p ) is identically constant on [ − π, π ] if, and only if, λ ′ j ( p ) = ℑ ( e ip h AW j ( p ) , W j ( p ) i C N ) = 0 , for all p ∈ [ − π, π ] . In this case, the spectral band Σ j = λ j ([ − π, π ]) degenerates into a single point which is an infinitelydegenerate eigenvalue of H . Proof:
We know that, for all p ∈ [ − π, π ], λ j ( p ) = h h ( p ) W j ( p ) , W j ( p ) i C N . Since k W j ( p ) k = 1 then h W ′ j ( p ) , W j ( p ) i C N + h W j ( p ) , W ′ j ( p ) i C N = 0 . Thus λ ′ j ( p ) = h h ( p ) ′ W j ( p ) , W j ( p ) i C N = h i ( − e − ip A + e ip A ∗ ) W j ( p ) , W j ( p ) i C N = − ℑ (cid:0) e ip h AW j ( p ) , W j ( p ) i C N (cid:1) . The proof is complete.
Remark
In particular, the j th spectral band Σ j = λ j ([ − π, π ]) is not degenerate if, and only if,there is p ∈ [ − π, π ] such that λ ′ j ( p ) = − ℑ (cid:0) e ip h AW j ( p ) , W j ( p ) i C N (cid:1) = 0. In this case, σ p ( H ) ∩ Σ j isfinite and σ sc ( H ) ∩ Σ j = ∅ . Moreover, Σ j contains an eigenvalues e of H if and only if there exists i = j such that λ i ( q ) = e for all q ∈ [ − π, π ]. Finally, if for all j = 1 , · · · , N there is p ∈ [ − π, π ] suchthat λ ′ j ( p ) = 0, then the spectrum of H is purely absolutely continuous, i.e. σ p ( H ) = σ sc ( H ) = ∅ .We deduce immediately the following corollary: 5 orollary 2.1 If A is positive definite then λ ( p ) , · · · , λ N ( p ) are even on [ − π, π ] , decreasing on [0 , π ] and for all j = 1 , · · · , N , we have λ ′ j ( p ) = − h AW j ( p ) , W j ( p ) i C N sin p , for all p ∈ [ − π, π ] . In particular, Σ j = [ λ j ( π ) , λ j (0)] , κ ( λ j ) = { λ j ( π ) , λ j (0) } and the spectrum of H is purely absolutelycontinuous with σ ( H ) = ∪ ≤ j ≤ N [ λ j ( π ) , λ j (0)] . Corollary 2.2 If A is a symmetric invertible matrix then λ ( p ) , · · · , λ N ( p ) are non constant evenfunctions on [ − π, π ] . In particular, the spectrum of H is purely absolutely continuous. Proof
Assume by contradiction that λ ( p ) = c for all p ∈ [ − π, π ] for some constant c . Then for any p ∈ [ − π, π ] there exists W ( p ) an eigenvector of h ( p ) associated to c so that h ( p ) W ( p ) = cW ( p ). Hencefor any p ∈ [ − π, π ], (2 cos p ) AW ( p ) = [( c − B ) A − ] AW ( p ), which means that [ − , ⊂ σ (( c − B ) A − )which is impossible. Corollary 2.3
Assume that A is a symmetric matrix such that for any eigenvector w of B , h Aw, w i 6 =0 . Then the conclusion of Corollary 2.2 holds true. Proof
Here remark that h ( π/
2) = B . So λ j ( π/
2) is an eigenvalue of B and W j ( π/
2) is an associatedeigenvector. So λ ′ j ( π/
2) = − h AW j ( π/ , W j ( π/ i 6 = 0, by hypothesis. The proof is complete. Corollary 2.4
Assume that for any eigenvector w of i ( A − A ∗ )+ B one has h ( A + A ∗ ) w, w i 6 = 0 . Then λ ( p ) , · · · , λ N ( p ) are non constant on [ − π, π ] . In particular, the spectrum of H is purely absolutelycontinuous and there exist α < β , · · · , α n < β n such that σ ( H ) = ∪ ≤ j ≤ N [ α j , β j ] . Proof
It is enough to remark that h ( − π/
2) = i ( A − A ∗ ) + B .We close this section by few explicit examples to illustrate these consideration and our commentjust after Proposition 1.1 on the spectrum of H . Example 2.1
Assume that A and B are both diagonals with diagonal elements A jj = a j ≥ and B jj = b j ∈ R . In this case, the eigenvalues of the reduced operators h ( p ) are given by λ j ( p ) = b j + 2 a j cos p . Therefore, we immediately see that by playing with the parameters a j and b j one mayrealize whatever we said with the associated spectral bands Σ j = [ b j − a j , b j + 2 a j ] . Example 2.2
Let A = (cid:18) (cid:19) and B = (cid:18) b aa − b (cid:19) where a, b > . In this case, the eigenval-ues of h ( p ) are given, for all p ∈ [ − π, π ] , by λ j ( p ) = ( − j p b + ( a + 2 cos p ) with j = 1 , . Put α = p b + ( a − and β = p b + ( a + 2) . Hence we have the following.1. If a ≥ then λ is increasing on [0 , π ] while λ is decreasing and κ ( H ) = { λ j (0) , λ j ( π ) } j =1 , .Moreover, the spectrum of H is purely absolutely continuous and consists of two spectral bandswith a gap between them of length α and σ ( H ) = [ − β, − α ] ∪ [ α, β ] .2. If < a < then the critical points of λ and λ are , π plus an additional point p ∈ ] π/ , π [ defined by cos( p ) = − a/ . In such case, σ ( H ) = [ − β, − b ] ∪ [ b, β ] and κ ( H ) = { λ j (0) , λ j ( p ) , λ j ( π ) } j =1 , . So we have two spectral bands with a gap between them of length b . Example 2.3
Let A = (cid:18) a (cid:19) , B = (cid:18) b a a − b (cid:19) where a , a > and b ∈ R . In this case, theeigenvalues of reduced operators h ( p ) are given by λ j ( p ) = ( − j q b + a + a + 2 a a cos p , j = 1 , . hen λ ( p ) is an increasing function of p ∈ [0 , π ] while λ ( p ) is decreasing and λ ( p ) < λ ( p ) , for all p ∈ ]0 , π [ . Therefore, κ ( H ) = { λ (0) , λ ( π ) , λ ( π ) , λ (0) } and σ ( H ) = [ λ (0) , λ ( π )] ∪ [ λ ( π ) , λ (0)] . This spectrum is purely absolutely continuous and consists of two spectral bands with a gap betweenthem of length L = λ ( π ) − λ ( π ) = 2 p b + ( a − a ) . That gap is degenerate (i.e. L = 0 ), if and onlyif, b = 0 and a = a . In such case, the two spectral bands have one common point λ ( π ) = λ ( π ) = 2 a which is a double eigenvalue of h ( π ) .Mention that λ and λ are identically constant on [ − π, π ] if, and only if, a a = 0 . For example, if a > and a = 0 then the spectrum of H consists of two infinitely degenerate eigenvalues ± p b + a . The following brief review on the conjugate operator theory is based on [1, 4, 5, 26]. Let A be aself-adjoint operator in a separable complex Hilbert space H and S ∈ B ( H ). Definition 3.1 (i) Let k ≥ be an integer and σ > . We say that S is of class C k ( A ) , respectivelyof class C σ ( A ) , if the map t S ( t ) = e − i A t Se i A t ∈ B ( H ) is strongly of class C k , respectively of class Λ σ on R .(ii) We say that S is of class C , ( A ) if Z k e − i A ε Se i A ε − S + e i A ε Se − i A ε S k dεε < ∞ . Remarks (i) We have the following inclusions C s ( A ) ⊂ C , ( A ) ⊂ C ( A ), for any s > S is of class C ( A ), if and only if, the sequilinear form defined on D ( A )by [ S, A ] = S A − A S has a continuous extension to H , which we identify with the associated boundedoperator in H (from the Riesz Lemma) that we denote by the same symbol. Moreover, [ S, i A ] = ddt | t =0 S ( t ) . (iii) Recall that h A i = (1 + A ) / . To prove that S is of class C σ ( A ) it is enough to show that h A i σ S is bounded in H , see for example the appendix of [6]. In particular, in the case where σ = 1 + θ for some θ >
0, the operator S is of class C θ ( A ) if one of the following conditions is true:(1) h A i σ S is bounded in H or(2) S is of class C ( A ) and h A i θ [ S, i A ] is bounded in H .In the sequel of this section, let H be a bounded self-adjoint operator which is at least of class C ( A ). Then [ H, i A ] defines a bounded operator in H that we still denote by the same symbol [ H, i A ].We define the open set ˜ µ A ( H ) of real numbers x such that, for some constant a >
0, a neighborhood∆ of x and a compact operator K in H , we have E H (∆)[ H, i A ] E H (∆) ≥ aE H (∆) + K. (3.2)The inequality (3.2) is called the Mourre estimate and the set of point x ∈ ˜ µ A ( H ) where it holds with K = 0 will be denoted by µ A ( H ). Theorem 3.1
The set ˜ µ A ( H ) contains at most a discrete set of eigenvalues of H and all these eigen-values are finitely degenerate. Moreover, µ A ( H ) = ˜ µ A ( H ) \ σ p ( H ) . Recall that K A := H / , is the Besov space associated to A defined by the norm k ψ k / , = k ˜ θ ( A ) ψ k + ∞ X j =0 j/ k θ (2 − j | A | ) ψ k . where θ ∈ C ∞ ( R ) with θ ( x ) > − < x < θ ( x ) = 0 otherwise, and ˜ θ ∈ C ∞ ( R ) with ˜ θ ( x ) > | x | < θ ( x ) = 0 otherwise. One has, see [4, 26]:7 heorem 3.2 Assume that H is of class C , ( A ) . Then H has no singular continuous spectrum in µ A ( H ) . Moreover, the limits ( H − x ∓ i − := lim µ → ( H − x ∓ iµ ) − exist locally uniformly on µ A ( H ) for the weak* topology of B ( K A , K ∗ A ) . We also have the following:
Theorem 3.3
Assume that H is of class C s + ( A ) for some s > / . Then the maps x
7→ h A i − s ( H − x ∓ i − h A i − s are locally of class Λ s − on µ A ( H ) . Moreover, for every ϕ ∈ C ∞ ( µ A ( H )) we have ||h A i − s e − iHt ϕ ( H ) h A i − s || ≤ C < t > − ( s − ) . Since k e − iHt ϕ ( H ) k = k ϕ ( H ) k , then by using the complex interpolation one obtains much more thanthis, see [4, 5, 26]. For example, if H is of class C ∞ ( A ), then, for any ϕ ∈ C ∞ ( µ A ( H )) and σ, ε > ||h A i − σ e − iHt ϕ ( H ) h A i − σ || ≤ C (1 + | t | ) − σ + ε , for all t ∈ R λ j Here we denote the operator of multiplication by a function f in L ([ − π, π ]) by the same symbol f orby f ( p ) when we want to stress the p -dependence.According to Section 2, H is unitarily equivalent to ⊕ Nj =1 λ j . So it is convenient to prove first theMourre estimate for each multiplication operator λ j . So, let us fix j ∈ { , · · · , N } and put λ = λ j .Recall that the set κ ( λ ) of critical values of λ is finite. Let ∆ ⊂ λ ([ − π, π ]) be a compact set, suchthat ∆ ∩ κ ( λ ) = ∅ . Obviously, on the set λ − (∆) one has | λ ′ ( p ) | ≥ c >
0. One may then construct afunction F ∈ C ∞ ([ − π, π ]) , such that F ( p ) λ ′ ( p ) ≥ c > λ − (∆). (4.1)For example, F = λ ′ perfectly fulfills this role. Nevertheless, in some concrete situations more appro-priate choices can be made, see next sections. We are now able to define our conjugate operator a to λ on L ([ − π, π ]) by, a = i { F ( p ) ddp + ddp F ( p ) } = iF ( p ) ddp + i F ′ ( p ) . (4.2)It is clear that a is an essentially self-adjoint operator in L ([ − π, π ]). Moreover, direct computationyields to [ λ, i a ] = F ( p ) λ ′ ( p ) . So λ is of class C ( a ), and by repeating the same calculation, we show that λ is of class C ∞ ( a ).Moreover, thanks to (4.1), a is strictly conjugate to λ on ∆: E λ (∆)[ λ, i a ] E λ (∆) = E λ (∆) (cid:0) F ( p ) λ ′ ( p ) (cid:1) E λ (∆) ≥ cE λ (∆) (4.3)From now on we put F = λ ′ . We have: Proposition 4.1
The operator λ is of class C ∞ ( a ) and a is locally strictly conjugate to λ on R \ κ ( λ ) ,i.e. µ a ( λ ) = R \ κ ( λ ) . Proof
First, recall that R \ σ ( λ ) ⊂ µ a ( λ ). So, if λ is constant then the assertion is trivial, since σ ( λ ) = κ ( λ ). Suppose that λ is non constant and let x ∈ σ ( λ ) \ κ ( λ ). Then choose a compact interval∆ about x sufficiently small so that ∆ ∩ κ ( λ ) = ∅ . The last discussion shows that a is strictly conjugateto λ on ∆, i.e. x ∈ µ a ( λ ). The proof is complete. 8 Mourre estimate for H The direct sum M = ⊕ ≤ j ≤ N λ j is a self-adjoint bounded operator in the Hilbert space ⊕ ≤ j ≤ N L ([ − π, π ]).Recall that κ ( H ) = ∪ ≤ j ≤ N κ ( λ j ) is finite. For any j = 1 , · · · , N , let us denote by a j the operatorgiven by (4.2), with F = F j constructed in the last section with λ = λ j , that is a j = i { F j ( p ) ddp + ddp F j ( p ) } = iF j ( p ) ddp + i F ′ j ( p ) . (5.1)and define the operator e A = ⊕ ≤ j ≤ N a j . (5.2) Proposition 5.1
The operator M is of class C ∞ ( e A ) and e A is locally strictly conjugate to M on R \ κ ( H ) , i.e. µ e A ( M ) = R \ κ ( H ) . Proof:
Let x ∈ R \ κ ( H ). By definition κ ( H ) = ∪ ≤ j ≤ N κ ( λ j ), and according to Proposition 4.1, µ a j ( λ j ) = R \ κ ( λ j ) . Hence, x ∈ ∩ ≤ j ≤ N µ a j ( λ j ). Then for any j = 1 , · · · , N , there exists a constant c j > j about x such that E λ j (∆ j )[ λ j , i a j ] E λ j (∆ j ) ≥ c j E λ j (∆ j ) . But, for any Borel set J , one has E M ( J ) = ⊕ ≤ j ≤ N E λ j ( J ) and [ M, i ˜ A ] = ⊕ ≤ j ≤ N [ λ j , i a j ], Hence, weimmediately conclude that, for ∆ = ∩ Nj =1 ∆ j and c j = min ≤ j ≤ N c j we have, E M (∆)[ M, i ˜ A ] E M (∆) ≥ cE M (∆) , that is, x ∈ µ ˜ A ( M ). The proof is complete.According to the Proposition 2.1, H = U − M U . Define the operator A by A = U − e A U . (5.3)By a direct application of the previous proposition, we get Corollary 5.1
The operator H is of class C ∞ ( A ) and A is locally strictly conjugate to H on R \ κ ( H ) , i.e. µ A ( H ) = R \ κ ( H ) . In particular, the spectrum of H is purely absolutely continuous on R \ κ ( H ). In fact, combiningthis corollary, Lemma 6.1, and the results of Section 3, we immediately get, Theorem 5.1 (i) The limits ( H − x ∓ i − := lim µ → ( H − x ∓ iµ ) − exist locally uniformly on R \ κ ( H ) for the weak* topology of B ( K , K ∗ ) .(ii) For any s > / , the maps x
7→ h N i − s ( H − x ∓ i − h N i − s ∈ B ( H ) are locally of class Λ s − .(iii) For any ϕ ∈ C ∞ ( R \ κ ( H )) and σ, ε > . ||h N i − σ e − iH t ϕ ( H ) h N i − σ || ≤ C (1 + | t | ) − σ + ε . H In this section we extend our analysis to the operator H . For we start by recalling that H = H + V where V is the difference operator acting in H by( V ψ ) n = ( A n − − A ) ∗ ψ n − + ( B n − B ) ψ n + ( A n − A ) ψ n +1 , for all n ∈ Z . (6.1)According to (1.2), V is a compact operator in H . In particular, σ ess ( H ) = σ ess ( H ) = ∪ j = Nj =1 λ j ([ − π, π ])The main result of this section is the extension, up to a compact operator, of the Mourre estimateobtained in the last section for H to H . In fact we will prove the following more general result,9 heorem 6.1 Let V be a self-adjoint compact operator on H (not necessarily of the form (6.1)) suchthat V < N > is also compact. Then the self-adjoint operator H = H + V is of class C ( A ) and A is locally conjugate to H on R \ κ ( H ) , i.e. ˜ µ A ( H ) = µ A ( H ) = R \ κ ( H ) . In particular, outside thefinite set κ ( H ) the eigenvalues of H + V are all finitely degenerate and their possible accumulationpoints are included in κ ( H ) . Proof of Theorem 6.1
According to the last section, the operator H is of class C ∞ ( A ) with A = U − e A U , and e A is defined by (5.1) and (5.2). So to prove Theorem 6.1 we first show that V is ofclass C ( A ) and that [ V, i A ] is a compact operator in H .According to Lemma 6.1 below, the operator h N i − A is bounded in H . Hence, V A = V h N i ( h N i − A )is a compact operator in H . By adjonction, A V is also a compact operator. Thus, the commutator[ V, i A ] is a compact operator in H . Moreover, for any Borel set J , E H ( J ) − E H ( J ) is a compactoperator. Hence, for any J ⊂ µ A ( H ), E H ( J )[ H, i A ] E H ( J ) = E H ( J )[ H , i A ] E H ( J ) + E H ( J )[ V, i A ] E H ( J )= E H ( J )[ H , i A ] E H ( J ) + K ≥ cE H ( J ) + K ≥ cE H ( J ) + K ′ for some c > K, K ′ . The proof is finished. Lemma 6.1
For every integer m > the operators h N i − m A m and A m h N i − m are bounded in H . Proof:
The proof can be done by induction. We only illustrate the idea for m = 2 and deal with h N i − A (the remaining cases are similar). Let us show that this operator is bounded in H .We check that A = U − e A U is given by( A ψ ) n = N X j =1 Z π − π (cid:20) G j ( p ) d f j dp ( p ) + + G j ( p ) df j dp ( p ) + G j ( p ) f j ( p ) (cid:21) W j ( p ) e − ipn dp, where f j is defined by (2.1), and G j , G j and G j are smooth functions on [ − π, π ]. Integrations byparts show that ( h N i − A ψ ) n is a finite sum of terms of the form h n i − ( an + bn + c ) Z π − π f j ( p ) W ( p ) e − ipn dp, where a, b and c are constants, and W is a smooth function. Since, on the one hand, h n i − ( an + bn + c )is bounded, and using, on the other hand, Plancherel’s theorem: || (cid:16) Z π − π f j ( p ) W ( p ) e − ipn dp (cid:17) n || l ( Z ) = || f j ( p ) W ( p ) || L ([ − π,π ]) ≤ C || f j ( p ) || L ([ − π,π ]) ≤ C || ψ || H , we obtain ||h N i − A ψ || H ≤ C ′ || ψ || H , for some constants C, C ′ > Proof of Theorem 1.1
First, the assumption (1.8) implies that V h N i is a compact operator in H . Hence Theorem 1.1 is an immediate consequence of Theorem 6.1 and Theorem 3.1. Proof of Theorem 1.2
Since condition (1.9) implies (1.8), ˜ µ A ( H ) = R \ κ ( H ). So, according toTheorem 3.2 and Lemma 6.1, we only have to prove that H is of class C , ( A ). As in the last proof,it is enough to show that V is of class C , ( A ). But this is true according to the appendix of [6]. Theproof of Theorem 1.2 is complete. Proof of Theorem 1.3
As in the precedent proof, according to Theorem 3.3, it is enough toshow that the condition (1.11) implies that V is of class C θ ( A ). But (1.11) ensures that the operator10 N i θ V is bounded in B ( H ). The preceding lemma shows that h A i θ V = ( h A i θ h N i − (1+ θ ) ) h N i θ V is bounded too. The proof of Theorem 1.3 is finished by the point (ii) of the remark given just afterDefinition 3.1. Remark
Here also it should be clear that Theorem 1.2 and Theorem 1.3 are valid for H = H + V where the perturbation V is any bounded operator in H that satisfies one of the following:(i) h N i θ V ∈ B ( H ) for some θ >
0, or(ii) V is a compact operator on H such that h N i θ [ V, A ] ∈ B ( H ). A is positive definite In this section we focus on the case where H is the block Jacobi operator defined by (1.1)-(1.2) suchthat A is positive definite. Then, thanks to the Corollary 2.1, we have:1. for all j = 1 , · · · , N , we have λ ′ j ( p ) = − < AW j ( p ) , W j ( p ) > C N sin p, for all p ∈ [ − π, π ] .
2. the eigenvalues λ ( p ) , · · · , λ N ( p ) are even on [ − π, π ] and decreasing on [0 , π ]. In particular,Σ j = [ λ j ( π ) , λ j (0)] and κ ( λ j ) = { λ j ( π ) , λ j (0) } .3. the spectrum of H is purely absolutely continuous and σ ( H ) = ∪ ≤ j ≤ N [ λ j ( π ) , λ j (0)] . Moreover, according to the last section, the operator H is of class C ∞ ( A ) and µ A ( H ) = R \ κ ( H ),with A = U − e A U , and e A is defined by (5.1) and (5.2). The advantage now is that, one may choose F j ( p ) = − sin p, j = 1 , · · · , N, in the definition of the operator A . Indeed, F j ( p ) λ ′ j ( p ) = ( − sin p ) λ ′ j ( p ) = 2 < AW j ( p ) , W j ( p ) > C N (sin p ) , which is strictly positive on ]0 , π [. But then, with this choice, one may easily show that A = D + L (7.2)for some bounded operator L in H and where D is defined in H by,( D ψ ) n = i ( n + 12 ) ψ n +1 − i ( n −
12 ) ψ n − (7.3)This allows us to handle the larger class of perturbations V satisfyinglim | n |→∞ | n | ( k A n +1 − A n k + k B n +1 − B n k ) = 0 . (7.4)Remark that the condition (1.8) implies (7.4) but not the inverse, think to the example A n = A + | n | ) A ′ , with any matrix A ′ . So we get the following more general result in comparison withTheorem 1.1: Theorem 7.1
If (7.4) holds then H is of class C ( A ) and ˜ µ A ( H ) = R \ { α j , β j } ≤ j ≤ N . In particular,the possible eigenvalues of H in R \ { α j , β j } ≤ j ≤ N are all finitely degenerate and cannot accumulateoutside { α j , β j } ≤ j ≤ N . roof The only point we have to verify is the compactness of [
V, i A ] in H under the conditions(1.2) and (7.4). Indeed, [ V, i A ] = [ V, i D ] + [ V, iL ] . But [
V, iL ] = i ( V L − LV ) is clearly a compact operator since each term is ( V being compact and L bounded). Moreover, a direct computation shows that the only nonzero matrix coefficients of thecommutator K = [ V, i D ] are: K n,n +2 = K ∗ n +2 ,n = ( n + )( A n +1 − A n ) − ( A n +1 − A ); K n,n +1 = K ∗ n +1 ,n = ( n + )( B n +1 − B n ); K n,n = ( n + )[( A n − A n − ) + ( A n − A n − ) ∗ ] + ( A n − − A ) + ( A n − − A ) ∗ . All these coefficients tend to zero at infinity, thanks to (1.2) and (7.4). Hence, [
V, i D ] is a compactoperator and the proof is complete. Remark:
In addition of (1.2) suppose that Z ∞ sup r< | n | < r (cid:16) | n | (cid:0) k A n − A n − k + k B n − B n − k (cid:1) + k A n − A k (cid:17) drr < ∞ . (7.5)Then according to the appendix of [6] we get H is of class C , ( A ) (in fact we have slightly more thanthat but we will not go in these details here). Remark that (7.5) follows if, for some θ >
0, we havesup n ∈ Z | n | θ (cid:16) | n | (cid:0) k A n − A n − k + k B n − B n − k (cid:1) + k A n − A k (cid:17) < ∞ . (7.6)In this case it is clear that h N i θ [ V, i A ] is bounded. Hence h A i θ [ V, i A ] = ( h A i θ h N i − θ ) h N i θ [ V, i A ] isbounded too. Consequently, H is of class C θ ( A ) so that, we have Theorem 7.2
Assume (1.2). Theorems 1.2, respectively Theorem 1.3, remain valid for this model ifwe replace the condition (1.9), respectively (1.11), by (7.5), respectively (7.6).
Example 7.1
Let us consider the Jacobi operator J acting in l ( Z , C ) by ( J ψ ) n = a n − ψ n − + b n ψ n + a n ψ n +1 , for all n ∈ Z . Here-above a n > and b n ∈ R . Assume that there exist a ± > , b ± ∈ R such that, lim n →±∞ a n = a ± and lim n →±∞ b n = b ± . (7.7) This model has been studied in [7] where the authors used some ideas of three body problem to establishtheir Mourre estimate. Here our present approach applies in straightforward way. Indeed, Let U : l ( Z ) → l ( N , C ) defined by ( U ψ ) n = ( ψ n , ψ − n +1 ) . It is clear, see [2], that U is unitary operator and U J U − = H where H is the block Jacobi operator realized by A n = (cid:18) a n a − n (cid:19) , n ≥ B n = (cid:18) b n b − n +1 (cid:19) , n ≥ and B := (cid:18) b a a b (cid:19) . In particular, one may deduce directly by applying the last discussion that σ ess ( J ) = [ b − − a − , b − +2 a − ] ∪ [ b + − a + , b + + 2 a + ] . Moreover, if lim n →±∞ | n | θ (cid:16) | n | (cid:0) | a n − a n +1 | + | b n − b n +1 | (cid:1) + | a n − a ± | ) = 0 , for some θ ≥ holds then σ p ( J ) has no accumulation points outside κ ( J ) = { b − ± a − , b + ± a + } . Moreover, if θ > then σ sc ( J ) = ∅ . Difference operators on cylindrical domains
Consider the configuration space X = Z × { , · · · , N } , for some integer N , and the difference operator J acting in l ( X ) by( J ψ ) n,k = a n − ,k ψ n − ,k + b n,k ψ n,k + a n,k ψ n +1 ,k + c n,k − ψ n,k − + c n,k ψ n,k +1 (8.8)initialized by ψ n, = ψ n,N +1 = 0 and where a n,k , c n,k are positive while b n,k are real numbers. Theorem 8.1
Assume that, there is some positive numbers a , · · · , a N , c , · · · , c N − and real numbers b , · · · , b N such that for all j = 1 , · · · , N , lim n →±∞ | a n,j − a j | + | b n,j − b j | + | c n,j − c j | = 0 , ( with the convention c n,N = c N = 0) . Then J is a bounded symmetric operator whose essential spectrum is of the form ∪ Nj =1 [ α j , β j ] for somenumbers α j < β j . If in addition, we have lim n →±∞ | n | ( | a n +1 ,j − a n,j | + | b n +1 ,j − b n,j | + | c n +1 ,j − c n,j | ) = 0 , then there is a self-adjoint operator A ′ such that ˜ µ A ′ ( J ) = R \ κ ( J ) with κ ( J ) = ∪ ≤ j ≤ N { α j , β j } . Inparticular, the possible eigenvalues of J are all finitely degenerate and cannot accumulate outside κ ( J ) .Finally, σ sc ( J ) = ∅ and conclusions of Theorems 1.2 and 1.3 hold for J if, for some θ > , we have sup n | n | θ (cid:16) | n | (cid:0) | a n +1 ,j − a n,j | + | b n +1 ,j − b n,j | + | c n +1 ,j − c n,j | (cid:1) + | a n,j − a j | (cid:17) < ∞ . Before to show this result let us fix some notations that we will use in the sequel of this paper. Let E ij be the elementary matrix whose only non zero entry is 1 placed at the intersection of the i th ligneand the j th column and diag( d , · · · , d N ) = d . . . d . . . . . . . . . . . . . . .0 . . . d N (8.9)tridiag( { α i } ≤ i ≤ N − , { β i } ≤ i ≤ N ) = β α . . . α β α . . . α β α . . . . . . . . . . . . . . . . . .0 . . . α N − β N (8.10) Proof of Theorem 8.1
Let U : l ( Z ×{ , · · · , N } ) → l ( Z , C N ) defined by ( U ψ ) n = ( ψ n, , · · · , ψ n,N ) . It is clear that U is unitary operator and U J U − = H ( { A n } , { B n } ) where A n = diag( a n, , · · · , a n,N ) , B n = tridiag( { c n,i } ≤ i ≤ N − , { b n,i } ≤ i ≤ N ) (8.11)forall n ∈ Z . By hypothesis the matrices A n and B n satisfy (1.2) with A = diag( a , · · · , a N ) and B = tridiag( { c i } ≤ i ≤ N − , { b i } ≤ i ≤ N ) (8.12)Since A is positive definite, the results of the last section apply directly to the operator H ( { A n } , { B n } ).In addition, the conjugate operator of J is given by A ′ = U − A U where A is defined by (7.2) and(7.3). The proof is finished. 13 xample 8.1 To have explicit formulae, assume in (8.12) that a i = a > , b i = 0 and c i = c > .In this case, h ( p ) is N by N Toeplitz matrix whose spectrum is given by λ j ( p ) = 2 a cos p + 2 cǫ j with ǫ j = cos ( N − j + 1) πN + 1 , j = 1 , · · · , N. Here again, we observe explicitly σ ( H ) = ∪ Nj =1 [ λ j ( π ) , λ j (0)] . Remark that j th gap is non trivial, ifand only if, L j = λ j +1 ( π ) − λ j (0) = 2 c ∆ j − a > where ∆ j = ǫ j +1 − ǫ j > . Then j th gap is non trivial, if and only if, c > a/ ∆ j . Example 8.2
Discrete wave propagator.
Consider on X = Z ×{ , · · · , N } the discrete Laplacian ∆ given by (∆ ψ ) n,m = ψ n +1 ,m + ψ n − ,m + ψ n,m +1 + ψ n,m − , for all x = ( n, m ) ∈ X with the initial boundary condition ψ ,m = ψ n, = ψ n,N +1 = 0 . Let ρ : X → ]0 , ∞ [ be the localpropagation speed. The discrete wave equation is given by ∂ u∂t = ρ ∆ u. Clearly, the change of variable u = ρv transforms the last equation to ∂ v∂t = ρ ∆ ρv, Hence the spectral analysis of the operator H = ρ ∆ ρ on H = l ( X ) is of interest for discrete waveequation. Assume that the cylinder X is stratified, i.e. for some constant ρ , · · · , ρ N > , one has ρ ( n, m ) − ρ m → as n → ∞ . Hence H = H ( { A n } , { B n } ) given in (8.11) with a n,m = ρ ( n, m ) ρ ( n + 1 , m ) , b n,m = 0 and c n,m = ρ ( n, m ) ρ ( n, m + 1) . Of course we assume that ρ = ρ N +1 = 0 . This case can be clearly studied by Theorem 8.1 since a n,m → a m = ρ m and c n,m → c m = ρ m ρ m +1 as | n | → ∞ . But we will not state the corresponding result separately.For example, if the stratification is simple in the sense that ρ is constant, say ρ m = 1 for all m .Then the essential spectrum of H has no gaps since the spectral bands Σ j and Σ j +1 overlap. Indeed,with the notations of example 8.1 we have: L j = λ j +1 ( π ) − λ j (0) = 2∆ j − ≤ here ∆ j = ǫ j +1 − ǫ j > A is singular The simplest situation one may consider is a naive modification of the example considered in theprevious section, namely where B is a tridiagonal matrix and A has only one nonzero coefficient onthe diagonal. More specifically, let a , a , · · · , a N > b , · · · , b N ∈ R be given and H be the blockJacobi operator defined by (1.1) where A n and B n satisfy (1.2) with A = a N E N,N and B = tridiag( { a i } ≤ i ≤ N − , { b i } ≤ i ≤ N ) (9.1)14 heorem 9.1
1. There exist α < β < α < β · · · < α N < β N such that the spectrum of H ispurely absolutely continuous and σ ( H ) = ∪ Nj =1 [ α j , β j ] and κ ( H ) = { α , β , α , β · · · , α N , β N } .2. If (1.8) holds then we have a Mourre estimate for H and H on R \ κ ( H ) , i.e. µ A ( H ) =˜ µ A ( H ) = R \ κ ( H ) . In particular, the eigenvalues of H are all finitely degenerate and cannotaccumulate outside κ ( H ) .3. If (1.9) holds then the conclusions of Theorems 1.2 and 1.3 are valid. Mention here that the spectrum of H has a band/gap structure, that is between each two successivebands Σ j = [ α j , β j ] and Σ j +1 = [ α j +1 , β j +1 ] there is a non trivial gap of length L j = α j +1 − β j > h ( p ) given here by h ( p ) = tridiag( { a i } ≤ i ≤ N − , { b , · · · , b N − , b N + 2 a N cos p } )Since h ( − p ) = h ( p ), it is enough to restrict ourselves to p ∈ [0 , π ]. Theorem 9.1 is an immediateconsequence of the following: Theorem 9.2
Let B N − = tridiag ( { a i } ≤ i ≤ N − , { b i } ≤ i ≤ N − ) and µ < µ · · · < µ N − its eigenval-ues. The eigenvalues λ ( p ) , · · · , λ N ( p ) are simple, decreasing on [0 , π ] and can be chosen so that, forall p ∈ [0 , π ] , λ ( p ) < µ < λ ( p ) · · · < µ N − < λ N ( p ) . (9.2) In particular, λ j ([0 , π ]) = [ α j , β j ] with α j = λ j ( π ) and β j = λ j (0) . Moreover, κ ( λ j ) = { α j , β i } . Proof
Let p ∈ [0 , π ]. The matrix h ( p ) is symmetric tridiagonal with positive off-diagonal elementsand its characteristic polynomial satisfies D ( x, p ) = det( h ( p ) − x ) = ( b N + 2 a N cos p − x ) D N − ( x ) − a N − D N − ( x ) , where D N − j ( x ) = det( B N − j − x ) and B N − j is obtained from h ( p ), and so from B , by eliminatingthe last j rows and columns. Moreover, it is known that D ( x, p ) has N distincts real roots that areseparated by the N − D N − ( x ). So (9.2) is proved. Moreover, direct calculationshows that ∂D∂p ( λ j ( p ) , p ) = − a N (sin p ) D N − ( λ j ( p )) = 0 , for all p ∈ ]0 , π [ . Thus by differentiating the identity D ( λ j ( p ) , p ) = 0 we get λ ′ j ( p ) ∂D∂x ( λ j ( p ) , p ) + ∂D∂p ( λ j ( p ) , p ) = 0and so λ ′ j ( p ) is non zero on ]0 , π [. Finally, because the matrix A is non negative we have, for all j = 1 , · · · , N , λ ′ j ( p ) = − < AW j ( p ) , W j ( p ) > C N sin p ≤ , for all p ∈ [0 , π ] . Hence λ ′ j ( p ) < p ∈ ]0 , π [. We see that κ ( λ i ) = { λ j (0) , λ j ( π ) } . The proof is finished. Remark:
Mention that if we put the nonzero coefficient at the first place of the diagonal insteadof the last one then exactly the same phenomenon happens. In contrast, if we put it in a differentplace of the diagonal then some drastic changes may arise. For example, assume that B is the sameand A = a N E N − ,N − . Hence h ( p ) = tridiag( { a i } ≤ i ≤ N − , { b , · · · , b N − , b N − + 2 a N cos p, b N } )In this case, on one hand D ( x, p ) = det( B − x ) + 2 a N cos p ( b N − x ) D N − ( x ) . On the other handdet( B − x ) = ( b N − x ) D N − ( x ) − a N − D N − ( x ) . Hence, D ( x, p ) = ( b N − x ) (cid:16) D N − ( x ) + 2 a N cos p ( b N − x ) D N − ( x ) (cid:17) − a N − D N − ( x ) . Now let us choose first a , · · · a N − and b , · · · , b N − then pick b N among the eigenvalues of B N − . So b N − x divide D N − ( x ). Hence, b N is a root of D ( x, p ) for all p and therefore H has a degeneratespectral band and b N is an infinitely degenerate eigenvalue of H .15 xample 9.1 Let N = 2 so that A = a E , , B = tridiag ( { a } , { b , b } ) where a , a > and b , b ∈ R . It is easy to check that σ ( H ) = [ λ ( π ) , λ (0)] ∪ [ λ ( π ) , λ (0)] and that λ (0) < b < λ ( π ) .Let us consider now the inverse problem of this model. More specifically, let α < β , α < β begiven and find a , a > and b , b ∈ R such that the corresponding H satisfies σ ( H ) = [ α , β ] ∪ [ α , β ] . First, if there is a solution then, according to Theorem 9.1 and Theorem 9.2, α , α are theeigenvalues of h ( π ) and β , β are the eigenvalues of h (0) . Thus α + α = b + b − a , α α = b ( b − a ) − a β + β = b + b + 2 a , β β = b ( b + 2 a ) − a Hence a = ( β − α ) + ( β − α ) and a b = β β − α α gives b while b + b ) = α + α + β + β leads to b . Finally we get a = 2 b b − α α − β β which gives a . Of course we have to make surthat b b − α α − β β > which holds if and only if β < α .
10 A second case where A is singular: Periodic Jacobi operators In this part we look at the case where the only nonzero entry of A is put at the lower left corner and B is a tridiagonal matrix. It turns out that scalar Jacobi operators with periodic coefficients are coveredby this model. More precisely, let H = H ( A, B ) defined by (1.3) with A = a N E N, , B = tridiag( { a i } ≤ i ≤ N − , { b i } ≤ i ≤ N ) . (10.1)Then we have Theorem 10.1
1. There exist α < β ≤ α < β · · · ≤ α N < β N such that the spectrum of H ispurely absolutely continuous and σ ( H ) = ∪ Nj =1 [ α j , β j ] .
2. Define the critical set κ ( H ) = { α , β , α , β · · · , α N , β N } . There exists a conjugate operator A for H i.e. µ A ( H ) = R \ κ ( H ) . In particular, conclusions of Theorem 5.1 hold for H .3. Let V be a symmetric compact operator. Then the sum H = H + V defines a bounded self-adjoint operator in H whose essential spectrum coincides with σ ( H ) . Moreover, if h N i V iscompact then ˜ µ A ( H ) = R \ κ ( H ) . In particular, the eigenvalues of H are all finitely degenerateand cannot accumulate outside κ ( H ) .4. If h N i θ V is bounded for some θ > then the conclusions of Theorems 1.2 and 1.3 are validfor H . A consequence of this theorem is the following result on Jacobi operators with periodic coefficients,
Corollary 10.1
Let J be the periodic Jacobi operator given in l ( Z ) by ( J ψ ) n = a n ψ n +1 + b n ψ n + a n − ψ n − such that a n > and b n ∈ R with a j + N = a j and b j + N = b j . Then assertions of Theorem 10.1 holdtrue when we replace H by J . Proof
Consider the unitary operator U : l ( Z ) → l ( Z , C N ) defined by ( U ψ ) n = ( ψ ( n − N +1 , · · · , ψ nN ).One may show that U J U − = H ( A, B ) where
A, B are given by (10.1). The proof is complete.Of course spectral properties of J are well known, see [28] for example. The point here is toobtain the Mourre estimate for these operators and also to show how our general approach works inthis context.Here again spectral analysis of H is based on the study of the eigenvalues of the reduced operators h ( p ) given here by h ( p ) = b a · · · a N e ip a b a · · · · · · a N − b N − a N − a N e − ip · · · a N − b N , ∀ p ∈ [ − π, π ] . (10.2)16ince h ( − p ) is the transpose of h ( p ), it is enough to restrict ourselves to p ∈ [0 , π ]. Theorem 10.1 isan immediate consequence of the following: Theorem 10.2
The eigenvalues λ ( p ) , · · · , λ N ( p ) , of h ( p ) are monotonic on [0 , π ] and can be chosenso that λ ( p ) < λ ( p ) · · · < λ N ( p ) , for all p ∈ ]0 , π [ . In fact, if λ j is increasing (respectively decreasing)then λ j +1 is decreasing (respectively increasing). Moreover, for p = 0 or π the eigenvalues λ j ( p ) maybe double and λ ( π ) < λ (0) ≤ λ (0) < λ ( π ) ≤ · · · ≤ λ N ( π ) < λ N (0) if N is odd ; (10.3) λ (0) < λ ( π ) ≤ λ ( π ) < λ (0) ≤ · · · ≤ λ N ( π ) < λ N (0) if N is even . (10.4) In particular, there exist α < β ≤ α < β · · · α N − < β N − ≤ α N < β N such that λ j ([0 , π ]) =[ α j , β j ] and κ ( λ j ) = { α j , β j } . If for some ≤ j ≤ N , β j = α j +1 then such a value is a doubleeigenvalue of h ( p ) for p = 0 or π . Notice that h ( p ) for p = 0 and π are exactly the periodic and anti-periodic Jacobi matrices studiedin [20, 23] where the assertion (10.3)-(10.4) are proved. More generally, eigenvectors of h ( p ) , p ∈ [0 , π ]are nothing but the so-called Bloch solutions studied in the Floquet theory developed for periodicJacobi operators. In this sense, Theorem 10.2 is well known with different proofs, see [11, 20, 23, 28].We will give however a proof, mainly for completeness, but also because it is elementary and basedon standard linear algebra. It will be given in few lemmas. Lemma 10.1
For every p ∈ ]0 , π [ , the eigenvalues λ ( p ) , · · · , λ N ( p ) of h ( p ) are all simple, while for p = 0 , π they are at most of multiplicity two. Moreover, they are all monotonic on (0 , π ) . Proof
Let u = ( x , · · · , x N ) be an eigenvector of h ( p ) corresponding to an eigenvalues λ . This isequivalent to, ( b − λ ) x + a x + a N e ip x N = 0 a i − x i − + ( b i − λ ) x i + a i x i +1 = 0 , ≤ i ≤ N − a N e − ip x + a N − x N − + ( b N − λ ) x N = 0 (10.5)We see that for all 2 ≤ i ≤ N − x i is a linear combination of x and x N so that λ is at most ofmultiplicity two. Moreover, if x = 0 then p = 0 or p = π . Indeed, if x = 0 then x = − ( a N /a ) e ip x N .Moreover, using simultanousely the equations j = 2 , , · · · , N −
1, we get that x N = ce ip x N for some c ∈ R . This means that x N = 0 or e ip ∈ R . Since u is an eigenvector of h ( p ), u = 0 so that x N = 0and therefore e ip ∈ R ,or equivalently p = 0 or p = π . Similarly one may prove that if x N = 0 then p = 0 or p = π .Now let p ∈ (0 , π ) and assume that e ip x x N is real. Then multiplying our system by x weget from the first equation that x x is real too. Similarly, using simultanousely the equations j =2 , , · · · , N −
1, we get x x N is real which is impossible. Thus ℑ ( e ip x x N ) = 0 and so λ ′ ( p ) = 0according to the proposition 2.2. This completes the proof. Lemma 10.2
Let a = ( − N − a a · · · a N . The characteristic polynomial of h ( p ) is given by, D ( x, p ) := det( h ( p ) − x ) = ∆( x ) + 2 a cos p, (10.6)∆( x ) = det( B − x ) − a N det( ˙ B − x ) (10.7) where ˙ B is the matrix obtained from B by removing the first and last rows and columns with theconvention det( ˙ B − x ) = 1 in the case where N = 2 . Proof
Direct calculation, see however [17] where the formula is already used.
Remark
We emphasize that, according to (10.6), if one of the a j ’s is zero then D ( x, p ) = ∆( x )which is independent of p . In this case, the spectrum of h ( p ) is independent of p so that the spectrumof H consists of N infinitely degenerate eigenvalues. This provides one with a general class of blockJacobi matrices with this kind of spectra. 17 emma 10.3 The derivative ∆ ′ of ∆ has N − roots µ < µ < · · · < µ N − and the λ j ’s may bechosen so that, for all j = 1 , · · · , N ,(i) ( − j ∆ ′ ( λ j ( p )) > for all p ∈ ]0 , π [ ;(ii) λ j (]0 , π [) ⊂ ] µ j − , µ j [ with µ = −∞ and µ N = + ∞ .(iii) For any j = 1 , · · · , N , ( − N + j − λ j ( p ) is an increasing function of p ∈ [0 , π ] and κ ( λ j ) ⊂{ λ j (0) , λ j ( π ) } . Proof (i) Let us denote by λ ( π/ < λ ( π/ < · · · < λ N ( π/
2) the distinct eigenvalues of h ( π/ x ) = D ( x, π/ j = 1 , · · · , N − µ j ∈ ] λ j ( π/ , λ j +1 ( π/ ′ ( µ j ) = 0 , Moreover, since ∆ is a polynomial of the form∆( x ) = ( − N x N + lower terms, it is clear that ( − j ∆ ′ ( λ j ( π/ > j = 1 , · · · N which givesus the sign of ∆ ′ on each ] µ j − , µ j [.(ii) Now assume that (ii) is not true for some j , say j = 1 (the other cases are similar). Then, bycontinuity of λ and the fact that λ ( π/ < µ , µ would belong to λ (]0 , π [). Therefore µ = λ ( p )for some p ∈ ]0 , π [. But then for such p one has: D ( µ , p ) = D ( λ ( p ) , p ) = ∂D∂x ( λ ( p ) , p ) = 0 . Here-above we used the fact that ∂D∂x ( x, p ) = ∆ ′ ( x ). This means that µ = λ ( p ) is a multipleeigenvalue of h ( p ) which, according to Lemma 10.1, implies that p = 0 or p = π . But this isimpossible since p ∈ ]0 , π [.(iii) Again using the fundamental equation (10.6) we get, D ( λ j ( p ) , p ) = 0 for all p ∈ (0 , π ) and j = 1 , · · · , N . By differentiating this identity with respect to p and using ∂D∂p ( x, p ) = − a sin p, we get λ ′ j ( p ) = 2 a sin p ∆ ′ ( λ j ( p )) for all p ∈ (0 , π ) and j = 1 , · · · , N . So the sign of λ ′ j on ]0 , π [ is that of a ∆ ′ on ] µ j − , µ j [. As a = ( − N − a a · · · a N and according to theLemma 10.3 where the sign of ∆ ′ has been found, we see that ( − N + j − λ ′ j ( p ) > p ∈ ]0 , π [. Example 10.1
Let N = 2 so that A = a E , , B = tridiag ( { a } , { b , b } ) where a , a > and b , b ∈ R . In this case, it is easy to see on one hand that h (0) has two simple eigenvalues. On theother hand, h ( π ) has a double eigenvalues if, and only if, b = b and a = a , in such case the doubleeigenvalue is µ = ( b + b ) / b / .In the second part of this example we consider the inverse problem of this model. More specifically,let α < β ≤ α < β and find a , a > and b , b ∈ R such that the corresponding H satisfies σ ( H ) = [ α , β ] ∪ [ α , β ] . First, if there is a solution then, according to Theorem 10.1, α , β are theeigenvalues of h (0) and β , α are the eigenvalues of h ( π ) . Thus α + β = b + b = α + β , b b = ( a − a ) + α β and α β = α β − a a In particular, the numbers α , β , α , β are not completely arbitrary. In fact, once the identity α + β = α + β holds we get a a = ( β − α ) − ( α − β ) ; and b and b exist if, and only if, thediscriminant ( α − β ) − a − a ) ≥ . Here again we see that the gap is degenerate, i.e. α − β = 0 if, and only if, a = a = ( β − α ) / and b = b = α + β . All that is known for general periodicJacobi matrices, see [8, 17, 23]. The goal here is to show the usefulness of the identity (10.6) and tocontrast with the model considered in the example 9.1 where the inverse problem has a unique solution. A is a singular indefinite matrix Remark first that the assertions of the previous Theorem 10.2 remain valid if we replace A by a N E N .Here we study the combination of the two cases: we modify the last model by putting non zerocoefficients in the two corners of A . While the band/gap structure of the spectrum of H may breakdown our Mourre estimate still holds. More precisely, let N ≥ a , · · · , a N +1 > b , · · · , b N ∈ R .Let H = H ( A, B ) defined by (1.1) where A = a N E N + a N +1 E N and B = trid( { a i } ≤ i ≤ N − , { b i } ≤ i ≤ N ) (11.1)Notice that if N ≥ A is a singular indefinite matrix, since its spectrum is nothing but { , ±√ a N a N +1 } . We have, Theorem 11.1
1. There exist α < β , α < β , · · · , α N < β N such that the spectrum of H ispurely absolutely continuous and σ ( H ) = ∪ Nj =1 [ α j , β j ] ,
2. If (1.8) holds then we have a Mourre estimate for H on R \ κ ( H ) , i.e. µ A ( H ) = R \ κ ( H ) ( κ ( H ) is finite and contains { α , β , · · · , α N , β N } ). In particular, conclusions of Theorem 5.1hold for H It should be clear that one may deduce results for H = H + V for a class of compact perturbations V , but we will not state them separately.Again we will look at the eigenvalues of h ( p ) given in this case by h ( p ) = b a · · · z p a b a · · · · · · a N − b N − a N − z p · · · a N − b N with z p = a N e ip + a N +1 e − ip . Since h ( − p ) is the transpose of h ( p ), it is enough to study the operators h ( p ) for p ∈ [0 , π ]. Theorem11.1 is a consequence of the following: Theorem 11.2
The eigenvalues λ ( p ) , · · · , λ N ( p ) , of h ( p ) are non constant [0 , π ] . They are all simplefor all p ∈ [0 , π ] except for finite set of p ’s for which some of them may be double. Such set is reducedto { , π } if a N = a N +1 . The proof of Theorem 11.2 will be done in few steps that we state in lemmas.
Lemma 11.1 (i) The eigenvalues of h ( π/ are all simple.(ii) If a N = a N +1 then for all p ∈ ]0 , π [ , the eigenvalues of h ( p ) are all simple, while for p = 0 , π someof them may be double. Proof
Simple adaptation of the proof of Lemma 10.1.
Lemma 11.2
Let a ′ = ( − N − a a · · · a N − ( a N + a N +1 ) . The characteristic polynomial of h ( p ) isgiven by, D ( x, p ) = ∆( x ) − a N a N +1 det( ˙ B − x ) cos p + 2 a ′ cos p, (11.2) where, with ˙ B as in Lemma 10.2, ∆( x ) = det( B − x ) − ( a N − a N +1 ) det( ˙ B − x ) (11.3) Proof
Direct calculations.
Lemma 11.3
The derivative ∆ ′ of ∆ has N − roots µ < µ < · · · < µ N − and the λ j ’s may bechosen so that, for all j = 1 , · · · , N , ( − j ∆ ′ ( λ j ( π/ > . roof Again remark that ∆( x ) = D ( x, π/
2) and denote by λ ( π/ < λ ( π/ < · · · < λ N ( π/
2) thedistinct eigenvalues of h ( π/ x ) = D ( x, π/ j = 1 , · · · , N − µ j ∈ ] λ j ( π/ , λ j +1 ( π/ ′ ( µ j ) = 0 , Moreover, since ∆( x ) = ( − N x n + lower order, itis clear that ( − j ∆ ′ ( λ j ( π/ > j = 1 , · · · N . The proof is finished. Lemma 11.4
There is a neighborhood of π/ on which the eigenvalues λ j ( p ) of h ( p ) are monotonic. Proof
Let us fix j = 1 , · · · , N and put µ = −∞ , µ N = + ∞ . According to the Lemma 11.3, λ j ( π/ ∈ ] µ j − , µ j [ and ( − j ∆ ′ ( λ j ( p )) >
0. We also have ∂D∂p ( λ j ( π/ , π/
2) = − a ′ ( a N + a N +1 ) = 0 , Now after differentiation of the identity D ( λ j ( p ) , p ) = 0 at p = π/ λ ′ j ( π/
2) = 2 a ′ ( a N + a N +1 )∆ ′ ( λ j ( π/ = 0 . The proof is complete.
Example 11.1
Let A = a E , + a E , , B = tridiag ( { a , a } , { b i = 0 } ) , with a , a , a + a > and a , a ≥ . (i) If a = a then h ( p ) has only simple eigenvalues for all p ∈ (0 , π ) . Moreover, h (0) has a doubleeigenvalue if, and only if, a = a = a + a , which is equivalent to h ( π ) has a double eigenvalue. Insuch case, − ( a + a ) (resp. a + a ) is a double eigenvalues of h (0) (resp. h ( π ) ). (ii) Assume that a = a . In this case, h ( p ) = (2 a cos p ) A + B . Moreover, λ ′ j ( p ) = − a sin p h AW j ( p ) , W j ( p ) i = − a x j z j sin p where ( x j , y j , z j ) are the coordinate of W j ( p ) . This derivative vanishes if, and only if, p = 0 , π or x j z j = 0 . By writing the eigenvalue problem for h ( p ) we find out that, x j z j = 0 if, and only if,(i) x j = 0 leading to λ j = ± a , ± cos p ± = ∓ a / a , provided a ≤ a or;(ii) z j = 0 leading to λ j = ± a , ± cos p ′± = ∓ a / a , provided a ≤ a .Moreover, h ( p ) has a double eigenvalue if, and only if, a = a < a and p = p ± = p ′± .Let us arrange the eigenvalues of h ( p ) so that λ ( π/
2) = − p a + a , λ ( π/
2) = 0 , λ ( π/
2) = p a + a . Using the preceding formula of λ ′ j one may prove that λ ′ ( π/ < , λ ′ ( π/ > and λ ′ ( π/ < . In particular,1. if a > a and a > a then, for all j , λ ′ j ( p ) never vanishes on (0 , π ) so they have the signof λ ′ j ( π/ . Hence, the spectrum of H is absolutely continuous and σ ( H ) = [ λ ( π ) , λ (0)] ∪ [ λ (0) , λ ( π )] ∪ [ λ ( π ) , λ (0)] . Moreover, λ (0) < λ (0) otherwise one may find p ∈ [0 , π/ suchthat λ ( p ) = λ ( p ) which contradicts the simplicity of the eigenvalues. Similarly, λ ( π ) < λ ( π ) so that we have two gaps.2. If a ≤ a < a then there is two points p ± given by cos p ± = ∓ a / a for which h ( p ± ) has ± a as a simple eigenvalue whose corresponding eigenvectors satisfy x j z j = 0 . Hence κ ( H ) = {± a , λ j (0) , λ j ( π ) , j = 1 , , } . For example, if a = a = 1 and a = 2 one may deducethat the spectrum of H is purely absolutely continuous with two gaps and, for some α > , σ ( H ) = [ − α, − ∪ [ − / , / ∪ [1 , α ] .
3. Finally if a = a < a then,there is two points p ± given by cos p ± = ∓ a/ a for which h ( p ± ) has ± a as a double eigenvalue whose corresponding eigenvectors satisfy x j z j = 0 . Hence onemay deduce that we only have overlapping spectral bands.
12 An example where A is a lower triangular matrix: Symmetricfourth-order difference operators Let a n , b n > v n ∈ R . Consider the difference operator J = J ( a n , b n , v n ) acting in l ( Z ) by( J x ) n = a n x n +2 + b n x n +1 + v n x n + b n − x n − + a n − x n − . a, b, c, d > σ ≥
0, one haslim | n |→∞ | n | σ ( | a n − a | + | a n +1 − b | + | b n +1 − c | ) + | b n − d | + | v n | ) = 0 , Then J is a bounded symmetric operator. Moreover, We have, Theorem 12.1 (i) There exist α < β and α < β such that σ ess ( J ) = [ α , β ] ∪ [ α , β ] .(ii) If σ ≥ , then there is a self-adjoint operator A ′ such that ˜ µ A ′ ( J ) = R \ κ ( J ) for some finiteset κ ( J ) . In particular, all possible eigenvalues of J are finitely degenerate and cannot accumulateoutside κ ( J ) .(iii) Finally, if σ > then J has no singular continuous spectrum and conclusions of Theorems 1.2and 1.3 hold for J . Proof
Consider the unitary operator U : l ( Z ) → l ( Z , C ) defined by ( U x ) n = ( x n , x n +1 ). Onemay verify that U J U − = H ( A n , B n ) with A n = (cid:18) a n b n +1 a n +1 (cid:19) , B = (cid:18) v n b n b n v n +1 (cid:19) . Ourhypothesis means that H = H ( A n , B n ) is a compact perturbation of H = H ( A, B ) defined by (1.3)with A = (cid:18) a c b (cid:19) , B = (cid:18) dd (cid:19) . In this case, h ( p ) = (cid:18) a cos p z p z p b cos p (cid:19) , with z p = d + ce ip . One may easily prove that, h ( p ) has a double eigenvalue if, and only if, p = π, a = b and c = d .Moreover, for any p ∈ (0 , π ), h ( p ) has two simple eigenvalues given by λ j ( p ) = ( a + b ) cos p + ( − j p ( a − b ) cos p + 4 dc cos p/ d − c ) , j = 1 , . Hence our general method applies directly here also. The proof is complete.
Example 12.1
Suppose that a = b and c = d . Then λ is decreasing and generates the spectral band Σ = [ − a, a + d )] . In contrast, we have:1. if K := d a > then λ is increasing and Σ = [2( a − d ) , − a ] . In this case, κ ( J ) = ∂ Σ ∪ ∂ Σ where ∂ Σ i consists of the edges of Σ i ; and σ ( J ) = [2( a − d ) , − a ] ∪ [ − a, a + d )] = [2( a − d ) , a + d )] (the spectral bands touch in one point − a which is a double eigenvalue of h ( π ) ).2. if K ≤ then λ has a critical point at p given by cos p / K . In this case, λ ( p ) = − a (1 + 2 K ) and there is two possibilities. If a ≤ d ≤ a then Σ = [ − a (1 + 2 K ) , − a ] sothat the two bands Σ and Σ touch in one point. In contrary, if < d ≤ a then Σ = [ − a (1 +2 K ) , a − d )] so that Σ ∩ Σ = [ − a, a − d )] . In both cases κ ( J ) = ∂ Σ ∪ ∂ Σ ∪ { λ ( p ) } and σ ( J ) = [ − a (1 + 2 K ) , a + d )] .Of course in this case J is a compact perturbation of J = J ( a n = a, b n = d, v n = 0) that one maystudy directly. Indeed, by Fourier transform, J is unitarily equivalent to the multiplication operatorby the function f ( p ) = 2 d cos p + 2 a cos 2 p acting in L ( − π, π ) . Example 12.2
Suppose that a = b and d = c . Then here again λ is decreasing and Σ = [ − a + | d − c | , a + d + c ] and1. if a < dc d + c ) := K then λ is increasing and Σ = [2 a − ( d + c ) , − a − | d − c | ] . In this case, κ ( J ) = ∂ Σ ∪ ∂ Σ and σ ( H ) = [2 a − ( d + c ) , − a − | d − c | ] ∪ [ − a + | d − c | , a + d + c ] (a spectralgap of length | d − c | ).2. if a > dc | d − c | := K then λ is decreasing and Σ = [ − a − | d − c | , a − ( d + c )] . In this case, κ ( J ) = ∂ Σ ∪ ∂ Σ and σ ( H ) = [ − a − | d − c | , a − ( d + c )] ∪ [ − a + | d − c | , a + d + c ] . Onemay have a non trivial gap in some cases (e.g. d = 10 , c = 1 and a = 1 ) and overlapping bandsin others (e.g. d = 2 , c = 1 and a = 3 / ). . If K ≤ a ≤ K then λ has a critical point at p given by cos p = dc a − d + c dc . It generates thespectral band Σ = [ α, β ] where α = min( λ (0) , λ ( p ) , λ ( π )) and β = max( λ (0) , λ ( p ) , λ ( π )) .In this case, κ ( J ) = ∂ Σ ∪ ∂ Σ ∪ {− a − K } and σ ( H ) = [ α, β ] ∪ [ − a, a + d )] . For example,let c = 2 , d = 1 . Then K ≤ a ≤ K means that / ≤ a ≤ . Σ = [ − / , − and Σ = [ − , and we have a spectral gap. Σ = [ − , − and Σ = [ − , and we have overlapping spectralbands. Example 12.3
It remains to study the case where a = b . To be simple we only discuss the followingtwo situations.1. If c = d = 0 then λ ( p ) = 2 a cos p and λ ( p ) = 2 b cos p so that we have two spectral bands onecontained in the another.2. In contrast if d = 0 and c > then both λ ′ j s are monotonic and one may choose a, b, c to getoverlapping bands or bands with a spectral gap between them. Here κ ( J ) is the set of the edgesof the spectral bands.
13 Further developments