About essential spectra of unbounded Jacobi matrices
aa r X i v : . [ m a t h . SP ] J un ABOUT ESSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES
GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJANAbstract. We study spectral properties of unbounded Jacobi matrices with periodically modulated or blendedentries. Our approach is based on uniform asymptotic analysis of generalized eigenvectors. We determine whenthe studied operators are self-adjoint. We identify regions where the point spectrum has no accumulation points.This allows us to completely describe the essential spectrum of these operators. Introduction
Consider two sequences a = ( a n : n ∈ N ) and b = ( b n : n ∈ N ) such that a n > and b n ∈ R for all n ≥ . Let A be the closure in ℓ ( N ) of the operator acting on sequences having finite support by the matrix © « b a . . . a b a . . . a b a . . . a b ... ... ... . . . ª®®®®®®¬ . The operator A is called Jacobi matrix . Recall that ℓ ( N ) is the Hilbert space of square summable complexvalued sequences with the scalar product h x , y i ℓ ( N ) = ∞ Õ n = x n y n . The most throughly studied are bounded Jacobi matrices, see e.g. [32]. Let us remind that the Jacobi matrix A is bounded if and only if the sequences a and b are bounded. In this article we are exclusively interested in unbounded Jacobi matrices. We shall consider two classes: periodically modulated and periodically blended.The first class has been introduced in [17] and systematically studied since then. To be precise, let N be apositive integer. We say that A has N -periodically modulated entries if there are two N -periodic sequences ( α n : n ∈ Z ) and ( β n : n ∈ Z ) of positive and real numbers, respectively, such thata) lim n →∞ a n = ∞ , b) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) a n − a n − α n − α n (cid:12)(cid:12)(cid:12)(cid:12) = , c) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) b n a n − β n α n (cid:12)(cid:12)(cid:12)(cid:12) = . This class contains sequences one can find in many applications. It is also rich enough to allow building anintuition about the general case. In particular, in this class there are examples of Jacobi matrices with purelyabsolutely continuous spectrum filling the whole real line (see [15, 17, 35, 38, 41]), having a bounded gap inabsolutely continuous spectrum (see [5–7, 9, 10, 12, 14, 19, 27]), having absolutely continuous spectrum onthe half-line (see [4, 8, 16, 18, 23–26, 33]), having purely singular continuous spectral measure with explicitHausdorff dimension (see [2]), having a dense point spectrum on the real line (see [2]), and having an emptyessential spectrum (see [11, 29–31, 42]).
Mathematics Subject Classification.
Primary: 47B25, 47B36. Secondary: 42C05, 39A22.
Key words and phrases.
Jacobi matrix, orthogonal polynomials, essential spectrum, discrete spectrum, discrete Levinson’s typetheorems.
The second class, that is blended Jacobi matrices (see Definition 2.4), has been introduced in [1] as anexample of unbounded Jacobi matrices having absolutely continuous spectrum equal to a finite union ofcompact intervals. It has been further studied in [39, 41] in the context of orthogonal polynomials.Before we formulate the main results of this paper, let us introduce some definitions. In our investigation,the crucial rôle is played by the transfer matrix defined as B j ( x ) = − a j − a j x − b j a j ! . We say that a sequence ( x n : n ∈ N ) of vectors from a normed vector space V belongs to D r ( V ) for a certain r ∈ N , if it is bounded, and for each j ∈ { , . . . , r } , ∞ Õ n = (cid:13)(cid:13) ∆ j x n (cid:13)(cid:13) rj < ∞ where ∆ x n = x n , ∆ j x n = ∆ j − x n + − ∆ j − x n , j ≥ . If X is the real line with Euclidean norm we abbreviate D r = D r ( X ) . Given a compact set K ⊂ C and anormed vector space R , we denote by D r ( K , R ) the case when X is the space of all continuous mappingsfrom K to R equipped with the supremum norm. Theorem A.
Suppose that A is a Jacobi matrix with N -periodically modulated entries. Let X ( x ) = lim n →∞ X nN ( x ) where X n ( x ) = B n + N − ( x ) B n + N − ( x ) · · · B n ( x ) . Assume that discr X ( ) > . If there are a compact set K ⊂ R with at least N + points, r ∈ N and i ∈ { , , . . . , N − } , so that (1.1) (cid:0) X nN + i : n ∈ N (cid:1) ∈ D r (cid:0) K , Mat ( , R ) (cid:1) , then A is self-adjoint and σ ess ( A ) = ∅ . Recall that a sufficient condition for self-adjointness of the operator A is the Carleman’s condition (seee.g. [28, Corollary 6.19]), that is(1.2) ∞ Õ n = a n = ∞ . The conclusion of Theorem A is in strong contrast with the case when discr X ( ) < . Indeed, if discr X ( ) < , then by [41, Theorem A], the operator A is self-adjoint if and only if the Carleman’scondition is satisfied. If it is the case then A is purely absolutely continuous and σ ( A ) = R .Under the Carleman’s condition, the conclusion of Theorem A for r = has been proven in [17] by showingthat the resolvent of A is compact. Furthermore, by [3, Theorem 8] (see also [42, Theorem 2.6]) it followsthat if a self-adjoint Jacobi matrix A is -periodically modulated with discr X ( ) > , then σ ess ( A ) = ∅ , i.e.the condition (1.1) is not necessary here. For a real matrix X we define its discriminant as discr X = ( tr X ) − X . By Mat ( d , R ) we denote the real matrices of dimension d × d with the operator norm. For a self-adjoint operator A we denote by σ ess ( A ) , σ ac ( A ) and σ sing ( A ) its essential spectrum, the essential spectrum and thesingular spectrum, respectively. SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 3
Theorem B.
Suppose that A is a Jacobi matrix with N -periodically blended entries. Let X ( x ) = lim n →∞ X n ( N + ) + ( x ) where X n ( x ) = B n + N + ( x ) B n + N ( x ) · · · B n ( x ) . If there are a compact set K ⊂ R with at least N + points, r ∈ N , and i ∈ { , , . . . , N } , so that (cid:0) X n ( N + ) + i : n ∈ N (cid:1) ∈ D r (cid:0) K , Mat ( , R ) (cid:1) , then A is self-adjoint and σ sing ( A ) ∩ Λ = ∅ and σ ac ( A ) = σ ess ( A ) = Λ where Λ = (cid:8) x ∈ R : discr X ( x ) < (cid:9) . For the proof of Theorem B see Theorem 5.1. Let us comment that in Theorem B, the absolute continuityof A follows by [41, Theorem B]. Moreover, by [39, Theorem 3.13] it stems that Λ is a union of N opendisjoint bounded intervals. For r = and under certain very strong assumptions, Theorem B has been provenin [1, Theorem 5].The following results concerns the case when discr X ( ) = . For the proof see Theorem 5.5. Theorem C.
Let A be a Jacobi matrix with N -periodically modulated entries, and let X n and X be defined asin Theorem A. Suppose that X ( ) = σ Id for any σ ∈ {− , } , and that there are two N -periodic sequences ( s n : n ∈ N ) and ( z n : n ∈ N ) , such that lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) α n − α n a n − a n − − s n (cid:12)(cid:12)(cid:12)(cid:12) = , lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) β n α n a n − b n − z n (cid:12)(cid:12)(cid:12)(cid:12) = . Let R n = a n + N − ( X n − σ Id ) . Then ( R k N : k ∈ N ) converges locally uniformly on R to R . If there are acompact set K ⊂ R with at least N + points and i ∈ { , , . . . , N } , so that (cid:0) R nN + i : n ∈ N (cid:1) ∈ D (cid:0) K , Mat ( , R ) (cid:1) , then A is self-adjoint and σ sing ( A ) ∩ Λ = ∅ and σ ac ( A ) = σ ess ( A ) = Λ where Λ = (cid:8) x ∈ R : discr R ( x ) < (cid:9) . In fact, Theorem C completes the analysis started in [36] where it has been shown that Λ ⊂ σ ac ( A ) .Finally, we investigate the case when the Carleman’s condition (1.2) is not satisfied. Theorem D.
Let A be a Jacobi matrix with N -periodically modulated entries, and let X n and X be definedas in Theorem A. Suppose that X ( ) = σ Id for any σ ∈ {− , } , and that the Carleman’s condition is not satisfied. Assume that there are i ∈ { , , . . . , N − } and a sequence of positive numbers ( γ n : n ∈ N ) satisfying ∞ Õ n = γ n = ∞ , such that R nN + i ( ) = γ n ( X nN + i ( ) − σ Id ) converges to a non-zero matrix R i . Suppose that (cid:0) R nN + i ( ) : n ∈ N (cid:1) ∈ D (cid:0) K , Mat ( , R ) (cid:1) . Then(i) if discr R i < , then A is not self-adjoint;(ii) if discr R i > , then σ ess ( A ) = ∅ provided A is self-adjoint. GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN
In fact, in Theorem 6.3 we characterize when A is self-adjoint. To illustrate Theorem D, in Section 6.2we consider the N -periodically modulated Kostyuchenko–Mirzoev’s class. In this context we can preciselydescribe when the operator A is self-adjoint.In our analysis the basic objects are generalized eigenvectors of A . Let us recall that ( u n : n ∈ N ) is ageneralized eigenvector associated with x ∈ C , if for all n ≥ (cid:18) u n u n + (cid:19) = B n ( x ) (cid:18) u n − u n (cid:19) for a certain ( u , u ) , ( , ) . The spectral properties of A are intimately related to the asymptotic behaviorof generalized eigenvectors. For example, A is self-adjoint if and only if there is a generalized eigenvectorassociated with some x ∈ R , that is not square-summable. In another vein, the theory of subordinacy(see [20]) describes spectral properties of a self-adjoint A in terms of asymptotic behavior of generalizedeigenvectors. In particular, it has been shown in [30] that the subordinacy theory together with some generalproperties of self-adjoint operators imply the following: if K ⊂ R is a compact interval such that for each x ∈ K there is a generalized eigenvector ( u n ( x ) : n ∈ N ) associated with x ∈ K , so that(1.3) ∞ Õ n = sup x ∈ K | u n ( x )| < ∞ , then σ ess ( A ) ∩ K = ∅ . In [30], for some class of Jacobi matrices the condition (1.3) has been checked with ahelp of uniform discrete Levinson’s type theorems. In this article we take similar approach. In particular, inTheorems 4.1 and 4.4, we prove our uniform Levinson’s theorems. They improve the existing results knownin the literature. More precisely, Theorem 4.1 with r ≥ in the case of negative discriminant, improves thepointwise theorem [13, Theorem 3.1]. The case of positive discriminant for r > has not been studied before,even pointwise. Concerning the uniformity, Theorem 4.1 improves [29], where for r = it was assumedthat the limiting matrix is constant. Our analysis shows that this condition can be dropped (see the commentafter proof of Theorem 4.4). We prove uniformity by constructing explicit diagonalization of the relevantmatrices. The case of positive discriminant provides more technical challenges than the negative one. If theCarleman’s condition is not satisfied, our Levinson’s type theorems allowed us to study asymptotic behaviorof generalized eigenvectors on the whole complex plane for a large class of sequences a and b . In particular,our results cover the asymptotic recently obtained by Yafaev in [43], see Corollary 6.2 for details. Let usemphasize that our approach is different than used in [43].The organization of the paper is as follows. In Section 2 we collect basic properties and definitions. Inparticular, we prove axillary results concerning periodically modulated and blended Jacobi matrices. InSection 3 we describe Stolz classes, and prove results necessary to show in Section 4 our Levinson’s typetheorems which might be of independent interest. In Section 5 we apply them to deduce Theorems A, Band C. Finally, in Section 6 we prove Theorem D, and study the Kostyuchenko–Mirzoev’s class of Jacobimatrices in details. Notation. By N we denote the set of positive integers and N = N ∪ { } . Throughout the whole article, wewrite A . B if there is an absolute constant c > such that A ≤ cB . We write A ≍ B if A . B and B . A .Moreover, c stands for a positive constant whose value may vary from occurrence to occurrence. Acknowledgment.
The first author was partially supported by the Foundation for Polish Science (FNP) andby long term structural funding – Methusalem grant of the Flemish Government.
SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 5 Preliminaries
Given two sequences a = ( a n : n ∈ N ) and b = ( b n : n ∈ N ) of positive and real numbers, respectively,we define k th associated orthonormal polynomials as p [ k ] ( x ) = , p [ k ] ( x ) = x − b k a k , a n + k − p [ k ] n − ( x ) + b n + k p [ k ] n ( x ) + a n + k p [ k ] n + ( x ) = xp [ k ] n ( x ) , n ≥ . We usually omit the superscript if k = . Suppose that the Jacobi matrix A corresponding to the sequences a and b is self-adjoint. Let us denote by E A its spectral resolution of the identity. Then for any Borel subset B ⊂ R , we set µ ( B ) = h E A ( B ) δ , δ i ℓ ( N ) where δ is the sequence having on the th position and elsewhere. The polynomials ( p n : n ∈ N ) forman orthonormal basis of L ( R , µ ) .In this article, we are interested in Jacobi matrices associated to two classes of sequences that are defined interms of periodic Jacobi parameters. The latter are described as follows. Let ( α n : n ∈ Z ) and ( β n : n ∈ Z ) betwo N -periodic sequences of real and positive numbers, respectively. Let ( p n : n ∈ N ) be the correspondingpolynomials, that is p ( x ) = , p ( x ) = x − β α ,α n p n − ( x ) + β n p n ( x ) + α n p n + ( x ) = x p n ( x ) , n ≥ . Let B n ( x ) = (cid:18) − α n − α n x − β n α n (cid:19) , and X n ( x ) = N + n − Ö j = n B j ( x ) , n ∈ Z where for a sequence of square matrices ( C n : n ≤ n ≤ n ) we have set n Ö k = n C k = C n C n − · · · C n . Periodic modulation.Definition 2.1.
We say that the Jacobi matrix A associated to ( a n : n ∈ N ) and ( b n : n ∈ N ) has N -periodically modulated entries, if there are two N -periodic sequences ( α n : n ∈ Z ) and ( β n : n ∈ Z ) ofpositive and real numbers, respectively, such that(a) lim n →∞ a n = ∞ , (b) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) a n − a n − α n − α n (cid:12)(cid:12)(cid:12)(cid:12) = , (c) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) b n a n − β n α n (cid:12)(cid:12)(cid:12)(cid:12) = . For a Jacobi matrix A with N -periodically modulated entries, we set X n = N + n − Ö j = n B j . Then for each i ∈ { , , . . . , N − } the sequence ( X j N + i : j ∈ N ) has a limit X i . In view of [39, Proposition3.8], we have X i ( x ) = X i ( ) for all x ∈ C . GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN
Proposition 2.2.
Let N be a positive integer and σ ∈ {− , } . Let A be a Jacobi matrix with N -periodicallymodulated entries so that X ( ) = σ Id . Suppose that there are two N -periodic sequences ( s n : n ∈ Z ) and ( z n : n ∈ Z ) , such that lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) α n − α n a n − a n − − s n (cid:12)(cid:12)(cid:12)(cid:12) = , lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) β n α n a n − b n − z n (cid:12)(cid:12)(cid:12)(cid:12) = , then for each i ∈ { , , . . . , N − } the sequence (cid:0) a ( k + ) N + i − ( X k N + i − σ Id ) : k ∈ N (cid:1) converges locallyuniformly on C to R i , and tr R i = − σ lim k →∞ (cid:0) a ( k + ) N + i − − a k N + i − (cid:1) . Proof.
According to [36, Proposition 9], we have(2.1) R i ( x ) = α i − C i ( x ) + α i − D i where C i ( x ) = x © « − α i − α i (cid:16) p [ i + ] N − (cid:17) ′ ( ) (cid:16) p [ i ] N − (cid:17) ′ ( )− α i − α i (cid:16) p [ i + ] N − (cid:17) ′ ( ) (cid:16) p [ i ] N (cid:17) ′ ( ) ª®¬ and(2.2) D i = N − Õ j = α i + j ( N − Ö m = j + B i + m ( ) ) (cid:18) s i + j z i + j (cid:19) ( j − Ö m = B i + m ( ) ) . In view of [36, Proposition 6],(2.3) tr C i ≡ . Since the trace is linear and invariant under cyclic permutations, by (2.2) we get(2.4) tr D i = N − Õ j = α i + j tr ((cid:18) s i + j z i + j (cid:19) N + j − Ö m = j + B i + m ( ) ) . Using [36, Proposition 3] N + j − Ö m = j + B i + m ( ) = − α i + j α i + j + p [ i + j + ] N − ( ) p [ i + j + ] N − ( )− α i + j α i + j + p [ i + j + ] N − ( ) p [ i + j + ] N − ( ) ! , thus tr ((cid:18) s i + j z i + j (cid:19) N + j − Ö m = j + B i + m ( ) ) = s i + j p [ i + j + ] N − ( ) + z i + j p [ i + j + ] N − ( ) = − σ α i + j α i + j − s i + j , (2.5)where the last equality follows by [36, formula (13)]. Inserting (2.5) into (2.4) results in tr D i = − σ N − Õ j = s i + j α i + j − . Hence, by (2.1) and (2.3), we get tr R i = α i − tr D i = − σα i − N − Õ j = s i + j α i + j − . Finally, by [40, Proposition 3], we obtain tr R i = − σ lim k →∞ (cid:0) a ( k + ) N + i − − a k N + i − (cid:1) , SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 7 which completes the proof. (cid:3)
Periodic blend.Definition 2.3.
The Jacobi matrix A associated to ( a n : n ∈ N ) and ( b n : n ∈ N ) has asymptotically N -periodic entries if there are two N -periodic sequences ( α n : n ∈ Z ) and ( β n : n ∈ Z ) of positive and realnumbers, respectively, such that(a) lim n →∞ (cid:12)(cid:12) a n − α n (cid:12)(cid:12) = ,(b) lim n →∞ (cid:12)(cid:12) b n − β n (cid:12)(cid:12) = . Definition 2.4.
The Jacobi matrix A associated with sequences ( a n : n ∈ N ) and ( b n : n ∈ N ) has a N -periodically blended entries if there are an asymptotically N -periodic Jacobi matrix ˜ A associated withsequences ( ˜ a n : n ∈ N ) and ( ˜ b n : n ∈ N ) , and a sequence of positive numbers ( ˜ c n : n ∈ N ) , such that(a) lim n →∞ ˜ c n = ∞ , and lim m →∞ ˜ c m + ˜ c m = ,(b) a k ( N + ) + i = ˜ a k N + i if i ∈ { , , . . . , N − } , ˜ c k if i = N , ˜ c k + if i = N + , (c) b k ( N + ) + i = ( ˜ b k N + i if i ∈ { , , . . . , N − } , if i ∈ { N , N + } . If A is a Jacobi matrix having N -periodically blended entries, we set X n ( x ) = N + n + Ö j = n B j ( x ) . By [39, Proposition 3.12], for each i ∈ { , , . . . , N − } , lim j →∞ B j ( N + ) + i ( x ) = B i ( x ) , locally uniformly with respect to x ∈ C , thus the sequence ( X j ( N + ) + i : j ∈ N ) converges to X i locallyuniformly on C where(2.6) X i ( x ) = (cid:18) i − Ö j = B j ( x ) (cid:19) C( x ) (cid:18) N − Ö j = i B j ( x ) (cid:19) , and C( x ) = (cid:18) − α N − α − x − β α (cid:19) . Moreover, we have the following proposition.
Proposition 2.5. lim j →∞ B − j ( N + ) ( x ) = (cid:18) (cid:19) , (2.7) lim j →∞ B j ( N + ) + N ( x ) = (cid:18) (cid:19) , (2.8) lim j →∞ B j ( N + ) + N + ( x ) = (cid:18) − (cid:19) . (2.9) locally uniformly with respect to x ∈ C . GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN
Proof.
The proposition easily follows from Definition 2.4. Indeed, we have B − j ( N + ) ( x ) = x − ˜ b j N ˜ c j − − ˜ a j N ˜ c j − ! , and B j ( N + ) + N ( x ) = − ˜ a j N + N − ˜ c j x ˜ c j ! , B j ( N + ) + N + ( x ) = − ˜ c j ˜ c j + x ˜ c j + ! . Thus using Definition 2.4(i) and boundedness of the sequence ( ˜ a n : n ∈ N ) , we can compute the limits. (cid:3)
3. Stolz class
In this section we define a proper class of slowly oscillating sequences which is motivated by [34], seealso [41, Section 2]. Let X be a normed space. We say that a bounded sequence ( x n ) belongs to D r , s ( X ) forcertain r ∈ N and s ∈ { , , . . . , r − } , if for each j ∈ { , . . . , r − s } , ∞ Õ n = (cid:13)(cid:13) ∆ j x n (cid:13)(cid:13) rj + s < ∞ . Moreover, ( x n ) ∈ D r , s ( X ) , if ( x n ) ∈ D r , s ( X ) and ∞ Õ n = k x n k rs < ∞ . Let us observe that D r , s ( X ) ⊂ D r , ( X ) , and D r , r − ( X ) = D , ( X ) . To simplify the notation, if X is the real line with Euclidean norm we shortly write D r , s = D r , s ( R ) . Given acompact set K ⊂ C and a normed vector space R , by D r , s ( K , R ) we denote the case when X is the space ofall continuous mappings from K to R equipped with the supremum norm. Moreover, given a positive integer N , we say that ( x n ) ∈ D Nr , s ( X ) if for each i ∈ { , , . . . , N − } , (cid:0) x nN + i : n ∈ N (cid:1) ∈ D r , s ( X ) . Lemma 3.1.
Let d and M be positive integers, and let K ⊂ C be a set with at least M + points. Supposethat ( x n : n ∈ N ) is a sequence of elements from Mat ( d , P M ) where P M denotes the linear space of complexpolynomials of degree at most M . If there are r ≥ and s ∈ { , , . . . , r − } so that for all z ∈ K , (cid:0) x n ( z ) : n ∈ N (cid:1) ∈ D r , s (cid:0) Mat ( d , C ) (cid:1) , then for every compact set K ⊂ C , (cid:0) x n : n ∈ N (cid:1) ∈ D r , s (cid:0) K , Mat ( d , C ) (cid:1) . Proof.
Let { z , z , . . . , z M } be a subset of K consisting of distinct points. By the Lagrange interpolationformula, we can write x n ( z ) = M Õ j = ℓ j ( z ) x n ( z j ) where ℓ j ( z ) = M Ö m = m , j z − z m z j − z m . Let K be a compact subset of C . Then there is a constant c > such that for any j ∈ { , , . . . , M } , sup z ∈ K | ℓ j ( z )| ≤ c . SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 9
Since, for each k ≥ , ∆ k x n ( z ) = M Õ j = ℓ j ( z ) ∆ k x n ( z j ) , we obtain sup z ∈ K (cid:13)(cid:13) ∆ k x n ( z ) (cid:13)(cid:13) ≤ c M Õ j = (cid:13)(cid:13) ∆ k x n ( z j ) (cid:13)(cid:13) , and the conclusion follows. (cid:3) The following lemma is well-known and its proof is straightforward.
Lemma 3.2.
For any two sequences ( x n ) and ( y n ) , and j ∈ N , ∆ j ( x n y n : n ∈ N ) n = j Õ k = (cid:18) jk (cid:19) ∆ j − k x n · ∆ k y n + j − k . Corollary 3.3 ( [41, Corollary 1]) . Let r ∈ N and s ∈ { , . . . , r − } .(i) If ( x n ) ∈ D r , ( X ) and ( y n ) ∈ D r , s ( X ) then ( x n y n ) ∈ D r , s ( X ) .(ii) If ( x n ) , ( y n ) ∈ D r , s ( X ) , then ( x n y n ) ∈ D r , s ( X ) . Lemma 3.4 ( [41, Lemma 2]) . Fix r ∈ N , s ∈ { , . . . , r − } and a compact set K ⊆ R . Let ( f n : n ∈ N ) ∈ D r , s ( K , R ) be a sequence of real functions on K with values in I ⊆ R and let F ∈ C r − s ( I , R ) . Then ( F ◦ f n : n ∈ N ) ∈ D r , s ( K , R ) . By Lemma 3.4, we easily get the following corollary.
Corollary 3.5.
Let r ∈ N . If ( x n ) ∈ D r , ( K , C ) , and < δ ≤ | x n ( x )| , for all n ∈ N and x ∈ K , then ( x − n : n ∈ N ) ∈ D r , ( K , C ) . The next theorem is the main result of this section.
Theorem 3.6.
Fix two integers r ≥ and s ∈ { , . . . , r − } , and a compact set K ⊂ R . Suppose that ( λ + n : n ∈ N ) and ( λ − n : n ∈ N ) are two uniform Cauchy sequences from D r , ( K , R ) so that for all x ∈ K and n ∈ N , (3.1) λ + n ( x ) λ − n ( x ) > , | λ + n ( x )| − | λ − n ( x )| ≥ δ > . Let ( X n : n ∈ N ) ∈ D r , s (cid:0) K , GL ( , R ) (cid:1) be such that (3.2) sup x ∈ K sup n ∈ N (cid:0) k X n ( x )k + k X − n ( x )k (cid:1) < ∞ . Then there are sequences ( µ + n : n ∈ N ) , ( µ − n : n ∈ N ) ∈ D r , ( K , R ) and ( Y n : n ∈ N ) ∈ D r , s + (cid:0) K , GL ( , R ) (cid:1) satisfying (3.3) (cid:18) λ + n λ − n (cid:19) X − n X n − = Y n (cid:18) µ + n µ − n (cid:19) Y − n , such that ( µ + n : n ∈ N ) and ( µ − n : n ∈ N ) are uniform Cauchy sequences with µ + n ( x ) µ − n ( x ) > , | µ + n ( x )| − | µ − n ( x )| ≥ δ ′ > , for all x ∈ K and n ∈ N . Moreover, (3.4) lim n →∞ sup x ∈ K (cid:13)(cid:13) Y n ( x ) − Id (cid:13)(cid:13) = . Proof.
Let D n = (cid:18) λ + n λ − n (cid:19) . We set W n = D n X − n X n − = D n (cid:0) Id − X − n ∆ X n − (cid:1) . By (3.2), we have sup K (cid:13)(cid:13) W n − D n (cid:13)(cid:13) = sup K (cid:13)(cid:13) D n X − n ∆ X n − (cid:13)(cid:13) ≤ c sup K (cid:13)(cid:13) ∆ X n − (cid:13)(cid:13) . Since ( X n ) ∈ D r , s (cid:0) K , GL ( , R ) (cid:1) , lim n →∞ sup K k ∆ X n k = , thus lim n →∞ sup K (cid:13)(cid:13) W n − D n (cid:13)(cid:13) = . In particular, W n has positive discriminant. Let µ + n and µ − n be its eigenvalues with | µ + n | > | µ − n | . Then lim n →∞ sup K (cid:12)(cid:12) µ + n − λ + n (cid:12)(cid:12) = , and lim n →∞ sup K (cid:12)(cid:12) µ − n − λ − n (cid:12)(cid:12) = , and hence ( µ + n : n ∈ N ) and ( µ − n : n ∈ N ) are a uniform Cauchy sequence satisfying (3.1). Setting X n = x ( n ) x ( n ) x ( n ) x ( n ) ! , and W n = w ( n ) w ( n ) w ( n ) w ( n ) ! , we obtain W n = X n λ + n (cid:0) x ( n − ) x ( n ) − x ( n − ) x ( n ) (cid:1) λ + n (cid:0) x ( n − ) x ( n ) − x ( n − ) x ( n ) (cid:1) λ − n (cid:0) x ( n − ) x ( n ) − x ( n − ) x ( n ) (cid:1) λ − n (cid:0) x ( n − ) x ( n ) − x ( n − ) x ( n ) (cid:1) ! . By (3.2) and Corollary 3.5, we have (cid:18) X n (cid:19) ∈ D r , , hence by Corollary 3.3(ii), we get that (cid:0) w ( n ) : n ∈ N (cid:1) , (cid:0) w ( n ) : n ∈ N (cid:1) ∈ D r , . Moreover, w ( n ) = λ + n det X n (cid:0) x ( n − ) x ( n ) − x ( n − ) x ( n ) (cid:1) = λ + n det X n (cid:16) (cid:0) x ( n ) − x ( n − ) (cid:1) x ( n ) − (cid:0) x ( n ) − x ( n − ) (cid:1) x ( n ) (cid:17) , and w ( n ) = λ − n det X n (cid:16) (cid:0) x ( n ) − x ( n − ) (cid:1) x ( n ) − (cid:0) x ( n ) − x ( n − ) (cid:1) x ( n ) (cid:17) , thus, by Corollary 3.3(i), (cid:0) w ( n ) : n ∈ N (cid:1) , (cid:0) w ( n ) : n ∈ N (cid:1) ∈ D r , s + . Next, we compute the eigenvalues. We obtain µ + n = w ( n ) + w ( n ) + σ n p discr W n , and µ − n = w ( n ) + w ( n ) − σ n p discr W n where σ n = sign ( w ( n ) ) , and discr W n = (cid:0) w ( n ) − w ( n ) (cid:1) + w ( n ) w ( n ) . SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 11
Since for all n sufficiently large(3.5) (cid:12)(cid:12) w ( n ) − w ( n ) (cid:12)(cid:12) ≥ (cid:12)(cid:12) λ + n − λ − n (cid:12)(cid:12) − (cid:12)(cid:12) w ( n ) − λ + n (cid:12)(cid:12) − (cid:12)(cid:12) w ( n ) − λ − n (cid:12)(cid:12) ≥ δ , by Lemma 3.4, we have ( µ + n ) , ( µ − n ) ∈ D r , ( K , R ) . It remains to compute the matrix Y n . Suppose that theequations(3.6) W n (cid:18) v + n (cid:19) = µ + n (cid:18) v + n (cid:19) and W n (cid:18) v − n (cid:19) = µ − n (cid:18) v − n (cid:19) both have solutions, then the matrix Y n = (cid:18) v − n v + n (cid:19) satisfies (3.3). Observe that equations (3.6) are equivalent to ( w ( n ) + v + n w ( n ) = µ + n , w ( n ) + v + n w ( n ) = µ + n v + n , and ( w ( n ) v − n + w ( n ) = µ − n v − n , w ( n ) v − n + w ( n ) = µ − n . (3.7)If σ n = then by (3.5), w ( n ) − w ( n ) − p discr W n ≤ − δ , otherwise w ( n ) − w ( n ) + p discr W n ≥ δ . Thus (cid:12)(cid:12) w ( n ) − w ( n ) − σ n p discr W n (cid:12)(cid:12) ≥ δ , and v + n = − w ( n ) w ( n ) − w ( n ) − σ n √ discr W n , and v − n = w ( n ) w ( n ) − w ( n ) − σ n √ discr W n , satisfy the systems (3.7). In view of (3.5), Corollary 3.5 and Corollary 3.3(i), we conclude that ( v + n ) , ( v − n ) ∈D r , s + ( K , R ) . Finally, Lemma 3.4 implies that ( Y n ) belongs to D r , s + (cid:0) K , GL ( , R ) (cid:1) . Because lim n →∞ sup K | v + n | = lim n →∞ sup K | v − n | = , we easily obtain (3.4). (cid:3) Corollary 3.7.
The sequences ( µ − n ) and ( µ + n ) converge to the same limit as ( λ − n ) and ( λ + n ) , respectively.
4. Levinson’s type theorems
In this section we develop discrete variants of the Levinson’s theorem. There are two cases we need todistinguish according to whether the limiting matrix has two different eigenvalues or not.4.1.
Different eigenvalues.Theorem 4.1.
Let ( X n : n ∈ N ) be a sequence of continuous mappings defined on R with values in GL ( , R ) that converges uniformly on a compact set K to the mapping X with discr X( x ) , and det X( x ) > foreach x ∈ K . If discr X > , we additionally assume that for all x ∈ K , (4.1) (cid:12)(cid:12) [X( x )] , − λ ( x ) (cid:12)(cid:12) > and (cid:12)(cid:12) [X( x )] , − λ ( x ) (cid:12)(cid:12) > where λ and λ are continuous functions on K so that λ ( x ) and λ ( x ) are eigenvalues of X( x ) . Let ( E n : n ∈ N ) be a sequence of continuous mappings defined on R with values in Mat ( , C ) such that (4.2) ∞ Õ n = sup K k E n k < ∞ . If ( X n : n ∈ N ) belongs to D r , (cid:0) K , GL ( , R ) (cid:1) for a certain r ≥ and η is a continuous eigenvalue of X , thenthere are continuous mappings Φ n : K → C , µ n : K → C , and v : K → C , satisfying Φ n + = ( X n + E n ) Φ n and lim n →∞ sup x ∈ K | µ n ( x ) − η ( x )| = , such that (4.3) lim n →∞ sup x ∈ K (cid:13)(cid:13)(cid:13)(cid:13) Φ n ( x ) Î n − j = µ j ( x ) − v ( x ) (cid:13)(cid:13)(cid:13)(cid:13) = whereas v ( x ) is an eigenvector of X( x ) corresponding to η ( x ) for each x ∈ K .Proof. Suppose that discr X( x ) > and det X( x ) > for all x ∈ K . In particular, tr X( x ) , for all x ∈ K .Let λ + and λ − denote the eigenvalues of X such that | λ + | > | λ − | , namely we set λ + ( x ) = tr X( x ) + σ p discr X( x ) , and λ − ( x ) = tr X( x ) − σ p discr X( x ) where σ = sign ( tr X) . Without loss of generality we can assume that (4.1) is satisfied with λ = λ + and λ = λ − , since otherwise we consider mappings conjugated by J = (cid:18) (cid:19) . Select δ > such that for all x ∈ K , (cid:12)(cid:12) [X( x )] , − λ + ( x ) (cid:12)(cid:12) ≥ δ and (cid:12)(cid:12) [X( x )] , − λ − ( x ) (cid:12)(cid:12) ≥ δ, and discr X( x ) ≥ δ , det X( x ) ≥ δ . Since ( discr X n : n ∈ N ) converges uniformly on K , there is M ≥ such that for all n ≥ M and x ∈ K ,(4.4) discr X n ( x ) ≥ δ det X n ( x ) ≥ δ . Hence, the matrix X n ( x ) has two eigenvalues λ + n ( x ) = tr X n ( x ) + σ p discr X n ( x ) , and λ − n ( x ) = tr X n ( x ) − σ p discr X n ( x ) , By increasing M , we can also assume that for all n ≥ M and x ∈ K ,(4.5) (cid:12)(cid:12) [ X n ( x )] , − λ + n ( x ) (cid:12)(cid:12) ≥ δ and (cid:12)(cid:12) [ X n ( x )] , − λ − n ( x ) (cid:12)(cid:12) ≥ δ. Then setting C n , = [ X n ] , λ + n −[ X n ] , [ X n ] , λ − n −[ X n ] , ! and D n , = (cid:18) λ + n λ − n (cid:19) , we obtain X n = C n , D n , C − n , . In view of (4.4) and (4.5), by Corollaries 3.5 and 3.3, the sequences ( C n , : n ≥ M ) and ( D n , : n ≥ M ) belong to D r , (cid:0) K , GL ( , R ) (cid:1) . If r ≥ , in view of (4.4) we can apply Theorem 3.6 to get two sequences ofmappings ( C n , : n ≥ M ) ∈ D r , (cid:0) K , GL ( , R ) (cid:1) , and ( D n , : n ≥ M ) ∈ D r , (cid:0) K , GL ( , R ) (cid:1) , such that D n , C − n , C n − , = C n , D n , C − n , , SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 13 and D n , = (cid:18) γ + n , γ − n , (cid:19) . Then for n ≥ M + , X n + X n = ( C n + , D n + , C − n + , )( C n , D n , C − n , ) = C n + , ( D n + , C − n + , C n , )( D n , C − n , C n − , ) C − n − , = C n + , ( C n + , D n + , C − n + , )( C n , D n , C − n , ) C − n − , = C n + , C n + , ( D n + , C − n + , C n , )( D n , C − n , C n − , )( C n − , C n − , ) − . By repeated application of Theorem 3.6 for k ∈ { , , . . . , r − } , we can find sequences(4.6) ( C j , k : j ≥ M + k ) ∈ D r , k (cid:0) K , GL ( , R ) (cid:1) , and ( D j , k : j ≥ M + k ) ∈ D r , (cid:0) K , GL ( , R ) (cid:1) , such that D j , k − C − j , k − C j − , k − = C j , k D j , k C − j , k . Hence, X n + X n = Q n + (cid:0) D n + , r − C − n + , r − C n , r − (cid:1) (cid:0) D n , r − C − n , r − C n − , r − (cid:1) Q − n − where Q m = C m , C m , · · · C m , r − . Notice that lim m →∞ Q m = [X] , λ + −[X] , [X] , λ − −[X] , ! uniformly on K .Let us now consider the recurrence equation Ψ k + = Q − k + ( X k + + E k + )( X k + E k ) Q k − Ψ k = (cid:0) Y k + R k + F k (cid:1) Ψ k , k ≥ M where Y k = D k + , r − D k , r − = (cid:18) γ + k + , r − γ + k , r − γ − k + , r − γ − k , r − (cid:19) , R k = ( D k + , r − C − k + , r − C k , r − )( D k , r − C − k , r − C k − , r − ) − D k + , r − D k , r − , and F k = Q − k + X k + E k Q k − + Q − k + E k + X k Q k − + Q − k + E k + E k Q k − . Since R k = − D k + , r − C − k + , r − (cid:0) ∆ C k , r − (cid:1) D k , r − − D k + , r − C − k + , r − C k , r − D k , r − C − k , r − (cid:0) ∆ C k − , r − (cid:1) , we easily see that k R k k ≤ c (cid:0)(cid:13)(cid:13) ∆ C k , r − (cid:13)(cid:13) + (cid:13)(cid:13) ∆ C k − , r − (cid:13)(cid:13)(cid:1) , which together with (4.6) implies that ∞ Õ k = M sup K k R k k < ∞ . In view of (4.2) we also get ∞ Õ k = M sup K k F k k < ∞ . Let us consider the case η = λ − . The sequence ( γ − n , r − : n ≥ M ) converges to λ − , thus there are n ≥ M and δ ′ > , so that for all n ≥ n , (cid:12)(cid:12)(cid:12)(cid:12) γ + n , r − γ − n , r − (cid:12)(cid:12)(cid:12)(cid:12) ≥ + δ | γ − n , r − | ≥ + δ ′ , thus for all k ≥ k ≥ n , k Ö j = k (cid:12)(cid:12)(cid:12)(cid:12) γ + j + , r − γ − j + , r − (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) γ + j , r − γ − j , r − (cid:12)(cid:12)(cid:12)(cid:12) ≥ ( + δ ′ ) ( k − k ) . In particular, ( Y k : k ≥ n ) satisfies the uniform Levinson’s condition, see [29, Definition 2.1]. Therefore, inview of [29, Theorem 4.1], there is a sequence ( Ψ k : k ≥ n ) such that lim n →∞ sup x ∈ K (cid:13)(cid:13)(cid:13)(cid:13) Ψ k ( x ) Î k − j = n γ − j + , r − ( x ) γ − j , r − ( x ) − e (cid:13)(cid:13)(cid:13)(cid:13) = where e = (cid:18) (cid:19) . In fact, in the proof of [29, Theorem 4.1] the author used supremum norm with respect to the parameter, thuswhen all the mappings in [29, Theorem 4.1] are continuous (or holomorphic) with respect to this parameter,the functions Φ k are continuous (or holomorphic, respectively).We are now in the position to define ( Φ n : n ≥ n ) . Namely, for x ∈ K and n ≥ n , we set Φ n ( x ) = ( Q k − ( x ) Ψ k ( x ) if n = k , ( X k ( x ) + E k ( x )) Q k − ( x ) Ψ k ( x ) if n = k + . As it is easy to check, ( Φ n : n ≥ n ) satisfies Φ n + = ( X n + E n ) Φ n . Observe that for v − = [X] , λ − −[X] , ! we obtain lim k →∞ sup x ∈ K (cid:13)(cid:13) Q k − ( x ) e − v − ( x ) (cid:13)(cid:13) = , and lim k →∞ sup x ∈ K (cid:13)(cid:13)(cid:13)(cid:13) X k ( x ) + E k ( x ) γ − k , r − ( x ) Q k − ( x ) e − v − ( x ) (cid:13)(cid:13)(cid:13)(cid:13) = . Therefore, (4.3) is satisfied for ( µ n : n ∈ N ) defined on K by the formula µ n = ( for n < n ,γ − n , r − for n ≥ n . This completes the proof. The reasoning when η = λ + is analogous.If discr X( x ) < for x ∈ K , the argument is simpler. First, let us observe that the matrix X( x ) has realentries, thus |[X( x )] , | > for all x ∈ K . Since ( X n : n ∈ N ) converges uniformly on K , there are δ > and M ≥ , such that for all n ≥ M and x ∈ K , discr X n ( x ) ≤ − δ, and |[ X n ( x )] , | > δ. Therefore, for each x ∈ K , the matrix X n ( x ) has two eigenvalues λ n and λ n where λ n ( x ) = tr X n ( X ) + i p | discr X n ( x )| . SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 15
Hence, setting C n , = λ n −[ X n ] , [ X n ] , λ n −[ X n ] , [ X n ] , ! , and D n , = (cid:18) λ n λ n (cid:19) , we obtain X n = C n , D n , C − n , . Moreover, ( C n , : n ≥ M ) and ( D n , : n ≥ M ) belong to D r , (cid:0) K , GL ( , C ) (cid:1) . If r ≥ , then by [41, Theorem1], there are two sequences of matrices ( C n , : n ≥ M ) ∈ D r , (cid:0) K , GL ( , C ) (cid:1) , and ( C n , : n ≥ M ) ∈ D r , (cid:0) K , GL ( , C ) (cid:1) , such that D n , C − n , C n , = C n , D n , C − n , , and D n , = (cid:18) γ n , γ n , (cid:19) . By repeated application of [41, Theorem 1], for k ∈ { , , . . . , r − } , we can find sequences ( C j , k : j ≥ M + k ) ∈ D r , k (cid:0) K , GL ( , C ) (cid:1) , and ( D j , k : j ≥ M + k ) ∈ D r , (cid:0) K , GL ( , C ) (cid:1) , such that D j , k − C − j , k − C j − , k − = C j , k D j , k C − j , k . Hence, X n + X n = Q n + (cid:0) D n + , r − C − n + , r − C n , r − (cid:1) (cid:0) D n , r − C − n , r − C n − , r − (cid:1) Q − n − , where Q m = C m , C m , · · · C m , r − . Notice that lim m →∞ Q m = λ −[X] , [X] , λ −[X] , [X] , ! uniformly on K .We next consider the recurrence equation Ψ k + = Q − k + ( X k + + E k + )( X k + E k ) Q k − Ψ k = ( Y k + R k + F k ) Ψ k , k ≥ M where Y k = D k + , r − D k , r − = (cid:18) γ k + , r γ k , r γ k + , r γ k , r (cid:19) , R k = ( D k + , r − C − k + , r − C k , r − )( D k , r − C − k , r − C k − , r − ) − D k + , r − D k , r − , and F k = Q − k + X k + E k Q k − + Q − k + E k + X k Q k − + Q − k + E k + E k Q k − . Suppose that η = λ . Since for all n ≥ M , (cid:12)(cid:12)(cid:12)(cid:12) γ n , r − γ n , r − (cid:12)(cid:12)(cid:12)(cid:12) = , the sequence ( Y k : k ≥ M ) satisfies the uniform Levinson’s condition. Therefore, by [29, Theorem 4.1], thereis a sequence ( Ψ k : k ≥ M ) such that lim k →∞ sup x ∈ K (cid:13)(cid:13)(cid:13)(cid:13) Ψ k ( x ) Î k − j = M γ j + , r − ( x ) γ j , r − ( x ) − e (cid:13)(cid:13)(cid:13)(cid:13) = . Hence, ( Φ n : n ≥ M ) defined by the formula Φ n ( x ) = ( Q k − ( x ) Ψ k ( x ) if n = k , ( X k ( x ) + E k ( x )) Q k − ( x ) Ψ k ( x ) if n = k + . together with µ n = ( for n < M ,γ n , r − for n ≥ M , satisfies (4.3). This completes the proof of the theorem. (cid:3) The following lemma guarantees that in the case of positive discriminant Theorem 4.1 can at least beapplied locally.
Lemma 4.2.
Suppose that X is a continuous mapping defined on a closed interval I ⊂ R with values in Mat ( , R ) that has positive discriminant on I . Let λ , λ : I → R , be continuous functions so that λ ( x ) and λ ( x ) are the distinct eigenvalues of X ( x ) . Then for each x ∈ I there is an open interval I x containing x suchthat(i) for all y ∈ I x ∩ I , (cid:0) [ X ( y )] , − λ ( y ) (cid:1) (cid:0) [ X ( y )] , − λ ( y ) (cid:1) , , or(ii) for all y ∈ I x ∩ I , (cid:0) [ X ( y )] , − λ ( y ) (cid:1) (cid:0) [ X ( y )] , − λ ( y ) (cid:1) , . Proof.
Let x ∈ I . Since discr X ( x ) > , we have λ ( x ) , λ ( x ) . By the continuity of X , it is enough to showthat (cid:0) [ X ( x )] , − λ ( x ) (cid:1) (cid:0) [ X ( x )] , − λ ( x ) (cid:1) , , or (cid:0) [ X ( x )] , − λ ( x ) (cid:1) (cid:0) [ X ( x )] , − λ ( x ) (cid:1) , . If neither of the conditions is met, we would have [ X ( x )] , = λ ( x ) and [ X ( x )] , = λ ( x ) , or [ X ( x )] , = λ ( x ) and [ X ( x )] , = λ ( x ) . Thus tr X ( x ) equals λ ( x ) or λ ( x ) , but neither of the situations is possible. (cid:3) The following corollary gives a Levinson’s type theorem in the case when the limit X is a constant matrix.It may be proved in much the same way as Theorem 4.1. Here, the condition (4.1) can be dropped since X isa constant matrix. Corollary 4.3.
Let ( X n : n ∈ N ) be a sequence of matrices belonging to D (cid:0) GL ( , R ) (cid:1) convergent to thematrix X . Let ( E n : n ∈ N ) be a sequence of continuous (or holomorphic) mappings defined on a compactset K ⊂ C with values in Mat ( , C ) , such that ∞ Õ n = sup K k E n k < ∞ . Suppose that discr X , and det X > . If η is an eigenvalue of X , then there are continuous (orholomorphic, respectively) mappings Φ n : K → C , satisfying Φ n + = ( X n + E n ) Φ n such that lim n →∞ sup x ∈ K (cid:13)(cid:13)(cid:13)(cid:13) Φ n ( x ) Î n − j = µ j − v (cid:13)(cid:13)(cid:13)(cid:13) = SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 17 where v is the eigenvector of X corresponding to η , and µ n is the eigenvalue of X n such that lim n →∞ | µ n − η | = . Perturbations of the identity.Theorem 4.4.
Let ( X n : n ∈ N ) be a sequence of continuous mappings defined on R with values in GL ( , R ) that uniformly converges on a compact interval K to σ Id for a certain σ ∈ {− , } . Suppose that there is asequence of positive numbers ( γ n : n ∈ N ) such that R n = γ n ( X n − σ Id ) converges uniformly on K to themapping R satisfying discr R( x ) , for all x ∈ K . If discr R > , we additionally assume that (4.7) ∞ Õ n = γ n = ∞ . Let ( E n : n ∈ N ) be a sequence of continuous mappings defined on R with values in Mat ( , C ) , such that (4.8) ∞ Õ n = sup K k E n k < ∞ . If ( R n : n ∈ N ) belongs to D , (cid:0) K , Mat ( , R ) (cid:1) and η is a continuous eigenvalue of R , then there are n ≥ and continuous mappings Φ n : K → C , µ n : K → C , and v : K → C satisfying Φ n + = ( X n + E n ) Φ n , such that (4.9) lim n →∞ sup x ∈ K (cid:13)(cid:13)(cid:13)(cid:13) Φ n ( x ) Î n − j = n (cid:0) σ + γ − j µ j ( x ) (cid:1) − v ( x ) (cid:13)(cid:13)(cid:13)(cid:13) = where for each x ∈ K , v ( x ) is an eigenvector of R( x ) corresponding to η ( x ) , and µ n ( x ) is an eigenvalue of R n ( x ) such that lim n →∞ sup x ∈ K (cid:12)(cid:12) µ n ( x ) − η ( x ) (cid:12)(cid:12) = . Proof.
Let us first consider the case of positive discriminant. There is δ > such that for all x ∈ K , discr R( x ) ≥ δ . Then the matrix R( x ) has two eigenvalues ξ + ( x ) = tr R( x ) + σ p discr R( x ) , and ξ − ( x ) = tr R( x ) − σ p discr R( x ) . Since ( R n : n ∈ N ) converges uniformly on K , there is M ≥ , such that for all n ≥ M and x ∈ K ,(4.10) discr R n ( x ) ≥ δ , and γ n ≥ δ. In particular, the matrix R n has two eigenvalues(4.11) ξ + n ( x ) = tr R n ( x ) + σ p discr R n ( x ) , and ξ − n ( x ) = tr R n ( x ) − σ p discr R n ( x ) . Now, let us consider the collection of intervals { I x : x ∈ K } determined in Lemma 4.2 for the mapping R .By compactness of K we can find a finite subcollection { I , . . . , I J } that covers K . Let us consider the case η = ξ − . It is clear that lim n →∞ sup x ∈ K (cid:13)(cid:13) ξ − n ( x ) − η ( x ) (cid:13)(cid:13) = . Suppose that on each K j = I j ∩ K , one can find Φ ( j ) n and v ( j ) so that(4.12) lim n →∞ sup x ∈ K j (cid:13)(cid:13)(cid:13)(cid:13) Φ ( j ) n ( x ) Î n − m = n (cid:0) σ + γ − m µ m ( x ) (cid:1) − v ( j ) ( x ) (cid:13)(cid:13)(cid:13)(cid:13) = . Let { ψ , . . . , ψ J } be the continuous partition of unity subordinate to the covering { I , . . . , I J } , that is ψ j is acontinuous non-negative function with supp ψ j ⊂ I j , so that J Õ j = ψ j ≡ . We set Φ n = J Õ j = Φ ( j ) n ψ j , and v = J Õ j = v ( j ) ψ j . Observe that v ( x ) is an eigenvector of R( x ) corresponding to η ( x ) for all x ∈ K . Moreover, since ψ j issupported inside I j , lim n →∞ sup x ∈ K (cid:13)(cid:13)(cid:13)(cid:13) Φ n ( x ) Î n − m = n (cid:0) σ + γ − m µ m ( x ) (cid:1) − v ( x ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ lim n →∞ J Õ j = sup x ∈ K j (cid:13)(cid:13)(cid:13)(cid:13) Φ ( j ) n ( x ) Î n − m = n (cid:0) σ + γ − m µ m ( x ) (cid:1) − v ( j ) ( x ) (cid:13)(cid:13)(cid:13)(cid:13) = . Therefore, it is sufficient to prove (4.12) for K = K j where j ∈ { , . . . , J } . To simplify the notation, we dropthe dependence on j . Without loss of generality, we can assume that for each x ∈ K , (cid:12)(cid:12) [R( x )] , − ξ + ( x ) (cid:12)(cid:12) ≥ δ, and (cid:12)(cid:12) [R( x )] , − ξ − ( x ) (cid:12)(cid:12) ≥ δ. Since ( R n : n ∈ N ) converges to R uniformly on K , there are M ≥ such that for all x ∈ K and n ≥ M ,(4.13) (cid:12)(cid:12) [ R n ( x )] , − ξ + n ( x ) (cid:12)(cid:12) ≥ δ, and (cid:12)(cid:12) [ R n ( x )] , − ξ − n ( x ) (cid:12)(cid:12) ≥ δ. Now, we can define C n = [ R n ] , ξ + n −[ R n ] , [ R n ] , ξ − n −[ R n ] , ! , and ˜ D n ( x ) = (cid:18) ξ + n ( x ) ξ − n ( x ) (cid:19) . Then R n ( x ) = C n ( x ) ˜ D n ( x ) C − n ( x ) , and in view of (4.13), (4.10), (4.11), Corollary 3.3 and Lemma 3.4, we conclude that(4.14) ( C n : n ≥ M ) ∈ D , (cid:0) K , GL ( , R ) (cid:1) . Notice that(4.15) lim n →∞ C n = [R] , ξ + −[R] , [R] , ξ − −[R] , ! uniformly on K . Since X n = σ Id + γ n R n , we obtain X n ( x ) = C n ( x ) D n ( x ) C − n ( x ) where D n = σ Id + γ n ˜ D n . Hence, eigenvalues of X n are(4.16) λ + n = σ + γ n ξ + n , and λ − n = σ + γ n ξ − n . Let us now consider the recurrence equation Ψ n + = C − n + ( X n + E n ) C n Ψ n = ( D n + F n ) Ψ n SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 19 where F n = − C − n + (cid:0) ∆ C n (cid:1) D n + C − n + E n C n . By (4.15), we easily see that k F n k ≤ c (cid:0) k ∆ C n k + k E n k (cid:1) , which together with (4.14) and (4.8) gives ∞ Õ n = M sup K k F n k < ∞ . Next, in view of (4.16), (4.11) and (4.10), for n ≥ M , (cid:12)(cid:12)(cid:12)(cid:12) λ + n λ − n (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) + γ n √ discr R n + σγ n ξ − n (cid:12)(cid:12)(cid:12)(cid:12) ≥ + √ discr R n γ n ≥ exp (cid:18) δ γ n (cid:19) , (4.17)after possibly enlarging M . Therefore, for all n > n ≥ M , n Ö n = n (cid:12)(cid:12)(cid:12)(cid:12) λ + n λ − n (cid:12)(cid:12)(cid:12)(cid:12) ≥ exp (cid:18) δ n Õ n = n γ n (cid:19) . Hence, (4.7) guarantees that the sequence ( D n : n ≥ M ) satisfies the uniform Levinson’s condition. Let usremind that we are considering η = ξ − . In view of [29, Theorem 4.1], there is a sequence ( Ψ n : n ≥ M ) suchthat lim n →∞ sup x ∈ K (cid:13)(cid:13)(cid:13)(cid:13) Ψ n ( x ) Î n − j = M λ − j ( x ) − e (cid:13)(cid:13)(cid:13)(cid:13) = . Now, for x ∈ K and n ≥ M , we set Φ n ( x ) = C n ( x ) Ψ n ( x ) . It is easy to verify that ( Φ n : n ≥ M ) satisfies Φ n + = ( X n + E n ) Φ n . Setting v = [R] , ξ − −[R] , ! by (4.15), we get lim n →∞ sup x ∈ K (cid:13)(cid:13) C n ( x ) e − v ( x ) (cid:13)(cid:13) = , which completes the proof of (4.9) for K = K j , and the case of positive discriminant follows.When discr R < on K , the reasoning is similar. Since the matrix R has real entries, [R( x )] , , for all x ∈ K . Therefore, for n ≥ M , we can set C n = ξ + n −[ R n ] , [ R n ] , ξ − n −[ R n ] , [ R n ] , ! where ξ + n ( x ) = tr R n ( x ) + i p | discr R n ( x )| , and ξ − n ( x ) = tr R n ( x ) − i p | discr R n ( x )| . Since (cid:12)(cid:12)(cid:12)(cid:12) λ + n λ − n (cid:12)(cid:12)(cid:12)(cid:12) = , the sequence ( D n : n ≥ M ) satisfies the uniform Levinson’s condition. The rest of the proof runs asbefore. (cid:3) The method of the proof used in Theorem 4.4, can be also applied in the case of different eigenvalues and r = . In particular, the condition (4.1) can be dropped.The proof of the following corollary is analogous to the proof of Theorem 4.4. Corollary 4.5.
Let ( X n : n ∈ N ) be a sequence of matrices in GL ( , R ) convergent to the matrix σ Id for a certain σ ∈ {− , } . Suppose that there is a sequence of positive numbers ( γ n : n ∈ N ) such that R n = γ n ( X n − σ Id ) converges to the matrix R satisfying discr R , . If discr R > , we additionally assume ∞ Õ n = γ n = ∞ . Let ( E n : n ∈ N ) be is a sequence of continuous (or holomorphic) mappings on a compact set K ⊂ C withvalues in Mat ( , C ) , such that ∞ Õ n = sup K k E n k < ∞ . If ( R n : n ∈ N ) belongs to D , ( Mat ( , R )) , and η is an eigenvalue of R , then there are n ≥ and continuous(or holomorphic, respectively) mappings Φ n : K → C , satisfying Φ n + = ( X n + E n ) Φ n , and such that lim n →∞ sup x ∈ K (cid:13)(cid:13)(cid:13)(cid:13) Φ n ( x ) Î n − j = n ( σ + γ − j µ j ) − v (cid:13)(cid:13)(cid:13)(cid:13) = where v is an eigenvector of R corresponding to η , µ n is the eigenvalue of R n such that lim n →∞ | µ n − η | = . In the following proposition we describe a way to estimate the denominator in (4.9).
Proposition 4.6.
Let ( X n : n ∈ N ) be a sequence of mappings defined on R with values in GL ( , R ) convergenton a compact set K to σ Id for a certain σ ∈ {− , } . Suppose that there is a sequence of positive numbers ( γ n : n ∈ N ) satisfying lim n →∞ Γ n = ∞ where Γ n = n Õ j = γ j , such that R n = γ n ( X n − σ Id ) converges uniformly on K to the mapping R . Assume that discr R( x ) > forall x ∈ K , and (4.18) ∞ Õ n = Γ n · sup x ∈ K k R n + ( x ) − R n ( x )k < ∞ . Then there is n such that for all n ≥ n , and x ∈ K , (4.19) n Ö j = n (cid:12)(cid:12) σ + γ − j µ − j ( x ) (cid:12)(cid:12) ≍ exp (cid:16) Γ n (cid:0) σ tr R( x ) − p discr R( x ) (cid:1) (cid:17) and (4.20) n Ö j = n (cid:12)(cid:12) σ + γ − j µ + j ( x ) (cid:12)(cid:12) ≍ exp (cid:16) Γ n (cid:0) σ tr R( x ) + p discr R( x ) (cid:1) (cid:17) where (4.21) µ − j = (cid:16) tr R j − σ q discr R j (cid:17) , and µ + j = (cid:16) tr R j + σ q discr R j (cid:17) . The implicit constants in (4.19) and (4.20) are independent of x and n . SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 21
Proof.
Since discr R > on K , there is n such that for all j ≥ n and x ∈ K , discr R j ( x ) > . Thus R j hastwo eigenvalues given by the formulas (4.21). By possible enlarging n , for all n ≥ n , we have log (cid:16) n Ö j = n (cid:12)(cid:12) σ + γ − j µ − j (cid:12)(cid:12) (cid:17) ≍ n Õ j = n γ j (cid:16) σ tr R j − q discr R j (cid:17) uniformly on K . Let A − n = σ tr R n − p discr R n , A −∞ = σ tr R − √ discr R . Since for m ≥ n , (cid:12)(cid:12) A − n − A − m (cid:12)(cid:12) · Γ n ≤ c ∞ Õ k = n (cid:13)(cid:13) R k + − R k (cid:13)(cid:13) · Γ n ≤ c ∞ Õ k = n (cid:13)(cid:13) R k + − R k (cid:13)(cid:13) · Γ k , we obtain(4.22) sup K (cid:12)(cid:12) A − n − A −∞ (cid:12)(cid:12) · Γ n ≤ c . Now, by the summation by parts, we get n Õ j = n γ j A − j = ( Γ n − Γ n − ) A −∞ + n Õ j = n ( Γ j − Γ j − )( A − j − A −∞ ) = Γ n A −∞ − Γ n − A − n + Γ n ( A − n − A −∞ ) + n − Õ j = n Γ j ( A − j − A − j + ) , thus, by (4.18) and (4.22), sup K (cid:12)(cid:12)(cid:12)(cid:12) n Õ j = n γ j A − j − A −∞ · Γ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ c . Hence, n Ö j = n (cid:12)(cid:12) σ + γ − j µ − j (cid:12)(cid:12) ≍ exp (cid:16) Γ n (cid:0) σ tr R − √ discr R (cid:1) (cid:17) , uniformly on K . The proof of (4.20) is similar. (cid:3)
5. Essential spectrum for positive discriminant
In this section we prove the main results of the paper.
Theorem 5.1.
Let N and r be positive integers and i ∈ { , , . . . , N } . Let A be a Jacobi matrix with N -periodically blended entries. If there is a compact set K ⊂ R with at least N + points so that (5.1) (cid:0) X n ( N + ) + i : n ∈ N (cid:1) ∈ D r , (cid:0) K , Mat ( , R ) (cid:1) , then A is self-adjoint and σ sing ( A ) ∩ Λ = ∅ and σ ac ( A ) = σ ess ( A ) = Λ where Λ = (cid:8) x ∈ R : discr X ( x ) < (cid:9) wherein X is given by the formula (2.6) . Proof.
Fix x ∈ R \ Λ . Let I be an open interval containing x such that I ⊂ R \ Λ . Since discr X = discr X i ,we have discr X i > on I . Thus the matrix X i has two different eigenvalues λ + and λ − . Since det X i ≡ ,we can select them in such a way that | λ − | < < | λ + | . Let I be an open interval determined by Lemma 4.2 for x and the mapping X i : I → Mat ( , R ) . Withoutloss of generality we can assume that, for all x ∈ I , (cid:12)(cid:12) [X i ( x )] , − λ − ( x ) (cid:12)(cid:12) > , and (cid:12)(cid:12) [X i ( x )] , − λ + ( x ) (cid:12)(cid:12) > . Let K = I . In view of Lemma 3.1,(5.2) (cid:0) X j ( N + ) + i : j ∈ N (cid:1) ∈ D r , (cid:0) K , GL ( , R ) (cid:1) . Now, by Theorem 4.1, there are sequences ( Φ ± n : n ≥ n ) and ( µ ± n : n ∈ N ) , such that(5.3) lim n →∞ sup x ∈ K (cid:13)(cid:13)(cid:13)(cid:13) Φ ± n ( x ) Î nj = µ ± j ( x ) − v ± ( x ) (cid:13)(cid:13)(cid:13)(cid:13) = . where v ± is a continuous eigenvector of X i corresponding to λ ± . We set φ ± = B − · · · B − n ( N + ) + i − Φ ± n , and for n ≥ ,(5.4) φ ± n + = B n φ ± n . Then for k ( N + ) + i ′ > n ( N + ) + i with i ′ ∈ { , , . . . , N + } , we have(5.5) φ ± k ( N + ) + i ′ = B − k ( N + ) + i ′ B − k ( N + ) + i ′ + · · · B − k ( N + ) + i − Φ ± k if i ′ ∈ { , , . . . , i − } , Φ ± k if i ′ = i , B k ( N + ) + i ′ − B k ( N + ) + i ′ − · · · B k ( N + ) + i Φ ± k if i ′ ∈ { i + , . . . , N + } . Since for i ′ ∈ { , . . . , i − } , lim k →∞ B − k ( N + ) + i ′ B − k ( N + ) + i ′ + · · · B − k ( N + ) + i − = B − i ′ B − i ′ − · · · B − i − , we obtain(5.6) lim k →∞ sup K (cid:13)(cid:13)(cid:13)(cid:13) φ − k ( N + ) + i ′ Î k − j = µ − j − B − i ′ B − i ′ − · · · B − i − v − (cid:13)(cid:13)(cid:13)(cid:13) = . Analogously, for i ′ ∈ { i + , . . . , N } , we get(5.7) lim k →∞ sup K (cid:13)(cid:13)(cid:13)(cid:13) φ − k ( N + ) + i ′ Î k − j = µ − j − B i ′ − B i ′ − · · · B i v − (cid:13)(cid:13)(cid:13)(cid:13) = . Lastly, by Proposition 2.5,(5.8) lim k →∞ sup K (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) φ − k ( N + ) Î k − j = µ − j − (cid:18) (cid:19) B − B − · · · B − i − v − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = and(5.9) lim k →∞ sup K (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) φ ± k ( N + ) + N + Î k − j = µ − j − (cid:18) (cid:19) B N − B N − · · · B i v − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = . Since ( φ ± n : n ∈ N ) satisfies (5.4), the sequence ( u ± n ( x ) : n ∈ N ) defined as u ± n ( x ) = ( h φ ± ( x ) , e i if n = , h φ ± n ( x ) , e i if n ≥ , SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 23 is a generalized eigenvector associated to x ∈ K . Observe that ( u , u ) , on K . Indeed, otherwise there is x ∈ K , so that φ ± ( x ) = , hence φ ± n ( x ) = for all n ∈ N . Therefore, v ± ( x ) = , which is impossible. Now, inview of (5.3) and (5.5), there are constants c > and δ > such that for all n ≥ n and x ∈ K , (cid:12)(cid:12) u + n ( N + ) + i − ( x ) (cid:12)(cid:12) + (cid:12)(cid:12) u + n ( N + ) + i ( x ) (cid:12)(cid:12) = (cid:13)(cid:13) φ + n ( N + ) + i ( x ) (cid:13)(cid:13) ≥ c n − Ö j = n | µ + j ( x )| ≥ c ( + δ ) n . Moreover, by (5.6)–(5.9), for all n ≥ n , i ′ ∈ { , , . . . , N + } , and x ∈ K , (cid:13)(cid:13) φ − n ( N + ) + i ′ ( x ) (cid:13)(cid:13) ≤ c n − Ö j = n | µ − j ( x )| ≤ c ( + δ ) − n . Consequently, for any x ∈ K , ∞ Õ n = | u + n ( x )| = ∞ which shows that A is self-adjoint (see [28, Theorem 6.16]). Moreover, ∞ Õ n = sup x ∈ K | u − n ( x )| < ∞ , thus by the proof of [30, Theorem 5.3], σ ess ( A ) ∩ K = ∅ . Therefore, for all x ∈ R \ Λ there is an open interval I containing x such that σ ess ( A )∩ I = ∅ . Consequently, σ ess ( A ) ⊆ Λ . In view of (5.2), [41, Theorem B] implies that A is purely absolutely continuous on Λ , and Λ ⊂ σ ac ( A ) . This completes the proof. (cid:3) Remark 5.2.
The proof of [37, Corollary 6.7] entails that (5.1) is satisfied for any compact set K ⊂ R , andall i ∈ { , , . . . , N } , provided that (cid:18) a n : n ∈ N (cid:19) , (cid:0) b n : n ∈ N (cid:1) ∈ D N + r , and (cid:18) a n ( N + ) + N a n ( N + ) + N + : n ∈ N (cid:19) ∈ D r , . Essentially the same reasoning as in the proof of Theorem 5.1 leads to the following theorem.
Theorem 5.3.
Let N and r be positive integers and i ∈ { , , . . . , N − } . Let A be a Jacobi matrix with N -periodically modulated entries. If | tr X ( )| > and there is a compact set K ⊂ R with at least N + points so that (5.10) (cid:0) X nN + i : n ∈ N (cid:1) ∈ D r , (cid:0) K , Mat ( , R ) (cid:1) , then A is self-adjoint and σ ess ( A ) = ∅ . Remark 5.4.
The proof of [41, Corollary 8] implies that (5.10) is satisfied for any compact set K ⊂ R , andall i ∈ { , , . . . , N − } , provided that (cid:18) a n − a n : n ∈ N (cid:19) , (cid:18) b n a n : n ∈ N (cid:19) , (cid:18) a n : n ∈ N (cid:19) ∈ D Nr , . We next consider the case when X has equal eigenvalues. Theorem 5.5.
Let N and r be positive integers and i ∈ { , , . . . , N − } . Let A be a Jacobi matrix with N -periodically modulated entries so that X ( ) = σ Id for a certain σ ∈ {− , } . Suppose that there are two N -periodic sequences ( s n : n ∈ N ) and ( z n : n ∈ N ) , such that lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) α n − α n a n − a n − − s n (cid:12)(cid:12)(cid:12)(cid:12) = , lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) β n α n a n − b n − z n (cid:12)(cid:12)(cid:12)(cid:12) = . Let R n = a n + N − (cid:0) X n − σ Id (cid:1) . Then ( R nN : n ∈ N ) converges to R locally uniformly on R . If there is a compact set K ⊂ R with at least N + points such that (5.11) (cid:0) R nN + i : n ∈ N (cid:1) ∈ D , (cid:0) K , Mat ( , R ) (cid:1) , then A is self-adjoint and σ sing ( A ) ∩ Λ = ∅ and σ ac ( A ) = σ ess ( A ) = Λ where Λ = (cid:8) x ∈ R : discr R ( x ) < (cid:9) . Proof.
In view of Proposition 2.2, there is c > such that for all k ≥ , a k N + i = k − Õ j = (cid:0) a ( j + ) N + i − a j N + i (cid:1) + a i ≤ c ( k + ) . Therefore, k N + i Õ n = a n ≥ k Õ k = a k N + i ≥ c k Õ k = k . Thus, the Carleman’s condition is satisfied, and so A is self-adjoint.Thanks to Lemma 3.1, for any compact set K ⊂ R , (cid:0) X j N + i : j ∈ N (cid:1) , (cid:0) R nN + i : n ∈ N (cid:1) ∈ D , (cid:0) K , Mat ( , R ) (cid:1) . Since discr R = discr R i , by [22, Criterion 5.8] together with [22, Proposition 5.7] and [22, Theorem 5.6],we conclude that A is purely absolutely continuous on Λ and Λ ⊂ σ ac ( A ) . Hence, it remains to show that σ ess ( A ) ⊂ Λ . To do so, we fix a compact set K ⊂ R \ Λ with non-empty interior. Since discr R i > on K ,for each x ∈ K the matrix R i ( x ) has two distinct eigenvalues ξ + ( x ) = tr R i ( x ) + σ p discr R i ( x ) , and ξ − ( x ) = tr R i ( x ) − σ p discr R i ( x ) . Moreover, ( discr R j N + i : j ∈ N ) converges uniformly on K , thus there are M ≥ and δ > such that for all j ≥ M and x ∈ K , discr R j N + i ( x ) ≥ δ. Therefore, R j N + i ( x ) has two distinct eigenvalues ξ + j ( x ) = tr R j N + i ( x ) + σ p discr R j N + i ( x ) , and ξ − j ( x ) = tr R j N + i ( x ) − σ p discr R j N + i ( x ) . Since X n = σ Id + a n + N − R n , the eigenvalues of X j N + i ( x ) are λ + j ( x ) = σ + ξ + j ( x ) a ( j + ) N + i − , and λ − j ( x ) = σ + ξ − j ( x ) a ( j + ) N + i − . By Theorem 4.4, there is a sequence ( Φ n : n ≥ n ) such that(5.12) lim n →∞ sup x ∈ K (cid:13)(cid:13)(cid:13)(cid:13) Φ n ( x ) Î n − j = n λ − j ( x ) − v − ( x ) (cid:13)(cid:13)(cid:13)(cid:13) = where v − is a continuous eigenvector of R i corresponding to ξ − . We set φ = B − · · · B − n Φ n , SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 25 and for n ≥ ,(5.13) φ n + = B n φ n . Then for k N + i ′ > n N + i with i ′ ∈ { , , . . . , N − } , we have φ k N + i ′ = B − k N + i ′ B − k N + i ′ + · · · B − k N + i − Φ k if i ′ ∈ { , , . . . , i − } , Φ k if i ′ = i , B k N + i ′ − B k N + i ′ − · · · B k N + i Φ k if i ′ ∈ { i + , . . . , N − } . Since for i ′ ∈ { , , . . . , i − } , lim k →∞ B − k N + i ′ B − k N + i ′ + · · · B − k N + i − = B − i ′ ( ) B − i ′ + ( ) · · · B − i − ( ) , we obtain(5.14) lim k →∞ sup K (cid:13)(cid:13)(cid:13)(cid:13) φ k N + i ′ Î k − j = n λ − j − B − i ′ ( ) B − i ′ + ( ) · · · B − i − ( ) v − (cid:13)(cid:13)(cid:13)(cid:13) = . Analogously, for i ′ ∈ { i + , . . . , N − } ,(5.15) lim k →∞ sup K (cid:13)(cid:13)(cid:13)(cid:13) φ k N + i ′ Î k − j = n λ − j − B i ′ − ( ) B i ′ − ( ) · · · B i ( ) v − (cid:13)(cid:13)(cid:13)(cid:13) = . Since ( φ n : n ∈ N ) satisfies (5.13), the sequence ( u n ( x ) : n ∈ N ) defined as u n ( x ) = ( h φ ( x ) , e i if n = , h φ n ( x ) , e i if n ≥ , is a generalized eigenvector associated to x ∈ K . By (5.12), (5.14) and (5.15), for each i ′ ∈ { , , . . . , N − } , n > n , and x ∈ K ,(5.16) | u nN + i ′ ( x )| ≤ c n − Ö j = n | λ − j ( x )| . Since ( R nN + i : n ∈ N ) converges to R i uniformly on K , and lim n →∞ a n = ∞ , there is M ≥ n , such that for n ≥ M , | tr R nN + i ( x )| + p discr R nN + i ( x ) a ( n + ) N + i − ≤ . Therefore, for n ≥ M , | λ − n ( x )| = + σ tr R nN + i ( x ) − p discr R nN + i ( x ) a ( n + ) N + i − . We next claim the following holds true.
Claim 5.6.
There are δ ′ > and M ≥ M such that for all n ≥ M and x ∈ K , (5.17) n σ tr R nN + i ( x ) − p discr R nN + i ( x ) a ( n + ) N + i − ≤ − − δ ′ . First observe that by the Stolz–Cesáro theorem and Proposition 2.2, we have(5.18) ≤ lim n →∞ a ( n + ) N + i − n = lim n →∞ (cid:0) a ( n + ) N + i − − a nN + i − (cid:1) = − σ tr R i ( x ) . Since ( R nN + i : n ∈ N ) converges to R i uniformly on K ,(5.19) lim n →∞ (cid:16) σ tr R nN + i ( x ) − p discr R nN + i ( x ) (cid:17) = σ tr R i ( x ) − p discr R i ( x ) . Thus, by (5.18) and (5.19) we get lim n →∞ n σ tr R nN + i ( x ) − p discr R nN + i ( x ) a ( n + ) N + i − = ( −∞ if tr R i = , − − √ discr R i − σ tr R i otherwise,which together with (5.18) implies (5.17).Now, using Claim 5.6, we conclude for all n ≥ M , sup x ∈ K | λ − n ( x )| ≤ − + δ ′ n . Consequently, by (5.16), there is c ′ > such that for all i ′ ∈ { , , . . . , N − } and n ≥ M , sup x ∈ K | u nN + i ( x )| ≤ c n − Ö j = n (cid:18) − + δ ′ j (cid:19) ≤ c ′ exp (cid:18) − + δ ′ ( n − ) (cid:19) . Hence, ∞ Õ n = sup x ∈ K | u n ( x )| < ∞ , thus by the proof of [30, Theorem 5.3] we conclude that σ ess ( A ) ∩ K = ∅ . Since K was any compact subsetof R \ Λ , we obtain σ ess ( A ) ⊆ Λ , and the theorem follows. (cid:3) Remark 5.7.
By [36, Proposition 9], the regularity (5.11) holds true for any compact set K ⊂ R , and all i ∈ { , , . . . , N − } , if (cid:18) α n − α n a n − a n − : n ∈ N (cid:19) , (cid:18) β n α n a n − b n : n ∈ N (cid:19) , (cid:18) a n : n ∈ N (cid:19) ∈ D N ( R ) .
6. Periodic modulations in non-Carleman setup
In this section we shall consider Jacobi matrices such that ∞ Õ n = a n < ∞ . Let us start with the following general observation.
Proposition 6.1.
Let N be a positive integer and X n ( z ) = n + N − Ö j = n B j ( z ) . Let K be a compact subset of C containing , and suppose that (6.1) sup n ≥ sup z ∈ K k B n ( z )k < ∞ . Then there is c > such that sup x ∈ K k X n ( z ) − X n ( )k ≤ c N − Õ j = a n + j . In particular, if (6.2) ∞ Õ n = a n < ∞ , SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 27 then ∞ Õ n = sup z ∈ K k X n ( z ) − X n ( )k < ∞ . Proof.
Let us notice that B j ( z ) − B j ( ) = (cid:18) za j (cid:19) , thus(6.3) k B j ( z ) − B j ( )k ≤ | z | a j . Since X n ( z ) − X n ( ) = N − Õ j = ( N − Ö m = j + B n + m ( ) ) (cid:0) B n + j ( z ) − B n + j ( ) (cid:1) ( j − Ö m = B n + m ( z ) ) , we have k X n ( z ) − X n ( )k ≤ N − Õ j = ( N − Ö m = j + (cid:13)(cid:13) B n + m ( ) (cid:13)(cid:13))(cid:13)(cid:13) B n + j ( z ) − B n + j ( ) (cid:13)(cid:13)( j − Ö m = (cid:13)(cid:13) B n + m ( z )k ) . Now the conclusion easily follows by (6.1) and (6.3). (cid:3)
The following corollary reproves the main result of [43] obtained by a different method.
Corollary 6.2 (Yafaev) . Suppose that the Carleman’s condition is not satisfied and (6.4) (cid:18) a n √ a n − a n + − n ∈ N (cid:19) ∈ ℓ and (cid:18) b n √ a n − a n : n ∈ N (cid:19) ∈ D . Let (6.5) q = lim n →∞ b n √ a n − a n . If | q | , , then for every z ∈ C there is a basis { u + ( z ) , u − ( z )} of generalized eigenvectors associated with z such that (6.6) u ± n ( z ) = (cid:18) n Ö j = λ ± j ( ) (cid:19) (cid:0) + ǫ ± n ( z ) (cid:1) where λ ± j ( ) is the eigenvalue of B j ( ) , and ( ǫ ± n ) is a sequence of holomorphic functions tending to zerouniformly on any compact subset of C .Proof. By [43, Lemma 2.1](6.7) (cid:18)r a n + a n : n ∈ N (cid:19) ∈ D and lim n →∞ r a n + a n ≥ . By Corollary 3.5 and Lemma 3.4 it implies(6.8) (cid:18) a n − a n : n ∈ N (cid:19) ∈ D . Next, we write(6.9) b n a n = b n √ a n − a n r a n − a n , thus by (6.4), (6.7) and Corollary 3.3(6.10) (cid:18) b n a n : n ∈ N (cid:19) ∈ D . In particular, by (6.8) and (6.10) we conclude that ( B n ( ) : n ∈ N ) ∈ D (cid:0) GL ( , R ) (cid:1) . Now, in view of Proposition 6.1 B n ( z ) = B n ( ) + E n ( z ) where for any compact set K ⊂ C , ∞ Õ n = sup z ∈ K k E n ( z )k < ∞ . By (6.8) and (6.10), there are r , s ∈ R , r = lim n →∞ a n − a n and s = lim n →∞ b n a n . Then the limit of ( B n ( ) : n ∈ N ) is B = (cid:18) − r − s (cid:19) . Notice that discr B = s − r = r (cid:16) s √ r − (cid:17) (cid:16) s √ r + (cid:17) . On the other hand, by (6.7), (6.5) and (6.9), we can easily deduce that r ∈ ( , ] and q = s √ r . Therefore, discr B , whenever | q | , . Fix a compact set K ⊂ C . If discr B > , then B has twoeigenvectors v + = (cid:18) λ + (cid:19) v − = (cid:18) λ − (cid:19) corresponding to the eigenvalues λ + = − s + √ s − r , λ − = − s − √ s − r . Since det B = r these eigenvalues are non-zero. Let us consider the system Φ n + = (cid:0) B n ( ) + E n (cid:1) Φ n . By Corollary 4.3, there is a sequence of mappings ( Φ ± n : n ≥ n ) so that(6.11) lim n →∞ sup z ∈ K (cid:13)(cid:13)(cid:13)(cid:13) Φ ± n ( z ) Î n − j = λ ± j ( ) − v ± (cid:13)(cid:13)(cid:13)(cid:13) = . Since B n is invertible for any n , we set φ ± = B − · · · B − n Φ ± n . Then for n ≥ , we define φ ± n + = B n φ ± n . Finally, to obtain a generalized eigenvector associated with z ∈ K , we set u ± n ( z ) = ( h φ ± ( z ) , e i if n = , h φ ± n ( z ) , e i if n ≥ . Now, by (6.11) we easily see that u ± n ( z ) = (cid:18) n − Ö j = λ ± j ( ) (cid:19) (cid:0) λ ± + ǫ ± n ( z ) (cid:1) SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 29 with lim n →∞ sup z ∈ K | ǫ ± n ( z )| = . Since ( λ ± j ( )) converges to λ ± , we obtain (6.6). When discr B < , the reasoning is analogous. (cid:3) Perturbation of the identity.Theorem 6.3.
Let N be a positive integer. Let A be a Jacobi matrix with N -periodically modulated entriesso that X ( ) = σ Id for a certain σ ∈ {− , } . Assume that there are i ∈ { , , . . . , N − } , and a sequenceof positive numbers ( γ n : n ∈ N ) satisfying ∞ Õ n = γ n = ∞ , such that R nN + i ( ) = γ n (cid:0) X nN + i ( ) − σ Id (cid:1) converges to the non-zero matrix R i . If (cid:0) R nN + i ( ) : n ∈ N (cid:1) belongs to D (cid:0) Mat ( , R ) (cid:1) , discr R i > , and (6.12) ∞ Õ n = a n < ∞ , then A is self-adjoint if and only if there is n ≥ , such that (6.13) ∞ Õ n = n n Ö j = n (cid:12)(cid:12)(cid:12)(cid:12) + σ tr R j N + i ( ) + p discr R j N + i ( ) γ j (cid:12)(cid:12)(cid:12)(cid:12) = ∞ . Moreover, if A is self-adjoint, then σ ess ( A ) = ∅ .Proof. Since discr R i > , there are δ > and n ∈ N , such that for all j ≥ n , discr R j N + i ( ) > δ. Hence, the matrix R j N + i ( ) has two distinct eigenvalues ξ + j ( ) = tr R j N + i ( ) + σ p discr R j N + i ( ) , and ξ − j ( ) = tr R j N + i ( ) − σ p discr R j N + i ( ) , thus the matrix X j N + i ( ) = σ Id + γ − j R j N + i ( ) has two eigenvalues λ + j ( ) = σ + ξ + j ( ) γ j , and λ − j ( ) = σ + ξ − j ( ) γ j . Let K be any compact subset of R . By Proposition 6.1, we can write X nN + i ( x ) = σ Id + γ n R nN + i ( ) + E nN + i ( x ) where ∞ Õ n = sup x ∈ K k E nN + i ( x )k < ∞ . Since ( R j N + i ( ) : j ∈ N ) belongs to D (cid:0) Mat ( , R ) (cid:1) , by Corollary 4.5, there are two sequences (cid:0) Φ − j : j ≥ n (cid:1) and (cid:0) Φ + j : j ≥ n (cid:1) satisfying Φ j + = (cid:0) X j N + i ( ) + E j N + i (cid:1) Φ j , and such that(6.14) lim n →∞ sup K (cid:13)(cid:13)(cid:13)(cid:13) Φ ± n Î n − j = n λ ± j ( ) − v ± (cid:13)(cid:13)(cid:13)(cid:13) = for certain v − , v + , . Let φ ± = B − B − · · · B − n Φ ± n . For n ≥ , we set φ ± n + = B n φ ± n . Then for k N + i ′ > n N + i with i ′ ∈ { , , . . . , N − } , we have φ ± k N + i ′ = B − k N + i ′ B − k N + i ′ + · · · B − k N + i − Φ ± k if i ′ ∈ { , , . . . , i − } , Φ ± k if i ′ = i , B k N + i ′ − B k N + i ′ − · · · B k N + i Φ ± k if i ′ ∈ { i + , . . . , N − } . Consequently, we obtain(6.15) lim k →∞ φ ± k N + i ′ Î nj = n λ ± j ( ) = B − i ′ ( ) B − i ′ + ( ) · · · B − i − ( ) v ± if i ′ ∈ { , , . . . , i − } v ± if i ′ = i , B i ′ − ( ) B i ′ − ( ) · · · B i ( ) v ± if i ′ ∈ { i + , . . . , N − } , uniformly on K . Let u ± n ( x ) = ( h φ ± ( x ) , e i if n = , h φ ± n ( x ) , e i if n ≥ . Then ( u + n ( x ) : n ∈ N ) and ( u − n ( x ) : n ∈ N ) are two generalized eigenvectors associated with x ∈ K . Sincetheir asymptotic behavior is different from each other, they are linearly independent.By (6.14) and (6.15), there is a constant c > such that for all n > n , and x ∈ K ,(6.16) (cid:12)(cid:12) u ± nN + i ( x ) (cid:12)(cid:12) + (cid:12)(cid:12) u ± nN + i − ( x ) (cid:12)(cid:12) = (cid:13)(cid:13) φ ± nN + i ( x ) (cid:13)(cid:13) ≥ c n − Ö j = n (cid:12)(cid:12) λ ± j ( ) (cid:12)(cid:12) . Moreover, for all n > n , i ′ ∈ { , , . . . , N − } , and x ∈ K ,(6.17) (cid:13)(cid:13) φ ± nN + i ′ ( x ) (cid:13)(cid:13) ≤ c n − Ö j = n (cid:12)(cid:12) λ ± j ( ) (cid:12)(cid:12) . Since | λ − j ( )| ≤ | λ + j ( )| , we obtain(6.18) ∞ Õ n = n + | u − n ( x )| ≤ c ∞ Õ n = n + | u + n ( x )| . Now, observe that if (6.13) is satisfied then by (6.16) the generalized eigenvector ( u + n ( x ) : n ∈ N ) isnot square-summable, hence by [28, Theorem 6.16], the operator A is self-adjoint. On the other-hand, if(6.13) is not satisfied, then by (6.17) and (6.18), all generalized eigenvectors are square-summable, thusby [28, Theorem 6.16], the operator A is not self-adjoint.Finally, let us suppose that A is self-adjoint. By the proof of [30, Theorem 5.3], if(6.19) ∞ Õ n = sup x ∈ K | u − n ( x )| < ∞ , then σ ess ( A ) ∩ K = ∅ , and since K is any compact subset of R this implies that σ ess ( A ) = ∅ . Therefore, tocomplete the proof of the theorem it is enough to show (6.19). Observe that E nN + i ( ) = , thus λ + j ( ) λ − j ( ) = det X j N + i ( ) = a j N + i − a ( j + ) N + i − , and so k Ö j = n λ − j ( ) λ + j ( ) = a n N + i − a ( k + ) N + i − . SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 31
Consequently, by (6.12), ∞ Õ k = n k Ö j = n (cid:12)(cid:12) λ − j ( ) λ + j ( ) (cid:12)(cid:12) = ∞ Õ k = n a n N + i − a ( k + ) N + i < ∞ , which together with | λ − j ( )| ≤ | λ + j ( )| implies that ∞ Õ k = n k Ö j = n (cid:12)(cid:12) λ − j ( ) (cid:12)(cid:12) < ∞ . Hence, by (6.17) we obtain (6.19), and the theorem follows. (cid:3)
By the similar reasoning one can prove the following theorem.
Theorem 6.4.
Let N be a positive integer. Let A be a Jacobi matrix with N -periodically modulated entriesso that X ( ) = σ Id for a certain σ ∈ {− , } . Assume that there are i ∈ { , , . . . , N − } and a sequence ofpositive numbers ( γ n : n ∈ N ) such that R nN + i ( ) = γ n (cid:0) X nN + i ( ) − σ Id (cid:1) converges to the non-zero matrix R i . If (cid:0) R nN + i ( ) : n ∈ N (cid:1) belongs to D (cid:0) Mat ( , R ) (cid:1) , discr R i < , and ∞ Õ n = a n < ∞ , then the operator A is not self-adjoint. Proposition 4.6 motivates us to the following notion. Given a sequence ( w n : n ∈ N ) such that w n > forall n ∈ N , we introduce the weighted Stolz class. We say that ( x n ) a bounded sequence from a normed vectorspace X belongs to D ( X ; w ) , if ∞ Õ n = k ∆ x n k w n < ∞ . Moreover, given a positive integer N , we say that x ∈ D N ( X ; w ) if for each i ∈ { , , . . . , N − } , (cid:0) x nN + i : n ∈ N (cid:1) ∈ D ( X ; w ) . Similar reasoning to [41, Corollary 1] leads to the following fact.
Proposition 6.5.
For any weight ( w n ) (i) If ( x n ) , ( y n ) ∈ D ( X ; w ) , then ( x n y n ) , ( x n + y n ) ∈ D ( X ; w ) .(ii) If ( x n ) ∈ D ( K , C ; w ) , and k x n ( t )k ≥ c > for all n ≥ N and t ∈ K , then ( x − n ) ∈ D ( K , C ; w ) . The following proposition describes a way to construct matrices ( R n : n ∈ N ) satisfying the hypotheses ofTheorems 6.3 and 6.4. Proposition 6.6.
Let N be a positive integer and i ∈ { , , . . . , N − } . Let A be a Jacobi matrix with N -periodically modulated entries so that X ( ) = σ Id for a certain σ ∈ {− , } . Assume that there is ( γ n : n ∈ N ) a sequence of positive numbers such that R nN + i ( ) = γ n (cid:0) X nN + i ( ) − σ Id (cid:1) converges tonon-zero matrix R i . If there are two N -periodic sequences ( ˜ s i ′ : i ′ ∈ N ) and ( ˜ z i ′ : i ′ ∈ N ) such that (6.20) ˜ s i ′ = lim n →∞ γ n (cid:16) α i ′ − α i ′ − a nN + i ′ − a nN + i ′ (cid:17) , and ˜ z i ′ = lim n →∞ γ n (cid:16) β i ′ α i ′ − b nN + i ′ a nN + i ′ (cid:17) , then (6.21) R i = N − Õ j = ( N − Ö m = j + B i + m ( ) ) (cid:18) s i + j ˜ z i + j (cid:19) ( j − Ö m = B i + m ( ) ) and (6.22) tr R i = − σ N − Õ j = ˜ s i + j α i + j α i + j − . Moreover, if there is a weight ( w n : n ∈ N ) so that for all i ′ ∈ { , , . . . , N − } , (6.23) (cid:18) γ n : n ∈ N (cid:19) , (cid:18) γ n (cid:16) α i ′ − α i ′ − a nN + i ′ − a nN + i ′ (cid:17) : n ∈ N (cid:19) , (cid:18) γ n (cid:16) β i ′ α i ′ − b nN + i ′ a nN + i ′ (cid:17) : n ∈ N (cid:19) ∈ D ( R ; w ) , then (6.24) (cid:0) R nN + i ( ) : n ∈ N (cid:1) ∈ D (cid:0) Mat ( , R ) ; w (cid:1) . Proof.
Since X n ( ) − X n ( ) = N − Õ i ′ = ( N − Ö m = i ′ + B n + m ( ) ) (cid:0) B n + i ′ ( ) − B n + i ′ ( ) (cid:1) ( i ′ − Ö m = B n + m ( ) ) , and X i ( ) = σ Id , we get(6.25) R nN + i ( ) = N − Õ i ′ = ( N − Ö m = i ′ + B i + m ( ) ) γ n (cid:0) B nN + i + i ′ ( ) − B i + i ′ ( ) (cid:1) ( i ′ − Ö m = B nN + i + m ( ) ) . Observe that(6.26) γ n (cid:0) B nN + i + i ′ ( ) − B i + i ′ ( ) (cid:1) = γ n α i + i ′− α i + i ′ − a nN + i + i ′− a nN + i + i ′ β i + i ′ α i + i ′ − b nN + i + i ′ a nN + i + i ′ ! , thus by (6.20) we obtain lim n →∞ γ n (cid:0) B nN + i + i ′ ( ) − B i + i ′ ( ) (cid:1) = (cid:18) s i + i ′ ˜ z i + i ′ (cid:19) . Now, (6.21) easily follows from (6.25). The proof of (6.22) is analogous to Proposition 2.2, cf. (2.4) and(2.5).We proceed to showing (6.24). By (6.23), for each i ′ ∈ { , , . . . , N − } , (cid:18) a nN + i ′ − a nN + i ′ : n ∈ N (cid:19) , (cid:18) b nN + i ′ a nN + i ′ : n ∈ N (cid:19) ∈ D ( R ; w ) , thus,(6.27) (cid:0) B nN + i ′ ( ) : n ∈ N (cid:1) ∈ D (cid:0) GL ( , R ) ; w (cid:1) . Moreover, in view of (6.26), the condition (6.23) implies that(6.28) (cid:16) γ n (cid:0) B nN + i + i ′ ( ) − B i + i ′ ( ) (cid:1) : n ∈ N (cid:17) ∈ D (cid:0) Mat ( , R ) ; w (cid:1) . Now, (6.27) and (6.28) together with (6.25) implies (6.24) (cid:3)
A periodic modulations of Kostyuchenko–Mirzoev’s class.
Let N be a positive integer. We say thata Jacobi matrix A associated to ( a n : n ∈ N ) and ( b n : n ∈ N ) belongs to N -periodically modulatedKostyuchenko–Mirzoev’s class, if there are two N -periodic sequences ( α n : n ∈ Z ) and ( β n : n ∈ Z ) ofpositive and real numbers, respectively, such that a n = α n ˜ a n (cid:16) + f n γ n (cid:17) > , and b n = β n α n a n where ( f n : n ∈ N ) is bounded sequence, and ( ˜ a n : n ∈ N ) is a positive sequence satisfying ∞ Õ n = a n < ∞ and lim n →∞ γ n (cid:16) − ˜ a n − ˜ a n (cid:17) = κ > for a certain positive sequence ( γ n : n ∈ N ) tending to infinity.This class contains interesting examples of Jacobi matrices giving rise to self-adjoint operators which donot satisfy the Carleman’s condition. Moreover, we formulate certain conditions under which the essential SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 33 spectrum is empty. This class has been studied in [12, 21, 29–31] in the case when N is an even integer, α n ≡ , β n ≡ , and ˜ a n = ( n + ) κ , γ n = n + for some κ > . Theorem 6.7.
Let N be a positive integer. Let A be a Jacobi matrix from N -periodically modulatedKostyuchenko–Mirzoev’s class so that X ( ) = σ Id for a certain σ ∈ {− , } . Suppose that there is a weight ( w n : n ∈ N ) , so that (6.29) (cid:18) γ n (cid:16) − ˜ a n − ˜ a n (cid:17) : n ∈ N (cid:19) , (cid:0) f n : n ∈ N (cid:1) , (cid:18) γ n − γ n : n ∈ N (cid:19) ∈ D N ( R ; w ) , and (6.30) lim n →∞ γ n − γ n = . Then for all i ∈ { , , . . . , N − } , the matrices R nN + i ( ) = γ nN (cid:0) X nN + i ( ) − σ Id (cid:1) converge to the non-zeromatrix R i , (6.31) (cid:0) R nN + i ( ) : n ∈ N (cid:1) ∈ D (cid:0) Mat ( , R ) ; w (cid:1) , and ∞ Õ n = sup z ∈ K (cid:13)(cid:13) X nN + i ( z ) − X nN + i ( ) (cid:13)(cid:13) < ∞ for every compact set K ⊂ C . Moreover, tr R i = − κσ N , and (6.32) R i = N − Õ j = α i + j − α i + j (cid:0) κ + f i + j − f i + j − (cid:1) ( N − Ö m = j + B i + m ( ) ) (cid:18) (cid:19) ( j − Ö m = B i + m ( ) ) where ( f n : n ∈ Z ) is N -periodic sequence so that (6.33) lim n →∞ | f n − f n | = . Proof.
To prove (6.31), we are going to apply Proposition 6.6. To do so, we need to check (6.23). In fact, itis enough to show that for any i ∈ { , , . . . , N − } ,(6.34) (cid:18) γ j N (cid:16) α i − α i − a j N + i − a j N + i (cid:17) : j ∈ N (cid:19) ∈ D ( R ; w ) . We write γ j N (cid:16) α i − α i − a j N + i − a j N + i (cid:17) = α i − α i γ j N (cid:16) − ˜ a j N + i − ˜ a j N + i (cid:16) + e j γ j N (cid:17) (cid:19) where(6.35) e j = γ j N (cid:18) + f j N + i − γ j N + i − + f j N + i γ j N + i − (cid:19) = γ j N γ j N + i − f j N + i − − f j N + i γ j N + i − γ j N + i + f j N + i γ j N + i . Thus(6.36) γ j N (cid:16) α i − α i − a j N + i − a j N + i (cid:17) = α i − α i γ j N (cid:16) − ˜ a j N + i − ˜ a j N + i (cid:17) − α i − α i ˜ a j N + i − ˜ a j N + i e j and by (6.29) we easily obtain (6.34).In view of (6.30) and (6.33), the formula (6.35) gives lim j →∞ e j N + i = f i − − f i . Thus, by (6.36) lim j →∞ γ j N (cid:16) α i − α i − a j N + i − a j N + i (cid:17) = α i − α i ( κ + f i − f i − ) . Fix a compact set K ⊂ C . Since the condition (6.2) is satisfied, by Proposition 6.1, for all z ∈ K , X j N + i ( z ) = σ Id + γ j R j N + i ( ) + E j N + i ( z ) where sup z ∈ K k E n ( z )k ≤ c ˜ a n . Finally, by Proposition 6.6, we obtain (6.31) and (6.32). (cid:3)
Examples of modulated sequences.
In this section we present examples of sequences ( ˜ a n : n ∈ N ) and ( γ n : n ∈ N ) satisfying the assumptions of Theorem 6.7. Example 6.8.
Let κ > and ˜ a n = ( n + ) κ and γ n = n + . Then γ n (cid:16) − ˜ a n − ˜ a n (cid:17) = κ + κ ( κ − ) n + O (cid:16) n (cid:17) . Example 6.9.
Let ˜ a n = ( n + ) log ( n + ) and γ n = n + . Then γ n (cid:16) − ˜ a n − ˜ a n (cid:17) = + n − n log n + O (cid:16) n log n (cid:17) . Proposition 6.10.
Suppose that the hypotheses of Theorem 6.7 are satisfied with γ n = n + . Assume that discr R > . Then(i) if − κ + N √ discr R > − , then the operator A is self-adjoint;(ii) if − κ + N √ discr R < − , then the operator A is not self-adjoint.Moreover, if the operator A is self-adjoint then σ ess ( A ) = ∅ .Proof. We shall consider the case (i) only as the reasoning in (ii) is similar. By Theorem 6.3 it is enough tocheck whether there is n ≥ so that the series(6.37) ∞ Õ n = n n Ö j = n (cid:12)(cid:12)(cid:12)(cid:12) + σ tr R j N ( ) + p discr R j N ( ) j N (cid:12)(cid:12)(cid:12)(cid:12) diverges. Let us select δ > so that(6.38) − κ + N p discr R − δ > − . By Theorem 6.7, tr R = − κσ N . Hence, there is j ∈ N such that for all j ≥ j , (cid:12)(cid:12)(cid:12)(cid:0) σ tr R j N ( ) + q discr R j N ( ) (cid:1) − (cid:0) − κ N + p discr R (cid:1)(cid:12)(cid:12)(cid:12) ≤ N δ. Thus, + σ tr R j N ( ) + p discr R j N ( ) j N ≥ + j N (cid:0) − κ N + p discr R − N δ (cid:1) , and so log (cid:18) n Ö j = j (cid:12)(cid:12)(cid:12)(cid:12) + σ tr R j N ( ) + p discr R j N ( ) j N (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≥ − c + (cid:16) − κ + N p discr R − δ (cid:17) n Õ j = j ≥ − c ′ + (cid:16) − κ + N p discr R − δ (cid:17) log n . SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 35
Therefore, n Ö j = j (cid:12)(cid:12)(cid:12)(cid:12) + σ tr R j N ( ) + p discr R j N ( ) j N (cid:12)(cid:12)(cid:12)(cid:12) ≥ cn − κ + N √ discr R − δ which, in view of (6.38), implies that the series (6.37) is divergent. (cid:3) Example 6.11.
For < τ < we set ˜ a n = e n τ and γ n = max { n − τ , } . Let m ∈ N be chosen so that − m − ≤ τ < − m − . Then − ˜ a n − ˜ a n = m − Õ j = (− ) j + j ! (cid:0) n τ − ( n − ) τ (cid:1) j + O (cid:0) n m ( τ − ) (cid:1) = m − Õ j = (− ) j + j ! n τ j (cid:16) − ( − n − ) τ (cid:17) j + O (cid:0) n m ( τ − ) (cid:1) . Since − ( − n − ) τ = τ n − − τ ( τ − ) n − + O (cid:0) n − (cid:1) , we obtain − ˜ a n − ˜ a n = n τ (cid:16) τ n − − τ ( τ − ) n − + O (cid:0) n − (cid:1) (cid:17) − m − Õ j = (− ) j j ! n τ j (cid:16) τ n − + O (cid:0) n − (cid:1) (cid:17) j + O (cid:0) n m ( τ − ) (cid:1) = τ n τ − − τ ( τ − ) n τ − + O (cid:0) n τ − (cid:1) − m − Õ j = (− ) j j ! n τ j (cid:16) τ j n − j + O (cid:0) n − j − (cid:1) (cid:17) + O (cid:0) n m ( τ − ) (cid:1) = τ n τ − − τ ( τ − ) n τ − − m − Õ j = (− τ ) j j ! n j ( τ − ) + O (cid:0) n τ − (cid:1) + O (cid:0) n m ( τ − ) (cid:1) . Hence, γ n (cid:18) − ˜ a n − ˜ a n (cid:19) = τ + τ ( − τ ) n − − m − Õ j = (− τ ) j j ! n −( j − )( − τ ) + O (cid:0) n − + τ (cid:1) + O (cid:0) n −( m − )( − τ ) (cid:1) . In particular, the assumptions of Theorem 6.7 are satisfied.For a given sequence ( γ n : n ∈ N ) , the following proposition provides an explicit sequence ( ˜ a n : n ∈ N ) satisfying the regularity assumptions of Theorem 6.7. Proposition 6.12.
Suppose that ( γ n : n ∈ N ) is a positive sequence such that lim n →∞ γ n = ∞ , and (cid:18) γ n : n ∈ N (cid:19) ∈ D N ( R ; w ) where w = ( w n : n ∈ N ) is a weight. For κ > we set ˜ a n = exp (cid:18) n Õ j = κγ j (cid:19) . Then lim n →∞ γ n (cid:16) − ˜ a n − ˜ a n (cid:17) = κ, and (cid:18) γ n (cid:16) − ˜ a n − ˜ a n (cid:17) : n ∈ N (cid:19) ∈ D N ( R ; w ) . Proof.
We have γ n (cid:16) − ˜ a n − ˜ a n (cid:17) = γ n (cid:18) − exp (cid:16) − κγ n (cid:17) (cid:19) = f (cid:16) γ n (cid:17) where f ( x ) = − e − κ x x . Observe that lim x → f ( x ) = κ. Moreover, f has analytic extension to R , thus by the mean value theorem (cid:12)(cid:12)(cid:12) f (cid:16) γ n + N (cid:17) − f (cid:16) γ n (cid:17)(cid:12)(cid:12)(cid:12) ≤ c (cid:12)(cid:12)(cid:12) γ n + N − γ n (cid:12)(cid:12)(cid:12) , from which the conclusion follows. (cid:3) The following proposition settles the problem when the Carleman’s condition is satisfied in terms of thegrowth of the sequence ( γ n : n ∈ N ) . Proposition 6.13.
Suppose that ( γ n : n ∈ N ) and ( ˜ a n : n ∈ N ) are positive sequences satisfying lim n →∞ γ n = ∞ , and lim n →∞ γ n (cid:16) − ˜ a n − ˜ a n (cid:17) = κ > . Then(i) if lim n →∞ γ n n = , then Í ∞ n = a n < ∞ ;(ii) if lim n →∞ γ n n = ∞ , then Í ∞ n = a n = ∞ .Proof. We shall prove (i) only, as the proof of (ii) is similar. Let r n = γ n (cid:16) − ˜ a n − ˜ a n (cid:17) . There is n such that for n ≥ n , γ n n ≤ κ ≤ r n . Hence, for j ≥ n , ˜ a j − ˜ a j = − r j γ j ≤ − j , and so ˜ a n − ˜ a n = n Ö j = n ˜ a j − ˜ a j ≤ n Ö j = n (cid:18) − j (cid:19) . Consequently, for a certain c > , ˜ a n − ˜ a n ≤ cn − , which implies that ∞ Õ n = a n < ∞ . (cid:3) The following proposition has a proof similar to Proposition 6.10.
SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 37
Proposition 6.14.
Suppose that the hypotheses of Theorem 6.7 are satisfied for a sequence ( γ n : n ∈ N ) such that lim n →∞ γ n n = . Assume that discr R > . Then(i) if − κ + N √ discr R > then the operator A is self-adjoint;(ii) if − κ + N √ discr R < then the operator A is not self-adjoint.Moreover, if A is self-adjoint then σ ess ( A ) = ∅ . Construction of the modulating sequences.
In this section we present examples of sequences ( α n : n ∈ N ) and ( β n : n ∈ N ) for which one can compute tr R and discr R .The first example illustrates that the sign of discr R may be positive or negative. Example 6.15.
Let N = , and α n ≡ , and β n ≡ . Then σ = and R = (cid:18) − κ − f + f κ − f + f − κ + f − f − κ + f − f (cid:19) . Consequently, tr R = − κ and discr R = (cid:16) f + f + f − f f − f f − f f (cid:17) − κ . In particular, taking f = f = and f = t , we obtain sign ( discr R ) = | t | > √ κ, | t | = √ κ, − | t | < √ κ. In the following example, discriminant of R is non-negative regardless of ( f n : n ∈ Z ) . Example 6.16.
Let N = , and α n ≡ , β n = ( (− ) n / n even , otherwise.Then σ = and R = (cid:18) − κ − f + f − f + f − f + f − f − κ + f − f + f − f (cid:19) . Consequently, tr R = − κ and discr R = (cid:18) Õ j = (− ) j f j (cid:19) ≥ . The following theorem provides a large class of modulating sequences for which discr R is alwaysnon-negative. Theorem 6.17.
Let N be an even integer and κ > . Let ( f n : n ∈ Z ) be N -periodic sequence of non-negativenumbers and ( α n : n ∈ Z ) be N -periodic sequence of positive numbers satisfying (6.39) α α · · · α N − = α α · · · α N − . Let B n denote the transfer matrix associated with sequences ( α n : n ∈ Z ) and β n ≡ . We set R = N − Õ j = α j − α j (cid:0) κ + f j − f j − (cid:1) ( N − Ö m = j + B m ( ) ) (cid:18) (cid:19) ( j − Ö m = B m ( ) ) . Then tr R = −(− ) N / N κ and discr R = (cid:18) N − Õ j = (− ) j f j (cid:19) . Proof.
Let N = M . By [36, Proposition 3], for all ℓ ≥ k ≥ we have(6.40) ℓ Ö m = k B m ( ) = − α k − α k p [ k + ] ℓ − k − ( ) p [ k ] ℓ − k ( )− α k − α k p [ k + ] ℓ − k ( ) p [ k ] ℓ − k + ( ) ! . Observe that for k ≥ and j ≥ , j + k − Ö m = j B m ( ) = k − Ö m = (cid:16) B j + m + ( ) B j + m ( ) (cid:17) = k − Ö m = − α j + m − α j + m − α j + m α j + m + ! = (− ) k α j + k − α j + k − . . . α j + α j + α j − α j α j + k − α j + k − . . . α j + α j + α j α j + ! . In particular, by (6.39), we obtain N − Ö m = B m ( ) = (− ) M Id . Moreover, by (6.40), for all j ≥ and n ≥ ,(6.41) p [ j ] n ( ) = ( (− ) k α j + k − α j + k − . . . α j + α j + α j α j + n = k , otherwise.Setting s j = κ + f j − f j − , by (6.32), we write R = N − Õ j = α j − α j s j ( N − Ö m = j + B m ( ) ) (cid:18) (cid:19) ( j − Ö m = B m ( ) ) . Therefore, by (6.40), R = N − Õ j = α j − α j s j − α j α j + p [ j + ] N − j − ( ) p [ j + ] N − j − ( )− α j α j + p [ j + ] N − j − ( ) p [ j + ] N − j − ( ) ! (cid:18) (cid:19) − α N − α p [ ] j − ( ) p [ ] j − ( )− α N − α p [ ] j − ( ) p [ ] j ( ) ! , and consequently, R = N − Õ j = α j − α j s j − α N − α p [ ] j − ( ) p [ j + ] N − j − ( ) p [ j + ] N − j − ( ) p [ ] j − ( )− α N − α p [ j + ] N − j − ( ) p [ ] j − ( ) p [ j + ] N − j − ( ) p [ ] j − ( ) ! . In view of (6.41), we have R = N − Õ j = α j − α j s j − α N − α p [ ] j − ( ) p [ j + ] N − j − ( ) p [ j + ] N − j − ( ) p [ ] j − ( ) ! . By considering even and odd j , the last formula can be written in the form R = M − Õ k = α k − α k s k (cid:18) − α N − α p [ ] k − ( ) p [ k + ] N − k − ( )
00 0 (cid:19) + M − Õ k = α k α k + s k + (cid:18) p [ k + ] N − k − ( ) p [ ] k ( ) (cid:19) . SSENTIAL SPECTRA OF UNBOUNDED JACOBI MATRICES 39
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Grzegorz Świderski, Department of Mathematics, KU Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven,Belgium & Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
E-mail address : [email protected] Bartosz Trojan, Instytut Matematyczny, Polskiej Akademii Nauk, ul. Śniadeckich 8, 00-696 Warszawa, Poland
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