Orthogonal Polynomials with Periodically Modulated Recurrence Coefficients in the Jordan Block Case II

We study Jacobi matrices with N-periodically modulated recurrence coefficients when the sequence of N-step transfer matrices is convergent to a non-trivial Jordan block. In particular, we describe asymptotic behavior of their generalized eigenvectors, we prove convergence of N-shifted Turán determinants as well as of the Christoffel–Darboux kernel on the diagonal. Finally, by means of subordinacy theory, we identify their absolutely continuous spectrum as well as their essential spectrum. By quantifying the speed of convergence of transfer matrices we were able to cover a large class of Jacobi matrices. In particular, those related to generators of birth–death processes.


I
Consider two sequences = ( : ∈ N 0 ) and = ( : ∈ N 0 ) such that > 0 and ∈ R for all ≥ 0. Let be the closure in ℓ 2 (N 0 ) of the operator acting by the matrix A = 0 0 0 0 . . . on finitely supported sequences.The operator is called Jacobi matrix and its Jacobi parameters are the sequences and .Recall that ℓ 2 (N 0 ) is the Hilbert space of square summable complex-valued sequences with the scalar product Its standard orthonormal basis will be denoted by ( : ∈ N 0 ).Namely, is the sequence having 1 on the th position and 0 elsewhere.Let us observe that the operator is always symmetric.However, if is unbounded, that is at least one of the sequences and is unbounded, it does not have to be self-adjoint.If it is self-adjoint, then one can define a Borel probability measure as where is the spectral resolution of the identity of .Then the sequence of polynomials ( : . is an orthonormal basis in 2 (R, ), that is the Hilbert space of square integrable complex-valued functions with the scalar product Moreover, the operator : ℓ 2 (N 0 ) → 2 (R, ) defined on the basis vectors by = is unitary and satisfies ( −1 ) ( ) = ( ) for every ∈ 2 (R, ) such that ∈ 2 (R, ), see [45,Section 6] for more details.It follows that the spectral properties of are intimately related to the properties of .For example, ess ( ) is the set of accumulation points of supp( ).Furthermore, if = ac + sing is the Lebesgue decomposition of into the absolutely continuous and the singular parts with respect to the Lebesgue measure, then ac ( ) = supp( ac ) and sing ( ) = supp( sing ).
An interesting class of unbounded Jacobi matrices is related to the so-called birth-death processes (see, e.g.[46]), that is stationary Markov processes with the discrete state space N 0 .According to [24] generators of birth-death processes correspond to the Jacobi parameters where positive sequences ( : ∈ N 0 ) and ( : ∈ N 0 ) are called the birth and death rates, respectively.The simplest case is when = +1 , which we call symmetric.In particular, we can consider the following example.Another interesting class of unbounded Jacobi matrices has been recently studied in [66].
In particular, in [66], spectral properties of has been described if ∈ ( 3 2 , ∞) and + 2 − 2 ≠ 0. Let us observe that in both examples the Jacobi parameters satisfy (1.1) lim The aim of this article is to study spectral properties of as well as the asymptotic behavior of the associated orthogonal polynomials ( : ≥ 0) for a large subclass of Jacobi parameters satisfying (1.1) containing sequences from Examples 1.1 and 1.2 as special cases.In fact, in this article we will go beyond (1.1) by allowing the sequences ( −1 ) and ( ) to be asymptotically periodic.To be more precise, given a positive integer, we say that Jacobi parameters ( : ∈ N 0 ) and ( : ∈ N 0 ) are -periodically modulated if there are two -periodic sequences ( : ∈ Z) and ( : ∈ Z) of positive and real numbers, respectively, such that (a) lim It turns out that spectral properties of -periodically modulated Jacobi matrices depend on the matrix 0 (0) where for any ≥ 0 we have set More specifically, we can distinguish four cases: I. if | tr 0 (0)| < 2, then under some regularity assumptions on Jacobi parameters one has that ( ) = R, and it is purely absolutely continuous, see e.g.[19,21,54,56,57]; II.if | tr 0 (0)| = 2, then we have two subcases: a) if 0 (0) is diagonalizable then under some regularity assumptions on Jacobi parameters there is a compact interval ⊂ R such that is purely absolutely continuous on R \ , and it is purely discrete in the interior of , see e.g.[5-7, 10, 11, 17, 18, 23, 44, 54, 55, 60]; b) if 0 (0) is not diagonalizable then the only situation which was known is the case when either the essential spectrum of is empty or it is a half-line, see e.g.[4,8,9,20,22,33,34,36,37,39,42,51,61,66]; III. if | tr 0 (0)| > 2, then under some regularity assumptions on Jacobi parameters the essential spectrum of is empty, see e.g.[16,21,38,60,64]; Observe that in case I the absolutely continuous spectrum fills the whole real line, whereas in the case III it is empty.This phenomenon was originally observed in [21] and it was called spectral phase transition of the first type.Notice that the case II corresponds to the point where the actual phase transition occurs.In fact, in [21, Section 5] the task of analyzing the case II was formulated as a very interesting open problem, whose analysis required finding new tools.Nowadays, the case II.a is quite well-understood, see [58,60].Therefore, in this article we are exclusively interested in the case II.b, which for = 1 and ≡ 1, ≡ −2 covers (1.1).
Let us introduce an auxiliary positive sequence = ( : ∈ N 0 ) tending to infinity.In Examples 1.1 and 1.2 we take = and = + 1, respectively.We say that -periodically modulated Jacobi parameters ( ), ( ) are -tempered if Let us recall that a sequence ( : About the sequence we assume that and Moreover, we impose that where = sign(tr 0 (0)).To formulate the main results of this paper, we need further definitions.For ∈ C and ∈ N 0 we define the transfer matrix by We use the convention that −1 := 1.Moreover, for a matrix = 11 12   21  22   we set [ ] = .The discriminant of is defined as discr = (tr ) 2 − 4 det .The first main result of this article identifies the absolutely continuous and the essential spectrum of the studied class of Jacobi matrices.
Theorem A. Let be a positive integer.Let ( ) be a sequence of positive numbers tending to infinity and satisfying (1.2) and (1.3).Let ( ) and ( ) be -tempered -periodically modulated Jacobi parameters such that 0 (0) is a non-trivial parabolic element.Suppose that (1.4) holds true with = sign(tr 0 (0)).
Then the limit exists and defines a polynomial of degree at most one.Let Let us emphasize that Theorem A excludes the case Λ − = Λ + = ∅, that is ≡ 0.Moreover, Theorem A implies that the operator is not semi-bounded if Λ + = ∅ and Λ − ≠ ∅, because Λ − = R = ( ).However, it is unclear under what hypotheses the operator is semi-bounded when Λ + ≠ ∅.Recall that in the case III a characterization of semi-boundedness of the operator was given in [38].
The condition (1.4) might look rather restrictive.However, it is always satisfied by Jacobi parameters studied in [61] as well as for generators of symmetric birth-death processes (cf.Remark 11.5).Hence, Theorem A can be applied to Jacobi parameters described in Example 1.1 where for = we get ( ) = .Moreover, if = 1 the condition (1.4) reduces to Therefore, Theorem A can be applied to Jacobi parameters given in Example 1.2 where for The proof of Theorem A uses the theory of subordinacy.It was first developed in [14] for one-dimensional Schrödinger operators on the real half-line, and later adapted to other classes of operators, see e.g. the survey [13] for more details.In particular, the extension to Jacobi matrices has been accomplished in [26].The theory of subordinacy links asymptotic behavior of generalized eigenvectors to spectral properties of Jacobi matrices.Let us recall that a sequence ( : ∈ N 0 ) is a generalized eigenvector associated to ∈ C, and corresponding to ∈ R 2 \ {0}, if the sequence of vectors By cl( ) we denote the closure of the set . satisfies We often write ( ( , ) : ∈ N 0 ) to indicate the dependence on the parameters.In particular, the sequence of orthogonal polynomials ( ( ) : ∈ N 0 ) is the generalized eigenvector associated to = (0, 1) and ∈ C. Motivated by [49,Section 8] it will be convenient to define (generalized) Christoffel-Darboux kernel by Suppose that is self-adjoint.According to [26,Theorem 3], if for some compact interval with non-empty interior ⊂ R, then the measure is absolutely continuous on , and ⊂ supp( ).Consequently, is absolutely continuous on , and ⊂ ac ( ).This theory became a standard approach to spectral analysis of Jacobi matrices.It has also been observed that by imposing some uniformity conditions to (1.8) more detailed information on the density of can be obtained, see the references in [13,Section 4].In the present article we shall show that for any compact interval ⊂ Λ − the following stronger version of (1.8) holds true (1.9) sup In view of [2] (see also [32] for a different proof in a more general setup) the condition (1.9) implies existence of positive constants 1 , 2 such that the density of , ′ , satisfies (1.10) 1 < ′ ( ) < 2 for almost all ∈ , with respect to the Lebesgue measure.Finally, in [47], the following consequence of subordinacy theory has been established: if is self-adjoint and for some ⊂ R there is a function then ∩ ess ( ) = ∅.In Theorem 4.1, with a help of a recently obtained variant of discrete Levinson's type theorem (see [60]), we show that (1.11) holds for every compact interval ⊂ Λ + .The fact that (1.9) holds for every compact interval ⊂ Λ − is a consequence of the following theorem.
Theorem B. Let be a positive integer.Let ( ) be a sequence of positive numbers tending to infinity and satisfying (1.2) and (1.3).Let ( ) and ( ) be -tempered -periodically modulated Jacobi parameters such that 0 (0) is a non-trivial parabolic element.Suppose that (1.4) holds true with = sign(tr 0 (0)).Set If it is the case, then the limit exists locally uniformly with respect to ( , ) ∈ Λ − × S 1 , and defines a continuous positive function.
By S 1 we denote the unit sphere in R 2 .
We set Since > 1, the Carleman condition is not satisfied, that is As it is easy to check, we can apply Theorems A and B to the above Jacobi parameters, which leads to the conclusion that the corresponding Jacobi operator is self-adjoint and ac ( ) = R.
Example 1.3 is inspired by examples given by Kostyuchenko-Mirzoev in [28] who provided Jacobi parameters giving rise to self-adjoint Jacobi operators violating the Carleman's condition.Later the original Kostyuchenko-Mirzoev class was somewhat extended and it was proven that one usually has ess ( ) = ∅, see e.g.[17, Section 2.2] and [60, Section 6.2].To the best of our knowledge Jacobi parameters described in Example 1.3 provide the first instances of Jacobi operators violating the Carleman's condition such that ac ( ) = R.In contrast, a construction of self-adjoint Jacobi matrices with ac ( ) = [0, ∞) violating Carleman's condition is well-known, see e.g.[9].
To prove Theorem B, we first determine asymptotic behavior of generalized eigenvectors.Then we apply a non-trivial averaging procedure to it.The asymptotic formula is given in the following theorem.
The proof of Theorem C is based on uniform diagonalization of transfer matrices which has been already used in [61].However, in the current setup we were not able to relate | ( , )| to the density of .Hence, in order to prove that ( , ) ≠ 0 provided [ (0)] 21 ≠ 0, we needed an additional argument based on a consequence of the following theorem (see Corollary 6.3 for details) which studies convergence of generalized -shifted Turán determinants.The latter are defined as = ( , ) + −1 ( , ) − −1 ( , ) + ( , ) where ( ( , ) : ∈ N 0 ) is the generalized eigenvector associated to ∈ R, and corresponding to ∈ R 2 \ {0}.The (classical) shifted Turán determinants correspond to = (0, 1) .They were defined for the first time in [65] for = 1, and then generalized in [12] to ≥ 1.In [65] they were instrumental in studying the zeros of the Legendre polynomials where it was observed that they are non-negative on the support of their orthogonality measure, see also [25] for later developments.As it was shown in [40,Theorem 7.34] and [12,Theorem 6], if supp( ) is compact, the asymptotic behavior of shifted Turán determinants is usually closely related to the density of , see [30,31] and the survey [41].The extension of the above phenomena to measures with unbounded support has been accomplished in [54,55,57,61].For these reasons the following theorem is an important result on its own.Theorem D. Let be a positive integer.Let ( ) be a sequence of positive numbers tending to infinity and satisfying (1.2) and (1.3).Let ( ) and ( ) be -tempered -periodically modulated Jacobi parameters such that 0 (0) is a non-trivial parabolic element.Suppose that (1.4) holds true with = sign(tr 0 (0)).If Λ − ≠ ∅, then for each ∈ {0, 1, . . ., − 1} the limit exists locally uniformly with respect to ( , ) ∈ Λ − × S 1 and defines a continuous positive function.
Let us remark that the first order asymptotics of generalized eigenvectors provided by Theorem C is insufficient to prove (1.13).It is an open problem whether, similarly to [57][58][59]61], one can relate the value of (1.13) to the density of the measure .We hope to return to this problem in the future.
In this article, we also consider ℓ 1 -type perturbations of Jacobi parameters , satisfying hypotheses of Theorem A. Namely, in Section 10, we study Jacobi parameters ˜ , ˜ of the form where ( √ ), ( √ ) ∈ ℓ 1 .We show that for sequences ˜ and ˜ the analogues of Theorems A-C hold true.In particular, we can treat the following Jacobi parameters Example 1.4.For ∈ (1, ∞) and , ∈ R we set Jacobi parameters considered in Example 1.4 under the additional restrictions ∈ ( 3 2 , ∞) and , = O ( −2 ), have recently been studied in [66].
Before we close the introduction, let us mention some of the approaches used in the literature for analysis of the case II.b.In [9] it was observed that a certain class of Jacobi matrices related to birth-death processes can be studied by considering the restriction to a subspace of ℓ 2 of the square of Jacobi matrices belonging to the case I with ≡ 0. This method is particularly effective in describing ac ( ).Next, in [36], asymptotics of generalized eigenvectors was studied by the reduction to the analysis of a discrete variant of Ricatti equation, whereas in [42,51] the analysis was possible by applying Birkhoff-Adams theorem.Further, in [39] by adaptation of Kooman method (see [27]) and the approach of [1] it was possible to obtain asymptotic behavior of generalized eigenvectors for ∈ C \ {0} as well as continuity of the density of the measure .A very important class of methods is motivated by the technique introduced by Harris and Lutz in [15].In these methods for a given ∈ {0, 1, . . ., − 1} one consider the "change of variables" for some invertible matrices .Then by (1.7) and (1.5) the sequence (ì ) satisfies the equation The matrices are chosen in a way that one can apply to the system (1.15)Levinson's theorem.Then thanks to the relation (1.14), the asymptotics of (ì ) easily leads to the asymptotics of ( ì + : ≥ 0 ).The success of this approach depends on the properties of the matrices .In [20] the construction of these matrices were motivated by a formal WKB method in which, by means of an ansatz, one guesses the form of the solution.This approach was later extended in [4,22,33,37].It should be emphasized that the resulting matrices were complex-valued, oscillating and unbounded.In this work, we start by extending techniques which were successful in the prequel [61].Namely, we construct matrices such that the system (1.15) satisfies hypotheses of a uniform discrete Levinson's theorem so it belongs to Harris-Lutz paradigm.However, our matrices are very simple and explicit (see (3.1)), real and convergent (obviously to a singular matrix).These features lead to greater applicability of our approach than in the previous works.Since Jacobi parameters considered in this paper are more "singular" than in [61], we were forced to use a more general and delicate change of variables, so that we can exploit the condition (1.4) to "smooth them out".Using our change of variables, the spectral properties of on Λ + can be derived analogously to [61].On Λ − the situation is much more involved.Namely, in [61], in order to prove that is absolutely continuous on every compact ⊂ Λ − , we used an explicit sequence of probability measures ( : ∈ N) which converges weakly to , and such that the sequence of their densities converges uniformly on to a continuous positive function.In the present paper this approach does not work anymore.To get around of this issue we apply the subordinacy theory.This requires to analyze the asymptotic behavior of Christoffel-Darboux kernel which was possible thanks to the asymptotics obtained in Theorem C. All of this reduces the problem to study averages of highly oscillatory sums.For this reason we develop Lemma 8.1, which might be of independent interest.
The method of asymptotic analysis of generalized eigenvectors is similar to [61].However, in the present situation we had to find another argument showing positivity of the function | |.Previously, by using the convergence of densities of the sequence ( : ∈ N), we were able to explicitly compute the value of | | in terms of ′ .In the present work we use certain algebraic properties of together with Theorem D, see Claim 7.2 for details.Let us emphasize that the method of subordinacy gives the bound (1.10) only, which is weaker than the continuity of ′ .The drawback of the current approach compared to [61] is that we do not get a constructive method to approximate the density of .In the forthcoming article [62], by linking the asymptotic behavior of zeros of the polynomials ( : ∈ N 0 ) with the value of (1.12), we managed to prove that, under certain additional hypotheses, the density of the measure for Jacobi matrices satisfying Theorem B is a continuous positive function on Λ − .
The article is organized as follows: In Section 2 we fix notation and we formulate basic facts.Section 3 is devoted to our change of variables.In Section 4 we study spectral properties of on Λ + .Next, in Section 5 we describe uniform diagonalization of transfer matrices on Λ − , which is used in the rest of the article.The proof of Theorem D is presented in Section 6. Next, in Section 7, we prove Theorem C. Section 8 is devoted to the proof of Theorem B. In Section 9 we study the self-adjointness of .The extensions of Theorems A-C to ℓ 1 -type perturbations is achieved in Section 10.Finally, in Section 11, we present more concrete classes of sequences to illustrate results of this article.
Notation.By N we denote the set of positive integers and N 0 = N ∪ {0}.Throughout the whole article, we write if there is an absolute constant > 0 such that ≤ .We write ≍ if and .Moreover, stands for a positive constant whose value may vary from occurrence to occurrence.For any compact set , by (1) we denote the class of functions : → R such that lim →∞ = 0 uniformly on .
Acknowledgment.The first author was supported by long term structural funding -Methusalem grant of the Flemish Government.This work was completed while the first author was a postdoctoral fellow at KU Leuven.The authors would like to thank referees for their very valuable suggestions.

P
In this section we fix the notation which is used in the rest of the article.

Stolz class.
In this section we define a proper class of slowly oscillating sequences which is motivated by [52], see also [57,Section 2].Let be a normed space.We say that a sequence ( : ∈ N) of vectors from belongs to D ( ) where If is the real line with Euclidean norm we abbreviate D = D ( ).Given a compact set ⊂ C and a normed vector space , we denote by D ( , ) the case when is the space of all continuous mappings from to equipped with the supremum norm.Let us recall that D ( ) is an algebra provided is a normed algebra.Let be a positive integer.We say that a sequence ( : ∈ N) belongs to D ( ), if for any ∈ {0, 1, . . ., − 1}, Again, D ( ) is an algebra provided is a normed algebra.In what follows we shall use D 1 ( ) only.Lastly, for a sequence of square matrices ( : 0 ≤ ≤ 1 ) we set where is the transfer matrix defined as To indicate the dependence on the parameters, we write ( ( , ) : ∈ N 0 ).In particular, the sequence of orthogonal polynomials ( ( ) : ∈ N 0 ) is the generalized eigenvector associated to = 2 and ∈ C.
2.4.Periodic Jacobi parameters.By ( : ∈ Z) and ( : ∈ Z) we denote -periodic sequences of real and positive numbers, respectively.For each ≥ 0, let us define polynomials By we denote the Jacobi matrix corresponding to 0 0 0 0 . . .In this article we are mostly interested in tempered -periodically modulated Jacobi parameters, i.e. we assume that there is a sequence of positive numbers ( : ∈ N 0 ) tending to infinity and satisfying

and
(2.4) In view of (2.5), there are two -periodic sequence ( : ∈ Z) and Hence, there is ≥ 0, such that Let us observe that, if > 0 then with no lose of generality we can assume that = 1 and ≡ .Therefore, in what follows we shall assume that ∈ {0, 1}.
Let us define the -step transfer matrix by locally uniformly with respect to ∈ C. In the whole article we assume that the matrix 0 (0) is a non-trivial parabolic element of SL(2, R).Let 0 be a matrix so that (2.9) where (2.10) = sign(tr 0 (0)).
Hence, by the hypothesis (b), (2.7) and (2.6), and (2.23) follows.To see the last statement, we assume, contrary to our claim, that < 0. Then there is

T
In this section we freely use the notation introduced in Section 2. Fix ∈ {0, 1, . . ., − 1} and set where has been defined in (2.9) and (2.12), and Observe that Notice that Hence, by -periodicity of ( ), . Now, by (2.3) we obtain Similarly, -periodicity of ( ) and (2.3) leads to For fixed ∈ N, we have . . .
Similarly, we can find that Consequently, where ( ) is a sequence from D 1 , Mat(2, R) for any compact interval ⊂ Λ convergent to the zero matrix proving the formula (3.3).
Proof.In the following argument, we denote by ( ) and (E ) generic sequences tending to zero from D 1 and D 1 , Mat(2, R) , respectively, which may change from line to line.
Observe that by (1.2) Hence, Consequently, (3.9) Next, for ∈ , we have Using (3.10) we can write Hence, and so To find the asymptotic of the first factor, we write where Since by (3.7) Moreover, Analogously, we can find that Next, we write Similarly, we get and so Consequently, by (3.11)-(3.14)we obtain (3.15) Next, we observe that e = 1 + , sinh = 1 + , thus in view of (2.6), for each ′ ∈ {0, 1, . . ., − 1} we have We write We observe that and Hence, by (1.4) and (2.16), Next, let us notice that Therefore, by (2.6), we get and Analogously, and In view of (2.13), (2.14) and (2.16), we have thus by (3.16) and (3.17) we get and Hence, Summarizing, we obtain By (3.9) and (2.18), we have Therefore, Finally, by (3.15) we get which finishes the proof.locally uniformly on Λ where is defined in (2.17).
Proof.We write , thus by Theorems 3.1 and 3.2, we obtain
Using (5.1), ( ) has two eigenvalues ( ) and ( ) where For the proof, we write and so where the last estimate follows by [57, Proposition 1], proving Claim 5.1.Next, we show the following statement.
Corollary 6.3.Suppose that the hypotheses of Theorem 6.1 are satisfied.Then for any compact ⊂ Λ − there is a constant > 1 such that for any generalized eigenvector ì associated with ∈ and corresponding to ∈ S 1 , we have where ì = −1 ì + .
Proof.By (6.1) and Theorem 3.2 we have Hence, by (6.4), we have Observe that uniformly on .By the fact that ˜ is uniformly convergent on S 1 × to a positive function, the conclusion follows.

A
In this section we study the asymptotic behavior of generalized eigenvectors.We keep the notation introduced in Section 5.
On the contrary, let us suppose that there are ∈ S 1 , ∈ and a sequence ( : Setting ì = 1 1 + 2 2 , we have Hence, by (7.9), we obtain In view of (5.2), ì ( , ) is a real vector, thus by taking imaginary parts of the bracket, we conclude that We claim that (8.4) To see this, we observe that by the Stolz-Cesàro theorem, Since there is > 0 such that sup uniformly with respect to ( , ) ∈ × S 1 .Observe that (8.5) is an easy consequence of Lemma 8.1, provided we show the following statement.
Since lim Let us observe that, by Theorem 3.2, proving the claim.
To complete the proof of the theorem we write Hence, by (8.4) and the theorem follows.
Proof.If Λ − = ∅ then = 0 and so Λ + = R.Let = 0 and = {0}.We can repeat the first part of the proof of Theorem 4.1.Now, by (4.6) and (4.8), there are 1 ≥ 0 and > 0 such that for all ≥ 1 , (9.2) Moreover, for all ≥ 1 and ′ ∈ {0, 1, . . ., − 1}, By (4.3) we obtain (9.4) Hence, the operator is self-adjoint if and only if there is 0 > 0 such that (9.5) Indeed, if (9.5) is satisfied then by (9.2) the generalized eigenvector ( + (0) : ∈ N 0 ) is not square-summable, thus by [45,Theorem 6.16], the operator is self-adjoint.On the other hand, if (9.5) is not satisfied, then by (9.3) and (9.4), all generalized eigenvectors associated to 0 are square-summable, thus by [45,Theorem 6.16], the operator is not self-adjoint.The second part of the theorem follows by Theorem 4.1.Since ( : ∈ N) approaches infinity, there is 0 ≥ 1 such that Next, we observe that Let us consider the case (i).Because ( (0) : ∈ N) converges to R 0 (0), there is 1 ≥ 0 such that for all Consequently, that is (9.5) is not satisfied and so the operator is not self-adjoint.The reasoning in the case (ii) is analogous.Namely, there is 1 ≥ 0 such that for all ≥ 1 , −1 and so Therefore, (9.5) is satisfied and the operator is self-adjoint.In this section we show how to get the main results of the paper in the presence of certain size ℓ 1 perturbations.Let ( ˜ : ∈ N 0 ) and ( ˜ : ∈ N 0 ) be Jacobi parameters satisfying where ( : ∈ N 0 ) and ( : ∈ N 0 ) are -tempered -periodically modulated Jacobi parameters such that 0 (0) is a non-trivial parabolic element, and ( : ∈ N 0 ) and ( : ∈ N 0 ) are certain real sequences satisfying We follow the reasoning explained in [61, Section 9].Fix a compact set ⊂ R. Let us denote by (Δ ) any sequence of 2 × 2 matrices such that We notice that (10.1) where Moreover, for Suppose that ⊂ Λ + .Then, by Theorem 3.2, Since there is > 0 such that for all ∈ N, sup −1 +1 ≤ √ + , and sup ≤ , by setting where ( ) is a sequence from D 1 , Mat(2, R) convergent uniformly on to R , and If ( √ ) is sublinear and (sup Δ ) belongs to ℓ 1 , for each subsequence there is a further subsequence where ( : ∈ N 0 ) and ( : ∈ N 0 ) are -tempered -periodically modulated Jacobi parameters such that 0 (0) is a non-trivial parabolic element.Suppose that (1.4) holds true with = sign(tr 0 (0)).
for certain real sequences ( : ∈ N 0 ) and ( : ∈ N 0 ), then ess ( ˜ ) ∩ Λ + = ∅.Next, let us consider a compact set ⊂ Λ − .By Theorem 7.1 and Claim 5.1, there is > 0 such that for all ∈ N 0 , (10.4) and since det = −1 , we get Moreover, by (10.1) thus by (10.4) and (10.5) Hence, Next, let us introduce the following sequence of matrices , (10.5) and (10.6), we obtain Therefore, the sequence of matrices ( ) converges uniformly on to certain continuous mapping , and Observe that for each ∈ the matrix ( ) is non-degenerate.Indeed, we have Given ∈ S 1 , we set Then for = , the hypotheses of Theorem 3.2 are satisfied.Moreover, To see this, let us first observe that (11.8) = ˆ 1 1 + , which belongs to D 1 .Next, we write Moreover, (11.9) lim Next, by (11.3) This together with (11.8) implies that = lim →∞ ˆ exists.If we had > 0, then there would exist 0 ∈ N and a constant > 0 such that for all ≥ 0 On the other hand, we have Thus by the boundedness of ( − −1 : ∈ N) we get that for some ′ > 0 one has ≤ ′ ( + 1).It leads to a contradiction with (11.11).Hence, = 0, which easily gives the formula for .Proof.We start with the following Claim, which is inspired by [63,Lemma 2].
11.3.= 1.In this section we specify our results to = 1.

2. 2 .+ 1 2 *
Finite matrices.By Mat(2, C) and Mat(2, R) we denote the space of 2 × 2 matrices with complex and real entries, respectively, equipped with the spectral norm.Next, GL(2, R) and SL(2, R) consist of all matrices from Mat(2, R) which are invertible and of determinant equal 1, respectively.A matrix ∈ SL(2, R) is a non-trivial parabolic if it is not a multiple of the identity and | tr | = 2. Let ∈ Mat(2, C).By we denote the transpose of the matrix .Let us recall that symmetrization and the discriminant are defined as sym( ) = 1 2 , and discr( ) = (tr ) 2 − 4 det , respectively.Here * denotes the Hermitian transpose of the matrix .By { 1 , 2 } we denote the standard orthonormal basis of C 2 , i.e.

5 .
Tempered periodic modulations.Let be a positive integer.We say that Jacobi parameters ( : ∈ N 0 ) and ( : ∈ N 0 ) are -periodically modulated if there are two -periodic sequences ( : ∈ Z) and ( : ∈ Z) of positive and real numbers, respectively, such that (a) lim →∞