Sergey Naboko showed a keen interest in Jacobi operators, which led him to coauthor a considerable amount of papers on the subject. These papers are inescapable references in the spectral theory of Jacobi operators. His ideas and insights on the matter played a crucial role in the development of the know-how for revealing the spectral properties and finding examples of interesting spectral phenomena for Jacobi operators. Naboko’s approach to the spectral analysis of these operators is mainly based on the asymptotic analysis of the generalized eigenvectors. He developed multiple techniques and filigree methods for the analysis of the asymptotic behavior of such eigenvectors. Within this subject all papers (except for [1]) were written in collaboration with (in chronological order) S. Yakovlev, J. Janas, A. Laptev, O. Safronov, A. Boutet de Monvel, L. Silva, G. Stolz, M. Brown, R. Weikard, D. Damanik, E. Sheronova, I. Pchelintseva, S. Simonov, M. Marletta, R. Shterenberg, S. Kupin, I. Wood, and E. Judge. The process of doing mathematics with Sergey Naboko was always a process of friendship, sharing, and learning for us.

This note is intended to cover exhaustively the literature authored or coauthored by S. Naboko on Jacobi operators. We know for sure that there are papers to which S. Naboko contributed with important ideas, but did not agree to appear as a coauthor. S. Naboko was always open and eager to discuss questions concerning these operators and due to his recognized leadership in the area was constantly addressed to as an expert. For brevity, the list of references presented in this note includes only papers in which S. Naboko appeared as an author and it is ordered chronologically. Our account is also mostly chronological.

The Jacobi operator J is the operator in \(l_{2}(\mathbb {N})\) whose matrix representation with respect to the canonical basis \(\{\delta _{n}\}_{n\in \mathbb {N}}\) is

$$\displaystyle \begin{aligned} {} \begin{pmatrix} b_1 & a_1 & 0 & 0 & \cdots \\[1mm] a_1 & b_2 & a_2 & 0 & \cdots \\[1mm] 0 & a_2 & b_3 & a_3 & \\ 0 & 0 & a_3 & b_4 & \ddots\\ \vdots & \vdots & & \ddots & \ddots \end{pmatrix}\,, \end{aligned} $$
(1)

where \(a_{n}>0\) and \(b_{n}\in \mathbb {R}\) for \(n\in \mathbb {N}=\{1,2,\dots \}\). The sequences \(\{a_{n}\}_{n=1}^{\infty }\) and \(\{b_{n}\}_{n=1}^{\infty }\) are referred to as weights and the potential of (1). According to the definition of the matrix representation of a possibly unbounded symmetric operator, J is the minimal closed operator such that \(\delta _{n}\in \operatorname {\mathrm {dom}} J\) for all \(n\in \mathbb {N}\) and \(\left \langle \delta _{j},J\delta _{k}\right \rangle \) for \(j,k\in \mathbb {N}\) are the entries of the matrix (1). A Jacobi operator is either symmetric nonselfadjoint with deficiency indices \(n_{+}(J)=n_{-}(J)=1\) or it is selfadjoint (i.e. \(n_{+}(J)=n_{-}(J)=0\)). According to the Weyl’s alternative, the former case is called the limit circle case, while the latter corresponds to the limit point case. In the works referred to here, with the exception of [2], the sequences \(\{a_{n}\}_{n=1}^{\infty }\) and \(\{b_{n}\}_{n=1}^{\infty }\) are chosen in such a way that J turns out to be selfadjoint.

The sequence \(u=\{u_{n}\}_{n=1}^{\infty }\) being a solution to the equation

$$\displaystyle \begin{aligned} {} a_{n-1}u_{n-1}+b_{n}u_{n}+a_{n}u_{n+1}=zu_{n}\,,\quad n>1\,, \end{aligned} $$
(2)

is called a generalized eigenvector of J. The recurrence relation (2) can be written as a dynamical system

$$\displaystyle \begin{aligned} {} {\mathbf{u}}_{n+1}=B_{n}(z){\mathbf{u}}_{n}\,,\quad n>1\,, \end{aligned} $$
(3)

where

$$\displaystyle \begin{aligned} {} B_{n}(z):= \begin{pmatrix} 0 & 1 \\ \frac{a_{n-1}}{a_{n}} & \frac{z-b_{n}}{a_{n}} \end{pmatrix}\quad \text{and}\quad {\mathbf{u}}_{n}:=\binom{u_{n-1}}{u_{n}}\,, \end{aligned} $$
(4)

The matrix \(B_n(z)\) is called the transfer matrix of (3).

The first results by S. Naboko on the spectral properties of Jacobi operators are concerned with the discrete Schrödinger operators with decaying potentials, i.e. when in (1) \(a_{n}=1\) for all \(n\in \mathbb {N}\) and \(b_{n}\to 0\) as \(n\to \infty \). Under this assumption J is a bounded Jacobi operator and therefore it is selfadjoint. In the papers [3, 4], sufficient conditions on the sequence \(\{a_{n}\}_{n=1}^{\infty }\) are provided so that the elements of a certain given sequence of numbers inside the essential spectrum are eigenvalues of the corresponding operator. This result is obtained by ingenious transformations of the product of transfer matrices \(B_{n}(z)\) which lead to establishing estimates for norms of generalized eigenvectors. These estimates and the results of [5] allow one to obtain the mentioned sufficient conditions. In [3, 4] there are also conditions on the potential guaranteeing either dense pure point spectrum or absence of eigenvalues inside the essential spectrum. These results have a continuation in [6] where the approach of [3, 4] was applied to perturbed periodic Jacobi matrices to produce a possibly infinite number of eigenvalues embedded into the absolutely continuous spectrum at prescribed locations which are subject to certain conditions of rational independence.

In [7], by a different approach to the dynamical system (3) and without relying on [5], criteria for absence of eigenvalues of J when \(a_{n}\to 1\) and \(b_{n}\to 0\) as \(n\to \infty \) are provided. In this case, J is a compact perturbation of the discrete free Schrödinger operator. The case \(b_{n}\equiv 0\) is considered separately. In [7], the sharpness of these criteria is dealt with by providing examples and counterexamples showing the subtlety of the results. This kind of approach for illustrating results was a trademark of S. Naboko’s work.

S. Naboko was constantly concocting and implementing new methods for both the asymptotic analysis of generalized eigenvectors and the ways to translate this information into the spectral properties of the corresponding operator. The paper [8] considers the absolutely continuous spectrum of the operator J with \(b_{n}\equiv 0\) which S. Naboko used to refer to as the string operator (although it differs from Krein’s string operator). It is required in [8] that \(a_{n}\to 1\) as \(n\to \infty \) which implies that J is bounded and therefore selfadjoint. In this work, a Harris-Lutz type transform is used in the asymptotic analysis of the products of transfer matrices \(B_{n}(z)\) on the one hand and the Gilbert-Pearson-Khan subordinacy theory for characterizing the spectrum from the asymptotic behavior of generalized eigenvectors on the other. A solution \(\{u_n^-\}_{n=1}^{\infty }\) of the Eq. (2) is called subordinate, if for any other solution \(\{u_n\}_{n=1}^{\infty }\) of this equation one has

$$\displaystyle \begin{aligned} \frac{\sum_{k=1}^n|u_k^-|{}^2}{\sum_{k=1}^n|u_k|{}^2}\to0,\ n\to\infty. \end{aligned}$$

The existence of a subordinate solution at each point of an interval of the real line implies that the spectrum on this interval is purely point, absence of such a solution at each point of an interval implies that the interval is covered with purely absolutely continuous spectrum. The discreteness of the point spectrum cannot be determined by subordinacy theory and is established by other methods: estimating quadratic forms, controlling the uniformity of asymptotics of generalized eigenvectors, etc.

A further development of the results obtained in [8] is found in [9], where a discrete Schrödinger operator perturbed by a bounded Jacobi operator is studied. The results of [9] are stronger than the ones of [8] and the method used to obtain them is completely different; it is a “natural” generalization of the Deift-Killip approach to absolutely continuous spectrum of the one dimensional Schrödinger operator with square summable potential. This approach involves the use of trace formulae and is related to [10], which is also a generalization of results by Killip and Simon.

On the basis of the techniques used in [8], the paper [11] tackles the question of when the spectrum of the unbounded string operator fills the whole real line. This again makes use of a Harris-Lutz type transform and subordinacy theory with the additional innovative idea of arranging the product of transfer matrices by blocks of length determined by the particularities of the sequence \(\{a_{n}\}_{n=1}^{\infty }\). It turns out that the asymptotic behavior of the dynamical system (3) can be easier to analyze when considering its collective behavior over some periods. Later, a generalization of this method with blocks of variable length was used for studying the spectral properties of the so-called Mirzoev class of Jacobi operators.

In [12], an unbounded Jacobi operator J is seen as an additive perturbation of a string operator by a coupled diagonal operator. The weights of the string operator are growing and “modulated” by a periodic sequence \(\{c_{n}\}_{n=1}^{\infty }\). Thus, the resulting Jacobi operator has:

$$\displaystyle \begin{aligned} {} a_{n}:=c_{n}\mu_{n}\,,\quad b_{n}:=d\nu_{n}\,,\quad n\in\mathbb{N}\,, \end{aligned} $$
(5)

where \(\{\mu _n\}_{n=1}^{\infty }\) and \(\{\nu _{n}\}_{n=1}^{\infty }\) are certain sequences going to \(+\infty \). The growth of weights satisfies the Carleman condition so the operator is selfadjoint. Using the Stolz smoothness classes \(\mathcal {D}^{k,r}\) and \(\mathcal {D}^k\) and the generalized Behncke-Stolz lemma proved in [11], it is shown how the nature of the spectrum depends on the coupling constant d. Indeed, if d is in the absolutely continuous spectrum of the periodic operator \(J_{\mathrm {per}}\) with weights \(\{c_{n}\}_{n=1}^{\infty }\) and zero diagonal, then the spectrum of J is purely absolutely continuous, otherwise the spectrum is discrete. Here one has examples of multithreshold spectral phase transitions. Regarding the methodology, the generalized Behncke-Stolz lemma establishes that the absolute continuity of spectrum follows from the asymptotic estimate \(\|{\mathbf {u}}_n\|=O(1/\sqrt {a_n})\), \(n\to \infty \), for generalized eigenvectors. Speaking roughly, it is shown that in the case of dominating weights, the Behncke-Stolz sufficient condition holds, whereas in the case of dominance of the main diagonal the spectrum is discrete. This fact is proven by establishing a WKB asymptotics on the basis of a Levinson type theorem for discrete linear systems and then showing that the resolvent is compact.

A similar model to the one of [12] is studied in [13], where one has the same expression for the weights as in (5), but the potential is given by \(b_{n}:=d_{n}\nu _{n}\), where \(\{d_{n}\}_{n=1}^{\infty }\) is a periodic sequence. Milder conditions on the sequences \(\{\mu _n\}_{n=1}^{\infty }\) and \(\{\nu _{n}\}_{n=1}^{\infty }\) are considered. If M and N denote the least periods of sequences \(\{c_n\}_{n=1}^{\infty }\) and \(\{d_n\}_{n=1}^{\infty }\), then, in the space of parameters \((c,d)\in \mathbb R^{M+N}\), there are domains in which the spectrum of the operator is purely absolutely continuous and occupies the whole real line. The exterior of their union in the parameter space consists of domains in which the spectrum of the operator is discrete. Thus a spectral phase transition occurs at the boundaries of domains: the spectrum abruptly changes its structure. A separate article [14] is devoted to the question of semiboundedness for this class of operators. In [14], simple criteria for upper and lower semiboundedness of operator J are obtained. Whether J is upper or lower semibounded depends again on the spectrum of operator \(J_{\mathrm {per}}\).

Differential and difference equations have many similarities. One important analogy has to do with the asymptotic analysis of solutions to such equations. Indeed, every so often, the discrete linear system (3) allows one to use the so-called semiclassical analysis, i.e. it has a WKB asymptotics just as in the differential equations setting. To obtain asymptotic formulae for the behavior of solutions to (2) as \(n\to \infty \), one may recur to transformations of the system (3) reducing it to a form for which Levinson’s discrete asymptotic theorem is applicable in one of its many variants. In [12] and [13], it is possible to apply discrete Levinson type theorems to (3), although in [13], the grouping in blocks technique has to be used before applying the Levinson theorem. Another successful use of these techniques can be found in [15] which sets precedents for [16], later referred to on this account.

As a rule, the phase transition occurs in the so-called double root case of (3). This refers to the coincidence of roots of the characteristic equation or, from another point of view, the case of a multiple eigenvalue of the limit of transfer matrices (if this limit exists). This means that the limit matrix is similar to a Jordan block. In these cases, discrete Levinson type theorems cannot be used straightforwardly. Due to this, new methods were developed in [17,18,19,20,21] for finding the asymptotic behavior of generalized eigenvectors when the double root case takes place. The results resemble the ones obtained by the semiclassical method for ordinary differential equations; the behavior of solutions for large n is determined by lower terms in the expansion of the transfer matrix and can be “hyperbolic” or “elliptic”. Here one recurs to a chain of successive transformations of the dynamical system so that a Levinson type theorem is applicable to the transformed system. These transformations use analogies between difference and differential equations to find a correct Ansatz with the help of which one of the main steps of transforming the system to a simple form is carried out. This method is used in [17] for the matrix (1) having weights \(a_n=n+a\) and potential \(b_n=-2a_n\) for \(n\in \mathbb {N}\) (where \(a\in \mathbb R\) is a parameter), and as a result the asymptotic formula \(u_n^{\pm }\sim n^{-1/4}\exp (\pm 2\sqrt {(\lambda +1)n})\), \(n\to \infty \), was proved. The paper [19] deals with the case of weights \(a_n=n^{\alpha }(1+\mu _n)\) and main diagonal \(b_n=-2n^{\alpha }(1+\nu _n)\), where \(n^{\alpha /2}\mu _n,n^{\alpha /2}\nu _n\in l^1(\mathbb N)\) and \(\alpha \in (1/2,2/3)\). For the generalized eigenvectors, one has the following asymptotic formula

$$\displaystyle \begin{aligned} u_n^{\pm}\sim n^{-\alpha/4}\exp \left(\pm\left(\frac{\sqrt{\lambda}n^{1-\frac{\alpha}2}}{1-\frac{\alpha}2} +\frac{n^{\frac{\alpha}2}}{\sqrt{\lambda}}- \frac{\lambda^{\frac 32}n^{1-\frac{3\alpha}2}} {24\left(1-\frac{3\alpha}2\right)}\right)\right),\ n\to\infty. \end{aligned}$$

A more general case, i.e. \(\alpha \in (0,1)\), was studied later in [22] where the Kooman method was used to prove a formula expressing the spectral density of J in terms of some coefficients in the asymptotics of orthogonal polynomials (the Weyl-Titchmarsh formulae).

The papers [18] and [21] study the spectral properties of J with weights \(a_{n}=n^{\alpha }\) for \(n\in \mathbb {N}\) and periodically modulated diagonal entries so that \(b_n=bn^{\alpha }\) (\(b>0\)) for odd n and \(b_{n}=0\) for even n. This is yet another example of a critical situation. The difficulty with the non-smoothness of coefficients caused by the periodic modulation is overcome by grouping transfer matrices into pairs (cf. [11]), after which the analogy with a second-order differential equation and the WKB method becomes less obvious. For each case, the critical elliptic [18] and the critical hyperbolic [21], a progression of transformations is found so that the asymptotic analysis of the solutions to the system (3) is reduced to the asymptotic analysis á la Levinson.

In [20], another approach to finding the asymptotics of solutions to the recurrence relation (2) in the double root case was developed: the Kelly method. With its help, it is possible to obtain an asymptotic expansion which is locally uniform with respect to the spectral parameter for a matrix with the main diagonal \(b_n=n^{\alpha }\) and periodically modulated weights \(a_n=c_nn^{\alpha }\), where \(\{c_n\}_{n=1}^{\infty }\) is a sequence with period 2 and \(|c_1-c_2|=1\), \(c_1,c_2>0\). The generalized eigenvectors have asymptotics for \(\lambda >0\)

$$\displaystyle \begin{aligned} u_n^{\pm}\sim \exp\left(\pm\left( \sqrt{\frac{\lambda}{2c_1c_2}}\frac{n^{1-\frac{\alpha}2}}{1-\frac{\alpha}2}+O(n^{1-\alpha}) \right)\right),\ n\to\infty, \end{aligned}$$

locally uniform in \(\lambda \). From this result, it is deduced that the spectrum is discrete on the positive half-line.

S. Naboko also worked on the exact asymptotics of eigenvalues of Jacobi operators used for modeling phenomena in quantum and classical physics. This asymptotic analysis is relevant in physics to understand the role of physical parameters in the asymptotic expansions of large eigenvalues. A completely new method, somehow related to the transformation operator used in differential equations, is developed in [23] to obtain arbitrarily many terms in the asymptotic decomposition of large eigenvalues. This technique, which Naboko called the successive diagonalization technique, is quite general and is used in [24, 25] where for treating periodic modulations of a modified Jaynes-Cummings model the corresponding Jacobi matrix is seen as a block Jacobi matrix.

An inverse resonance problem for bounded Jacobi operators is solved in [26]. Strictly speaking, the operators considered in that paper constitute a wider class than the one of bounded Jacobi operators. Indeed, [26] studies the operator \(\mathcal J\) with \(\{a_{n}\}_{n=1}^{\infty }\) and \(\{b_{n}\}_{n=1}^{\infty }\) being bounded complex sequences such that \(a_{n}\ne 0\) and \(\operatorname {Re} a_{n}\ge 0\) for any \(n\in \mathbb {N}\). A similarity transformation is defined for the operator \(\mathcal J\) and it is shown that the Weyl m-function is the same for all operators obtained from one another by this similarity transformation. The main result of [26] establishes that the eigenvalues and resonances of \(\mathcal J\), under some complementary conditions on the sequences \(\{a_{n}\}_{n=1}^{\infty }\) and \(\{b_{n}\}_{n=1}^{\infty }\), determine uniquely these sequences. It is also shown that superexponentially decaying perturbations of the free discrete Schrödinger operator fall into the class of operators for which the main result holds. In [34], a similar resonance inverse problem is solved for unbounded Jacobi operators appearing in the context of quantum physics, namely, the sum of the creation and the annihilation operators perturbed by a compact potential. An inverse resonance problem focusing on stability and uniqueness is treated in [29].

S. Naboko was enthusiastic about constructing examples of selfadjoint Jacobi operators having predetermined spectral properties. Since any simple selfadjoint operator is unitarily equivalent to some Jacobi operator, any closed subset of \(\mathbb {R}\) can be realized as the spectrum of a Jacobi operator. However, finding out the weights and the potential sequences from the spectrum is not an easy task. The paper [29] gives examples of unbounded Jacobi operators having finitely many gaps in the essential spectrum. One should note that prior to [29], only few explicit examples were known of unbounded Jacobi operators having such spectral properties whereas typically the absolutely continuous spectrum of an unbounded Jacobi operator covers the whole line (cf. [12, 13]). The methodology in [29] is based on subordinacy theory and a grouping in blocks approach to (3) for the asymptotic analysis of generalized eigenvectors. Other examples of the same spectral structure are provided in [30] which is loosely motivated by a set of ideas by Last and Simon. A further development of the technique used in [30] can be found in [31]. The results in [31] can be seen as a generalization of Last and Simon approach to the essential spectrum of some operators.

In [32] and [33], the problem of embedded eigenvalues is considered in a setting similar to that of [6], but with a different approach. In contrast to [6], in which, starting from a given set of points inside the absolutely continuous spectrum of the unperturbed operator, a potential is constructed having a decay slightly slower than \(1/n\), for which these points are eigenvalues, the approach of [32, 33] is based upon a particular kind of potentials, namely, potentials of Wigner-von Neumann type, \(q_n:=c\sin {}(2\omega n+\delta )/n\), and sums of such terms. Each such term can produce eigenvalues at points \(\lambda \) such that \(\omega N\pm 2\theta (\lambda )\in 2\pi \mathbb Z\), where N is the least common period of the sequences \(\{a_n\}_{n=1}^{\infty }\) and \(\{b_n\}_{n=1}^{\infty }\) and \(\theta \) is the quasi-momentum corresponding to the periodic Jacobi matrix (which means that \(\exp (\pm i\theta (\lambda ))\) are eigenvalues of the monodromy matrix \(M_N(\lambda ):=B_N(\lambda )\cdots B_1(\lambda )\)). In the general case, critical points are not eigenvalues, but at each of them a subordinate generalized eigenvector exists. In [32], it is proven that an arbitrary point (with few exceptions) inside the absolutely continuous spectrum of the unperturbed operator can be made an eigenvalue by adding a potential of Wigner-von Neumann type to the sequence \(\{a_n\}_{n=1}^{\infty }\). In [33] it is shown that at countably many given points inside the absolutely continuous spectrum one can construct subordinate solutions by adding potential being an infinite sum of Wigner-von Neumann terms to \(\{a_n\}_{n=1}^{\infty }\). A similar technique was employed in [34] which deals with a differential Schrödinger operator on the half-line, but where the analysis is reduced to a discrete dynamical system resembling (3).

Another line of research of S. Naboko was obtaining estimates for the entries of the Green matrix of selfadjoint Jacobi operators in gaps of the essential spectrum and non-real values of the spectral parameter. This was done for scalar Jacobi matrices in [16] and [35] and for block Jacobi matrices in [36, 37] and [38]. These works use a refinement of the Combes-Thomas method and contain sharp decay estimates which are expressed in terms of the weights \(a_n\) and depend on the behavior of the sums \(\sum _{k=1}^n1/a_k\) or \(\sum _{k=1}^n1/\sqrt {a_k}\) (and \(\|A_k\|\) instead of \(a_k\) in the block matrix case, see below). The difference is determined by the finiteness of gaps: estimates of the first type hold for bounded gaps in the essential spectrum, while estimates of the second type hold for unbounded gaps. Moreover, in the case of an unbounded gap \((-\infty , d)\), it is additionally required that the operator is semibounded from below. In this case, the semiboundedness of J is crucial in this analysis as is illustrated by Naboko and Simonov [21] in which it is established for a nonsemibounded Jacobi operator that the asymptotic behavior of the generalized eigenvectors depends on the parameter b from the main diagonal and the estimate from [35] is violated for some values of \(b>0\). The results of [16] and [35] have applications in the study of random Jacobi operators and spectral phase transitions. The technique developed in [35] is used in [36, 37] to obtain decay estimates for the block entries of Green matrix in the case of selfadjoint block Jacobi operators when the blocks are bounded operators.

A block Jacobi operator \(\mathfrak J\) is defined by a sequence \(\{A_{n}\}_{n=1}^{\infty }\) of bounded and boundedly invertible operators in a separable Hilbert space \(\mathfrak {H}\) which usually is finite dimensional and a sequence \(\{B_{n}\}_{n=1}^{\infty }\) of bounded selfadjoint operators in \(\mathfrak {H}\). Thus the block Jacobi operator \(\mathfrak {J}\) is an operator in the space

$$\displaystyle \begin{aligned} l_{2}(\mathbb{N})=\bigoplus_{n=1}^\infty\mathfrak{H}_n\,, \end{aligned}$$

where for all \(n\in \mathbb {N}\)\(\mathfrak {H}_n=\mathfrak {H}\). Its matrix representation is

$$\displaystyle \begin{aligned} {} \begin{pmatrix} B_1 & A_1 & 0 & 0 & \cdots \\[1mm] A_1^{*} & B_2 & A_2 & 0 & \cdots \\[1mm] 0 & A_2^{*} & B_3 & A_3 & \\ 0 & 0 & A_3^{*} & B_4 & \ddots\\ \vdots & \vdots & & \ddots & \ddots \end{pmatrix}\,. \end{aligned} $$
(6)

For a block Jacobi operator \(\mathfrak J\), the deficiency indices satisfy \(0\le n_{+}(\mathfrak J), n_{-}(\mathfrak J) \le \dim \mathfrak H\). The transfer matrix \(\mathfrak {B}(z)\) of the dynamical system

$$\displaystyle \begin{aligned} A_{n-1}^*u_{n-1}+B_{n}u_{n}+A_{n}u_{n+1}=zu_{n}\,,\quad n>1\,, \end{aligned}$$

analogous to (3) is

$$\displaystyle \begin{aligned} \mathfrak{B}_n(z):= \begin{pmatrix} 0 & I \\ -A_{n}^{-1}A^{*}_{n-1} & A_{n}^{-1}(zI-B_{n}) \end{pmatrix}\,, \end{aligned}$$

where I is the identity in \(\mathfrak H\). Note that any block Jacobi operator with N-dimensional blocks can be seen as a block Jacobi operator of M-dimensional blocks where \(M=kN\), \(k\in \mathbb {N}\). This trick was actually used in [24, 25] for dealing with the modified Jaynes-Cummings model.

The greater complexity in the structure of block Jacobi operators means that the results of [36,37,38] can be used to obtain new examples of operators exhibiting spectral phase transitions. The broader class of block Jacobi operators attracted Naboko’s attention in his later works. Apart from [36,37,38], the papers [2, 39] also deal with block Jacobi operators. The point spectrum of periodic block Jacobi operators is studied in [39]. For the sequences of matrices \(\{A_{n}\}_{n=1}^{\infty }\) and \(\{B_{n}\}_{n=1}^{\infty }\) being N-periodic, define the monodromy matrix \(\mathfrak {M}(z):=\mathfrak {B}_{N}(z)\cdots \mathfrak {B}_{1}(z)\) and denote by \(\mathcal P_-(z)\) the Riesz projection corresponding to the eigenvalues of \(\mathfrak {M}(z)\) in the interior of the unit circle. If \(\lambda \) is in the point spectrum of \(\mathfrak {J}\) and \( \operatorname {\mathrm {rank}} \mathcal P_-(\lambda ) =1\), then \(\lambda \) is in the spectrum of the finite matrix

$$\displaystyle \begin{aligned} {} \widehat J:= \begin{pmatrix} B_1 & A_1 & 0 & \cdots & 0 \\ A_1^{*} & B_2 & A_{2} & \cdots & 0\\ 0 & A_{2}^{*}& B_{3}& \ddots & \vdots\\ \vdots & & \ddots & \ddots & A_{N-2} \\ 0 & \cdots & 0 & A_{N-2}^{*} & B_{N-1} \end{pmatrix}\,. \end{aligned} $$
(7)

It is also established in [39] that when the set of matrices \(\{A_{n},B_{n}\}_{n=1}^{N}\) is commutative, then all eigenvalues of \(\mathfrak J\) are also eigenvalues of \(\widehat J\) irrespectively of \( \operatorname {\mathrm {rank}} \mathcal P_-(\lambda )\). If \(\{A_{n},B_{n}\}_{n=1}^{N}\) is not commutative and \( \operatorname {\mathrm {rank}} \mathcal P_-(\lambda )>1\), then the eigenvalue \(\lambda \) of \(\mathfrak {J}\) is an eigenvalue of a modification of \(\widehat J\) which depends on the concrete value of \( \operatorname {\mathrm {rank}} \mathcal P_-(\lambda )\).

The work [2] is concerned with the conditions for discreteness of the spectrum of block Jacobi matrices in terms of its main diagonal entries. It turns out that sometimes the interplay between odd and even elements of the main diagonal sequence can be used to draw conclusions on discreteness independently of the weight sequence. Let \(n_{+}(\mathfrak {J}) = n_{-}(\mathfrak J)\). If, on the one hand, for some c and large n one has \(B_{2n}\le cI\) and, on the other, \(B_{2n-1}\to +\infty \) (in the sense that for every \(M>0\) there exists \(N\in \mathbb {N}\) such that for every \(n>N\)\(B_{2n-1}\ge MI\)), then any selfadjoint extension of \(\mathfrak J\) has discrete spectrum on the interval \((c,+\infty )\) accumulating only at \(+\infty \). This is directly applicable to the example considered in [18] and [21]. The result remains true if one permutes the roles of even and odd entries. The paper also contains an estimate of the counting function on subintervals of \((c,+\infty )\) as well as a refinement of the condition for discreteness of the spectrum on the positive half-line which involves the weight sequence. A simple condition of discreteness of the whole spectrum should also be mentioned: \(B_{2n-1}\to +\infty \) and \(B_{2n}\to -\infty \) as \(n\to \infty \) guarantees this. The results of [2] extend previous results in [12, 13, 17, 18].