Green matrix estimates of block Jacobi matrices II: Bounded gap in the essential spectrum

TL;DR

This paper derives decay bounds for Green matrices and eigenvectors of block Jacobi operators within spectral gaps, with refinements for commuting entries and illustrative examples.

Contribution

It introduces new decay estimates for block Jacobi matrices in spectral gaps, including cases with commuting entries and eigenvalues.

Findings

Decay bounds for Green matrices in spectral gaps

Refined estimates for commuting matrix entries

Examples illustrating the decay bounds

Abstract

This paper provides decay bounds for Green matrices and generalized eigenvectors of block Jacobi operators when the real part of the spectral parameter lies in a bounded gap of the operator's essential spectrum. The case of the spectral parameter being an eigenvalue is also considered. It is also shown that if the matrix entries commute, then the estimates can be refined. Finally, various examples of block Jacobi operators are given to illustrate the results.

Equations

$$\text{norm}{G_{mj}(\zeta )}_{B(\mathcal{K})}\le C_1\exp (-\gamma (\zeta )\underset{k=\min (m,j)}{\overset{\max (m,j)-1}{\text{scalebox}1.4\sum }}{\varphi }_{\delta }\left(\text{norm}{A_k}_{B(\mathcal{K})}\right))$$

$$\text{norm}{u_m}_{\mathcal{K}}\le C_2\exp (-\gamma (\zeta ){\text{scalebox}1.4\sum }_{k=1}^{m-1}{\varphi }_{\delta }\left(\text{norm}{A_k}_{B(\mathcal{K})}\right))$$

$$\text{norm}{\exp (\gamma (\zeta )\underset{k=\min (m,j)}{\overset{\max (m,j)-1}{\text{scalebox}1.4\sum }}{\varphi }_{\delta }(\text{absolutevalue}{A}_k))G_{mj}(\zeta )}_{B(\mathcal{K})}\le C$$

$$\mathcal{H}=\underset{m=1}{\overset{\infty }{⨁}}{\mathcal{K}}_m,$$

$$u=\{u_m{\}}_{m=1}^{\infty },u_m\in {\mathcal{K}}_m$$

$$\text{norm}{u}_{\mathcal{H}}^2=\sum _{m=1}^{\infty }\text{norm}{u_m}_{\mathcal{K}}^2.$$

$$\underset{m=1}{\overset{M}{⨁}}{\mathcal{K}}_m.$$

$$({\rm Y}u{)}_k$$

$$({\rm Y}u{)}_1$$

$$\mathrm{dom}(J):=\{u\in \mathcal{H}:{\rm Y}u\in \mathcal{H}\}$$

$$\left(\begin{array}{ccccc}{B}_1& A_1& 0& 0& \cdots \\ A_1^{\ast }& B_2& A_2& 0& \\ 0& A_2^{\ast }& B_3& A_3& \ddots \\ 0& 0& A_3^{\ast }& B_4& \ddots \\ ⋮& & \ddots & \ddots & \ddots \end{array}\right)$$

$$\mathrm{diag}\{C_m{\}}_{m=1}^{\infty }:=\left(\begin{array}{ccccc}{C}_1& 0& 0& 0& \cdots \\ 0& C_2& 0& 0& \\ 0& 0& C_3& 0& \ddots \\ 0& 0& 0& C_4& \ddots \\ ⋮& & \ddots & \ddots & \ddots \end{array}\right),$$

$$S\left(\begin{array}{c}{u}_1\\ u_2\\ u_3\\ ⋮\end{array}\right)=\left(\begin{array}{c}0\\ u_1\\ u_2\\ ⋮\end{array}\right)S^{\ast }\left(\begin{array}{c}{u}_1\\ u_2\\ u_3\\ ⋮\end{array}\right)=\left(\begin{array}{c}{u}_2\\ u_3\\ u_4\\ ⋮\end{array}\right).$$

$$\mathrm{diag}\{B_m{\}}_{m=1}^{\infty }+S\mathrm{diag}\{A_m^{\ast }{\}}_{m=1}^{\infty }+\mathrm{diag}\{A_m{\}}_{m=1}^{\infty }{S}^{\ast }$$

$$G_{jk}(\zeta ):=P_j(J-\zeta I{)}^{-1}{P}_k.$$

$$\text{norm}{G_{jk}(\zeta )}_{B(\mathcal{K})}=\text{norm}{G_{kj}(\overline{\zeta })}_{B(\mathcal{K})}.$$

$$\psi (x):=x^2{e}^x,\tilde{\psi }(x):=x{e}^x$$

$${\varphi }_{\delta }(x):=\{\begin{array}{ll}1/\delta & \text{ if }0\le x<\delta \\ 1/x& \text{ if }\delta \le x.\end{array}$$

$$w(x):=\sqrt{(x-r)(s-x)}>0.$$

$$d_{+}:$$

$$d_{-}:$$

$$\text{absolutevalue}\beta <\sqrt{d_{+}{d}_{-}},$$

$$\text{norm}{G_{mj}(\zeta )}_{B(\mathcal{K})}\le C\exp (-\gamma (\zeta ){\text{scalebox}1.4\sum }_{k=\min (m,j)}^{\max (m,j)-1}{\varphi }_{\delta }\left(\text{norm}{A_k}_{B(\mathcal{K})}\right)),$$

$$\gamma (\zeta ):=\min \left\{\delta {\psi }^{-1}\left(\frac{w^2(\Re \zeta )ϵ}{2\delta (s-r)}\right),\delta {\tilde{\psi }}^{-1}\left(\frac{w(\Re \zeta )(1-2ϵ)}{2\delta }\right)\right\}$$

$$\gamma (\zeta ):=\delta {\tilde{\psi }}^{-1}\left(\frac{w(\Re \zeta )ϵ(1-\eta )}{4\delta }\right)$$

$${\Phi }_m:=\{\begin{array}{ll}\exp \left(-\gamma \underset{k=1}{\overset{m-1}{\text{scalebox}1.2\sum }}{\varphi }_{\delta }\left(\text{norm}{A_k}_{B(\mathcal{K})}\right)\right)I,& m\le N,\\ \exp \left(-\gamma \underset{k=1}{\overset{N-1}{\text{scalebox}1.2\sum }}{\varphi }_{\delta }\left(\text{norm}{A_k}_{B(\mathcal{K})}\right)\right)I,& m>N,\end{array}$$

$$\Phi :=\mathrm{diag}\{{\Phi }_m{\}}_{m=1}^{\infty }.$$

$$F:={\Phi }^{-1}J\Phi -J.$$

$$F=S\mathrm{diag}\{{\Phi }_{m+1}^{-1}{A}_m^{\ast }{\Phi }_m-A_m^{\ast }\}+\mathrm{diag}\{{\Phi }_m^{-1}{A}_m{\Phi }_{m+1}-A_m\}{S}^{\ast }.$$

$${\Phi }^{-1}(J-\zeta I)\Phi =J+F-\zeta I.$$

$$X$$

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Green matrix estimates of block Jacobi matrices II:

Bounded gap in the essential spectrum 00footnotetext: Mathematics Subject Classification(2010): 41A10 47B36, 33E30 00footnotetext: Keywords: Block Jacobi operators; Generalized eigenvectors; Decay bounds.

Jan Janas

Institute of Mathematics

Polish Academy of Sciences (PAN)

Ul. Sw. Tomasza 30, 31-027

Krakow, Poland

najanas@cyf-kr.edu.pl

Sergey Naboko

Department of Mathematical Physics

Institute of Physics

St. Petersburg State University

Ulyanovskaya 1, St. Petersburg 198904, Russia

naboko@snoopy.phys.spbu.ru

Luis O. Silva

Department of Mathematical Sciences

University of Bath

Claverton Down, Bath BA2 7AY, U.K.

&

Departamento de Física Matemática

Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas

Universidad Nacional Autónoma de México

C.P. 04510, México D.F.

silva@iimas.unam.mx Supported by UNAM-DGAPA-PAPIIT IN110818 and SEP-CONACYT CB-2015 254062. On sabbatical leave from UNAM with the support of PASPA-DGAPA-UNAM

Abstract

This paper provides decay bounds for Green matrices and generalized eigenvectors of block Jacobi operators when the real part of the spectral parameter lies in a bounded gap of the operator’s essential spectrum. The case of the spectral parameter being an eigenvalue is also considered. It is also shown that if the matrix entries commute, then the estimates can be refined. Finally, various examples of block Jacobi operators are given to illustrate the results.

1 Introduction

This work gives decay bounds for the entries of the Green matrix corresponding to a self-adjoint block Jacobi operator when the real part of the spectral parameter lies in a bounded gap of the operator’s essential spectrum. These estimates also show how fast the sequence of generalized eigenvectors decays. In a previous paper [13], decay bounds for Green matrix entries were established for the case of semibounded block Jacobi operators. It turns out that, on the one hand, the technique used in [13] for unbounded gaps in the essential spectrum cannot be applied to the case of bounded gaps without substantial modifications. On the other hand, the estimates in the case of bounded gaps are completely different from the ones of unbounded gaps.

The results of this work provide a refinement of the Combes-Thomas method used for obtaining estimates of the Green function’s decay. A crucial fact of this refinement is the use of Lemma 3.1 which was already implemented in [14, 12, 21] and is in a certain sense behind the improvements of the original method (see [2]). Combes-Thomas type estimates (cf.[2, 6]) are used in the analysis of some discrete random models (see [16]). We remark that block type random operators have been studied earlier (e.g.[7, 8, 17]) and a Combes-Thomas type estimate for random block operators is given in [8, Lem. 5.7].

The decay bounds of the generalized eigenvectors established in this work have interesting applications to the study of spectral phase transition phenomena (see for instance [14]). Notably, block Jacobi matrices permit more freedom in the construction of models which exhibit multi-threshold spectral phase transitions.

In the Hilbert space $\mathcal{H}$ of square-summable sequences whose elements belong in turn to a Hilbert space $\mathcal{K}$, we consider the operator $J$ associated with a second order difference equation with operator coefficients acting in $\mathcal{K}$ (see Definition 1). The operator coefficients, $A_{n}$ and $B_{n}=B_{n}^{*}$ ($n\in\mathbb{N}$), are the entries of a block Jacobi matrix and, consequently, $J$ is called a block Jacobi operator. It is assumed that, for any $n\in\mathbb{N}$, $A_{n},B_{n}$ are bounded and defined on the whole space $\mathcal{K}$, *i.e.*they belong to $B(\mathcal{K})$. Additionally, the operators $A_{n},B_{n}$ ($n\in\mathbb{N}$) are required to satisfy conditions guaranteeing that $J=J^{*}$ on a proper domain and the existence of a gap in the essential spectrum of $J$. Under these assumptions, let us paraphrase the statements of Theorems 3.1 and 3.2. Denote by $G_{jk}(\zeta)$ the ($j,k$)-block entry of the resolvent’s matrix at the point $\zeta\in\mathbb{C}$ (Definition 2). If the real part of $\zeta$ is in the gap of the essential spectrum of $J$, then there are positive constants $C_{1}$ and $C_{2}$ such that

[TABLE]

when $\zeta$ is not an eigenvalue and

[TABLE]

when $\zeta$ is an eigenvalue. The function $\phi_{\delta}$ is given by (3.2) and corresponds to a regularization of the reciprocal function. The function $\gamma$ is given by (3.5) and (3.6), and the constants $C_{1}$ and $C_{2}$ do not depend on $m$ or $j$ (although they may depend on the spectral parameter $\zeta$). Apart from depending on $\zeta$ and the size of the gap, $\gamma$ depends on a parameter which permits certain optimization (Remark 2).

The decay bounds given above cover a wide range of block Jacobi operators. In these estimates, we provide an explicit expression for the decay coefficient, and establish how they depend on the off-diagonal entries of the block Jacobi operator. The fact that the bounds depend inversely on the growth of the off-diagonal entries of a Jacobi matrix was already shown by Combes-Thomas type estimates. The results of this work, on the one hand, generalize to the case of block matrix operators the corresponding decay bounds obtained in [14, 12] for the scalar case. On the other hand, our results extend the region of the spectral parameter so that it is a complex non necessarily real number.

When the entries of the block Jacobi operator commute, a subsequent refinement of the results given above can be obtained (see Section 4). Indeed, under the assumptions on $J$ considered above and the additional requirement that the entries commute pairwise (see in Theorem 4.1 the precise statement), one has

[TABLE]

for $\zeta$ being such that its real part is in the gap and it is not an eigenvalue. Here again $\gamma$ is given by (3.5) and (3.6), and the constant $C$ does not depend on $m$ or $j$. In this case of commuting entries, a result corresponding to the case when $\zeta$ is an eigenvalue is also established (Theorem 4.2). We remark that this operator way of estimating the decay of generalized eigenvectors make it possible to optimize the estimates of these vectors along any spatial direction in $\mathcal{K}$ by taking into account the matrix structure of the blocks.

Section 5 provides various examples which illustrate the results of this work. The asymptotic of generalized eigenvectors are obtained heuristically on the basis of methods which have been used to find the asymptotics of solutions to difference equations. We consider the examples of this section to be of independent interest.

Finally, in Section 6, we briefly describe an alternative approach to establishing decay bounds for block Jacobi operators. This approach makes use of the discrete nature of the problem and allows us to obtain more precise results in certain cases.

2 Preliminaries

2.1 Notation

By $\mathcal{H}$ we denote a separable infinite dimensional Hilbert space which is always decomposed as an infinite orthogonal sum:

[TABLE]

where $\mathcal{K}_{m}=\mathcal{K}$ for all $m\in\mathbb{N}$ and $\mathcal{K}$ is either an infinite or finite dimensional subspace of $\mathcal{H}$. A Hilbert space so decomposed is usually denoted by $l_{2}(\mathbb{N},\mathcal{K})$.

The symbol $\norm{\cdot}$ is used to denote the norm in $\mathcal{H}$, while the norm in $\mathcal{K}$ is denoted by $\norm{\cdot}_{\mathcal{K}}$. $B(\mathcal{H}$) and $B(\mathcal{K})$ denote the spaces of bounded linear operators defined on the whole space $\mathcal{H}$ and $\mathcal{K}$, respectively. The norms in $B(\mathcal{H})$ and $B(\mathcal{K})$ are denoted by $\norm{\cdot}_{B(\mathcal{H})}$ and $\norm{\cdot}_{B(\mathcal{K})}$, respectively.

A vector $u$ in $\mathcal{H}$ can be written as a sequence

[TABLE]

or as an infinite column vector, i.e.$u=(u_{1},u_{2},u_{3},\dots)^{\intercal}$. Note that

[TABLE]

Throughout this work, we use $I$ to denote the identity operator in the spaces $\mathcal{H}$ and $\mathcal{K}$ since it will cause no confusion to use the same letter for these operators. The orthogonal projector in $\mathcal{H}$ onto the subspace $\mathcal{K}_{m}$ is denoted by $P_{m}$ while the symbol $\widetilde{P}_{M}$ stands for the orthogonal projector onto

[TABLE]

Given a closed, densely defined operator $T$ in a Hilbert space, we denote by $\absolutevalue{T}$ the operator $(T^{*}T)^{1/2}$ given by applying the functional calculus to the self-adjoint operator $T^{*}T$. Finally, for any self-adjoint operator $A$, we denote its essential spectrum, *i.e.*the union of the continuous spectrum and the eigenvalues of infinite multiplicity, by $\sigma_{ess}(A)$.

2.2 Block Jacobi matrices

Let us turn to the definition of block Jacobi operators. For any sequence (2.1), consider the second order difference expressions

[TABLE]

where $B_{k}=B_{k}^{*}$ and $A_{k}$ are in $B(\mathcal{K})$ for any $k\in\mathbb{N}$.

Definition 1**.**

In $\mathcal{H}$, define the operator $J$ such that

[TABLE]

and $Ju=\Upsilon u$ for any $u\in\operatorname{dom}(J)$. Since $\operatorname{dom}(J)$ is dense in $\mathcal{H}$, the adjoint of $J$, $J^{*}$, is an operator. One verifies from the difference expression that $J^{*}\subset J$. The operator $J$ is said to be defined in the maximal domain given by the difference expression $\Upsilon$, whereas $J^{*}$ is the minimal closed operator associated with $\Upsilon$ (see [3, Chap. 7 Sec. 2.5]).

The block tridiagonal matrix

[TABLE]

can be regarded as the matrix representation of the operator $J^{*}$ when every element $u$ in $\mathcal{H}$ is written as in (2.1). (cf.[1, Sec. 47] where the matrix representation of an unbounded symmetric operator is given).

Remark 1**.**

In this work, we impose conditions on the sequences $\{A_{m}\}_{m=1}^{\infty}$ and $\{B_{m}\}_{m=1}^{\infty}$ so that $J$ is self-adjoint. A sufficient condition for this to happen is the generalized Carleman criterion [3, Chap. 7 Thm. 2.9], viz., if $\sum_{m=1}^{\infty}1/\norm{A_{m}}_{B(\mathcal{K})}=+\infty$, then $J$ is self-adjoint. A particular case of an operator satisfying the Carleman criterion is a bounded block Jacobi operator for which the sequences $\{\norm{A_{m}}_{B(\mathcal{K})}\}_{m=1}^{\infty}$ and $\{\norm{B_{m}}_{B(\mathcal{K})}\}_{m=1}^{\infty}$ are bounded [3, Chap. 7 Sec. 2.11].

Sometimes, the same notation is used for the matrix and the operator. This cannot lead to confusion when $J=J^{*}$ (in this case the maximal and minimal domains coincide) and in the case of diagonal matrices. Thus,

[TABLE]

where $C_{m}\in B(\mathcal{K})$ for any $m\in\mathbb{N}$, is used for denoting the operator and the matrix (the operator being $\bigoplus_{m=1}^{\infty}C_{m}$ with $C_{m}\in B(\mathcal{K})$ for all $m\in\mathbb{N}$).

In $\mathcal{H}$, consider the unilateral shift operator $S$ and its adjoint $S^{*}$ given by

[TABLE]

It can be verified that the operator

[TABLE]

coincides with $J$ when it is self-adjoint.

Definition 2**.**

Assume that the operator $J$ given in Definition 1 is self-adjoint. For any $\zeta$ in the resolvent set of $J$, define

[TABLE]

Note that by this definition $G_{jk}(\zeta)^{*}=G_{kj}(\overline{\zeta})$, and therefore

[TABLE]

The operator $G_{jk}(\zeta)$ can be regarded as the entry of the block matrix representation of the resolvent of $J$ at $\zeta$ corresponding to the $j$-th row and the $k$-th column. Accordingly, we refer to $\{G_{jk}(\zeta)\}_{j,k=1}^{\infty}$ as the block Green matrix corresponding to $J$ at $\zeta$.

3 Estimates in a bounded gap of the essential spectrum

This section establishes decay bounds for the Green matrix entries (Definition 2) corresponding to a self-adjoint Jacobi operator $J$ (Definition 1) with a bounded gap in the essential spectrum. The real part of the spectral parameter is assumed to be in this gap and we cover both cases; when this parameter is an eigenvalue and when it is not.

To streamline the writing of formulae, let us introduce the following functions. For $x>0$

[TABLE]

and

[TABLE]

Also, let $r,s\in\mathbb{R}$ be such that $r<s$. For $r<x<s$, define

[TABLE]

The next assertion is part of [14, Lem. 3.3] and turns out to be crucial for proving the results of this section. Earlier versions of the statement can be found in [21].

Lemma 3.1**.**

Let $T$ be an invertible self-adjoint operator such that

[TABLE]

If $V$ is a self-adjoint contraction and $\beta\in\mathbb{R}$ is such that

[TABLE]

then the operator $T+{\rm i}\beta V$ is continuously invertible.

One can find geometrical heuristics of this result in [14, 12]. It is worth mentioning that (3.4) cannot be replaced by the weaker condition $\absolutevalue{\beta}\leq\sqrt{d_{+}d_{-}}$ (see [12, Rem. 4]).

To streamline the statements of our results, consider the following hypothesis.

Hypothesis 1**.**

The operator $J$ given in Definition 1 is self-adjoint and there is an interval of the real line $(r,s)$ such that $(r,s)\cap\sigma_{ess}(J)=\emptyset$.

For the following assertion, bare in mind the functions given in (3.1), (3.2), and (3.3).

Theorem 3.1**.**

Assume that Hypothesis 1 holds true. Fix an arbitrary $\delta>0$, $\epsilon\in(0,1/2)$, and $\eta\in(0,1)$. If $\zeta\not\in\sigma(J)$ and $\real\zeta\in(r,s)$, then

[TABLE]

where

[TABLE]

when $\absolutevalue{\imaginary\zeta}\leq w(\real\zeta)\frac{\epsilon}{2}$, and

[TABLE]

otherwise. The constant $C$ depends neither on $m$ nor on $j$.

Proof.

For any fixed $N\in\mathbb{N}$, let

[TABLE]

where $\phi_{\delta}$ is given in (3.2) and the value of $\gamma$ is to be determined later. We assume $\Phi_{1}=I$. Clearly, $\Phi_{m}$ is a scalar operator for $m\in\mathbb{N}$. Consider the following operator in $\mathcal{H}$

[TABLE]

This operator is a contraction and depends on $\delta$, $\gamma$, and $N$. When needed, we indicate the dependence on $N$ explicitly, i. e., $\Phi=\Phi(N)$. Note that the operator $\Phi(N)$ is a boundedly invertible contraction for any finite $N$.

Let us introduce the operator:

[TABLE]

Using (2.4), one verifies that

[TABLE]

By (3.7), $F$ depends on $N$ and is in $B(\mathcal{H})$ since the sequences (2.1) in the range of $F$ have a finite number of nonzero elements. Note that (3.8) implies that

[TABLE]

Define

[TABLE]

where $E$ is the spectral measure of $J$. For $M\in\mathbb{N}$, we also introduce the following operators. Below, we take $M$ large enough (see (3.25), (3.28), (3.36)).

[TABLE]

where $\widetilde{P}_{M}$ is given in Section 2.1. Note that $Y$ is a compact operator and $\widetilde{J}$ depends on $N$ since $F$ also does.

Our aim is to show that $\widetilde{J}-\zeta I$ is continuously invertible for $\gamma$ indicated in the assertion of the theorem. Once the inverse of $\widetilde{J}-\zeta I$ exists and is bounded, the result is established by the next argumentation. Taking into account that $X+Y=J$ and (3.13), one obtains from (3.9) the equality

[TABLE]

Therefore, using the invertibility of $\widetilde{J}-\zeta I$ and the fact that $\zeta\not\in\sigma(J)$, one concludes that

[TABLE]

Left multiplying the last equality by $Y\widetilde{P}_{M}$ yields

[TABLE]

where we have used the identity $Q(I+Q)^{-1}=I-(I+Q)^{-1}$ formally valid for any linear operator $Q$. Inserting the expression for $\left[I+Y\widetilde{P}_{M}(\widetilde{J}-\zeta I)^{-1}\right]^{-1}$ from the last equality into (3.16), one arrives at

[TABLE]

In the right-hand side of the previous equality, the inverse of the operator in square brackets in (3.16) has been substituted by an explicit expression. The rigorous proof of the existence of this inverse follows by checking directly that the obtained expression is the left and right inverse simultaneously.

Clearly,

[TABLE]

Note that $\widetilde{C}$ does not depend on $N$. Thus,

[TABLE]

In view of Definition 2 and (3.7), by letting $N\to\infty$, one obtains

[TABLE]

for all $j,k\in\mathbb{N}$. The estimate of the theorem is proven by combining both scalar exponential factors in (3.18). Note that we have proven the estimate for $j\geq k$, but the other case is also covered by recurring to (2.5).

Let us turn to establishing the continuous invertibility of $\widetilde{J}-\zeta I$. We consider two mutually exclusive cases leading to the two different definitions of the function $\gamma$. If $\absolutevalue{\imaginary\zeta}\leq w(\real\zeta)\frac{\epsilon}{2}$, then one establishes the invertibility of $\widetilde{J}-\zeta I$ on the basis of Lemma 3.1 by choosing $\gamma$ appropriately. If $\absolutevalue{\imaginary\zeta}>w(\real\zeta)\frac{\epsilon}{2}$, then one can rely on the properties of dissipative (or anti-dissipative) operators to conclude that $\widetilde{J}-\zeta I$ is invertible for some values of $\gamma$.

First we assume that $\absolutevalue{\imaginary\zeta}\leq w(\real\zeta)\frac{\epsilon}{2}$. Observe that we have defined $T$ and $R$ so that

[TABLE]

Let us verify that the operators $T$ and $R$ satisfy the conditions of Lemma 3.1. To this end, we estimate the norms of the real and imaginary parts of $F$.

It follows from  (3.9) that

[TABLE]

Since the matrix of the operator $2\real F$ has only two block diagonals not necessarily zero and one diagonal is the adjoint of the other, one has

[TABLE]

which in view of (3.7) yields

[TABLE]

Using the elementary inequality

[TABLE]

which holds for any $x\geq 0$, one arrives at

[TABLE]

To obtain an appropriate estimate for $\norm{\real F}$, first note that, for any given $\delta,\epsilon>0$, if one defines

[TABLE]

then

[TABLE]

for all $\xi>0$. Indeed, it follows from (3.2) that, when $0<\xi\leq\delta$, the inequality (3.23) holds whenever

[TABLE]

Hence, (3.22) guarantees (3.23). In the case $\xi>\delta$, assuming (3.22), rewrite (3.23) using (3.2) as follows

[TABLE]

By (3.1), this inequality is equivalent to

[TABLE]

so, after dividing by $\gamma$, one verifies that it holds since $\xi>\delta$ and the function $te^{t}$ is monotone for $t>0$.

Comparing (3.21) to (3.23), one concludes that

[TABLE]

Since $Y$ is compact, one can take $M$ sufficiently large so that, simultaneously,

[TABLE]

Thus, taking into account that

[TABLE]

and combining (3.24) and (3.25), one obtains

[TABLE]

On the other hand, by (3.15), one has

[TABLE]

Make $M$ perhaps even larger than what was required in (3.25) so that

[TABLE]

To found a bound for the second term in the right-hand side of (3.27), one uses the argumentation for obtaining the inequality (3.19) from the expression (3.9). Thus,

[TABLE]

A straightforward computation yields

[TABLE]

from which, by means of the elementary inequality

[TABLE]

valid for any positive $x$, one obtains

[TABLE]

Proceeding as before, if one defines

[TABLE]

assuming $\delta>0$ and $\epsilon\in(0,1/2)$, then

[TABLE]

for all $\xi>0$. Comparing this inequality with (3.31), one concludes that

[TABLE]

Taking into account that $\absolutevalue{\imaginary\zeta}\leq w(\real\zeta)\frac{\epsilon}{2}$ and inserting (3.28) and (3.33) into (3.27), one arrives at

[TABLE]

where the second inequality is obtained from (3.26).

The estimates (3.24) and (3.25) show that $T$ is invertible. Clearly, $V:=R/\norm{R}$ is a self-adjoint contraction. Thus, since according to (3.34) $\beta:=\norm{R}$ satisfies the conditions of Lemma 3.1, one concludes that $\widetilde{J}-\zeta I$ is invertible.

Let us now consider the case

[TABLE]

As before, we first choose $M\in\mathbb{N}$ sufficiently large so that

[TABLE]

Now, if $\gamma$ is as in (3.6), then

[TABLE]

This is shown following the reasoning used to established that (3.23) holds under the assumption (3.22). Comparing (3.37) and (3.31), one obtains

[TABLE]

Since $\imaginary(\widetilde{J}-\zeta I)=R$, it follows from (3.15), (3.36) and (3.38) that

[TABLE]

Thus, if one considers the particular realization of (3.35) given by $\imaginary\zeta>w(\real\zeta)\frac{\epsilon}{2}$, then

[TABLE]

Thus the operator $\widetilde{J}-\zeta I+iw(\real\zeta)\frac{\epsilon\eta}{4}$ is anti-dissipative and therefore $\widetilde{J}-\zeta I$ is invertible. The case $\imaginary\zeta<-w(\real\zeta)\frac{\epsilon}{2}$ is treated analogously. ∎

Remark 2**.**

In the estimate given by Theorem 3.1, there are no constraints in the choice of the constant $\delta>0$. The function $\gamma(\zeta)$ yields a priori better estimates when $\delta\to\infty$. Note, however, that one does not obtain a more accurate estimate by letting $\delta\to\infty$ due to the dependence on $\delta$ of the function $\phi_{\delta}$.

Remark 3**.**

The hypothesis of Theorem 3.1 is fulfilled by a bounded from below self-adjoint operator $J$ with bounded from below essential spectrum. For these operators [13, Thm. 3.1] provides an estimate for the decay of Green matrix entries which is more precise than the one given by Theorem 3.1. Note that that boundedness from below is not required by Theorem 3.1. In [13, Thm. 3.1], this requirement was crucial for the proof. As shown in [20], the asymptotic behaviour of generalized eigenvectors of a non-bounded from below Jacobi operator can depend on the main diagonal in contrast to [13, Thm. 3.1] and Theorem 3.1. The results of [20] illustrate why the semiboundedness is essential in the case of [13, Thm. 3.1] and does not contradicts Theorem 3.1 due to the different order of estimates.

For the next theorem, we rely on [13, Lem. 3.1]. Here we reproduce the statement of the lemma for easy reference. Recall that $P_{1}$ is given in Section 2.1.

Lemma 3.2**.**

Let $J$ be the operator given in Definition 1 and $L$ be a compact operator in $\mathcal{K}$ with trivial kernel such that $\norm{L}_{B(\mathcal{K})}=1$. If $A_{m}$ has trivial kernel for all $m\in\mathbb{N}$ and $\zeta$ is in the discrete spectrum of $J$, then, for any $\tau>0$ sufficiently small, $\zeta$ is not in the spectrum of

[TABLE]

Theorem 3.2**.**

Assume that Hypothesis 1 holds true and that $\ker(A_{m})=\{0\}$ for any $m\in\mathbb{N}$. Fix an arbitrary $\delta>0$ and $\epsilon\in(0,1/2)$. If $\zeta\in\sigma_{p}(J)\cap(r,s)$ and $u$ is the corresponding eigenvector, then

[TABLE]

where $\gamma$ and $\phi_{\delta}$ are given by (3.5) and (3.2), respectively. The constant $C$ does not depend on $m$.

Proof.

By Lemma 3.2, one can choose $\tau>0$ such that $\zeta\not\in\sigma(J(\tau))$. Additionally, according to perturbation theory, $\sigma_{ess}(J)=\sigma_{ess}(J(\tau))$ (see [15, Chap. 4, Thm. 5.35 and Chap. 5, Thm. 4.11]). Therefore $J(\tau)$ satisfies the hypothesis of Theorem 3.1.

Now, if $u$ is a nonzero vector in $\ker(J-\zeta I)$, then

[TABLE]

Therefore

[TABLE]

which in turn implies

[TABLE]

where in the first inequality we use that $\norm{L}_{B(\mathcal{K})}=1$ and in the second we resort to Theorem 3.1. ∎

Corollary 3.1**.**

Assume that Hypothesis 1 holds and that $\norm{A_{k}}_{B(\mathcal{K})}\!\xrightarrow[k\to\infty]{}\infty$. Fix $\epsilon\in(0,1/2)$.

1. (a)

If $\zeta\not\in\sigma(J)$ and $\real\zeta\in(r,s)$, then

[TABLE]

where

[TABLE] 2. (b)

If $\ker(A_{n})$ is trivial for all $n\in\mathbb{N}$, $\zeta\in\sigma_{p}(J)\cap(r,s)$ and $u$ is the corresponding eigenvector, then

[TABLE]

The constant $C_{a}$ does not depend on $m$ and $j$, and $C_{b}$ does not depend on $m$.

Remark 4**.**

Perhaps the most relevant case in various theoretical applications corresponds to $\zeta$ being actually in the gap of the essential spectrum in item (a). This case admits further simplification. Indeed, it follows from (a) that if $\zeta\in(r,s)$, then, for arbitrarily small $\epsilon^{\prime}\in(0,1/2)$, one has

[TABLE]

Proof.

First note that the expressions inside the minimum in (3.5) monotonically grow as $\delta\to\infty$. Since the minimum has to be taken, one should consider only the expression that grows slower, namely,

[TABLE]

By choosing $\delta$ appropriately (essentially sufficiently large), one obtains

[TABLE]

Given $\epsilon$ and $\epsilon_{1}$, the choice of $\delta$ depends on $\zeta$.

If $0<t<\epsilon_{1}$, then

[TABLE]

where $\eta$ is arbitrarily small whenever $\epsilon_{1}$ is sufficiently small. Indeed, it follows from (3.1) that if $t=y\exp(y)$, then

[TABLE]

Thus, the assumption $t\leq\epsilon_{1}$ implies

[TABLE]

In view of (3.42), the choice of $\gamma$ in (3.5) can be replaced by

[TABLE]

Finally, observe that, for any $p\in\mathbb{N}$,

[TABLE]

where the second and third terms can be absorbed into a constant which does not depend on $p\gg 1$ since $\norm{A_{k}}_{B(\mathcal{K})}\xrightarrow[k\to\infty]{}\infty$. ∎

4 Estimates in the case of commuting entries

The results of the previous section admit a refinement when the hypotheses of Theorems 3.1 and 3.2 are complemented with the requirement that $A_{m},B_{m},A_{m}^{*}$ commute for $m\in\mathbb{N}$. This refinement permits to have different bound along different vectors in the space $\mathcal{K}$.

Hypothesis 2**.**

The system of operators $\{A_{m},B_{m},A^{*}_{m}\}_{m\in\mathbb{N}}$ given in section 2.2 commutes pairwise.

Theorem 4.1**.**

Assume that Hypotheses 1 and 2 hold true. Fix an arbitrary $\delta>0$, $\epsilon\in(0,1/2)$, and $\eta\in(0,1)$. If $\zeta\not\in\sigma(J)$ and $\real\zeta\in(r,s)$, then

[TABLE]

where $\gamma$ is given by (3.5) if $\absolutevalue{\imaginary(\zeta)}\leq w(\real\zeta)\frac{\epsilon}{2}$, and by (3.6) otherwise. The constant $C$ does not depend on $m$ and $j$.

Proof.

Consider the operators given in (3.11)–(3.15). Now, we modify the definition of the operator $\Phi$. For any fixed $N\in\mathbb{N}$, redefine the bounded operators on $\mathcal{K}$ given in the proof of Theorem 3.1:

[TABLE]

The bounded operator $\Phi$ on $\mathcal{H}$ is defined by

[TABLE]

Note that this operator differs from its counterpart of the proof of Theorem 3.1. Similar to what we had in the proof of Theorem 3.1, $\Phi$ depends on $N$ and $\Phi(N)$ is a boundedly invertible contraction for any finite $N$. Note that this time the block operator $\Phi_{m}$ is not a scalar operator.

Define the operator $F$ by (3.9) with the new sequence $\{\Phi_{m}\}_{m\in\mathbb{N}}$. Repeating the argumentation in the proof of Theorem 3.1, one arrives at (3.19). Using (4.1) and the fact that the system $\{A_{m},B_{m},A^{*}_{m}\}_{m\in\mathbb{N}}$ commutes pairwise, one obtains from (3.19) that

[TABLE]

Due to the inequality

[TABLE]

valid for any positive operator $Q$ and obtained from (3.20) by the spectral theorem, one derives from (4.2) the estimate

[TABLE]

Taking $\gamma$ as in (3.22), one concludes from (3.23) and the spectral theorem, that (3.24) holds.

Similarly, it follows from (3.30) that

[TABLE]

This inequality implies, by means of the operator inequality

[TABLE]

which holds for any positive operator due to the spectral theorem, that

[TABLE]

Thus, by choosing $\gamma$ as in (3.32), one verifies through the spectral theorem that (3.33) holds.

Since (3.24) and (3.33) take place, the operator $\widetilde{J}-\zeta I$ is invertible and therefore one has the estimate (3.17). Thus, in view of Definition 2 and (4.1), if $N\to\infty$, then

[TABLE]

for all $j,k\in\mathbb{N}$. The assertion of the theorem follows from this inequality by combining the operators on both sides of $G_{jk}$. In this proof, $j\geq k$, but the other case is also covered by recurring to (2.5). ∎

Theorem 4.2**.**

Assume that Hypotheses 1 and 2 hold true and that $\ker(A_{m})$ is trivial for any $m\in\mathbb{N}$. Fix an arbitrary $\delta>0$ and $\epsilon\in(0,1/2)$. If $\zeta\in\sigma_{p}(J)\cap(r,s)$ and $u$ is the corresponding eigenvector, then

[TABLE]

where $\gamma$ is given by (3.5). The constant $C$ does not depend on $m$.

Proof.

One follows the argumentation of the proof of Theorem 3.2. Resort to Lemma 3.2 and choose $\tau>0$ so that $\zeta\not\in\sigma(J(\tau))$. It follows from (3.39) that if $u$ is in $\ker(J-\zeta I)$, then (3.40) holds. Thus,

[TABLE]

One then obtains from this expression that

[TABLE]

For finishing the proof, it only remains to let $N\to\infty$ and note that $J(\tau)-\zeta I$ is continuously invertible. ∎

Corollary 4.1**.**

Fix $\epsilon\in(0,1/2)$. Assume that Hypotheses 1 and 2 hold true and $\norm{A_{m}^{-1}}_{B(\mathcal{K})}\mkern-1.0mu\xrightarrow[m\to\infty]{}0$ ($A_{m}$ invertible for $m\geq m_{0}\in\mathbb{N}$).

1. a)

If $\zeta\not\in\sigma(J)$ and $\real\zeta\in(r,s)$, then

[TABLE]

where $\widetilde{\gamma}(\zeta)$ is given by (3.41). 2. b)

If $\zeta\in\sigma_{p}(J)\cap(r,s)$ and $\ker(A_{m})$ is trivial for any $m\in\mathbb{N}$, then

[TABLE]

where $u$ is the corresponding eigenvector.

The constant $C_{a}$ does not depend on $m$ and $j$, and $C_{b}$ does not depend on $m$.

Proof.

We prove the claim in (a). The statement in (b) is proven analogously. First one uses the argumentation of Corollary 3.1 to simplify the expression of $\gamma(\zeta)$ when $\delta$ is large enough. Thus

[TABLE]

for all $j,k<N$. By letting $N\to\infty$, one obtains from this that

[TABLE]

Finally, using the argumentation in (3.43), one concludes that, for any $p\gg 1$,

[TABLE]

is a bounded operator whose norm is independent of $p\gg 1$. ∎

5 Examples

In this section, we consider concrete realizations of self-adjoint block Jacobi operators having finite gaps in the essential spectrum. The examples admit a straightforward calculation of the decay of the corresponding generalized eigenvectors although in some cases it is somehow involved. The estimates obtained in this way are compared with the ones given by Theorem 3.1.

Example 1. Let us first consider the block Jacobi operator given in Definition 1 so that, for any $n\in\mathbb{N}$, $B_{n}=0$ and $A_{n}=A_{n}^{(0)}$, where

[TABLE]

Here the sequence $\{\lambda_{n}\}_{n=1}^{\infty}$ of complex numbers is such that $\lambda_{n}\xrightarrow[n\to\infty]{}\infty$. We denote this block Jacobi operator by $J_{0}$. It can be decomposed as an infinite orthogonal sum of matrices:

[TABLE]

where the matrix $\boldsymbol{0}$ is a one-dimensional matrix. From this decomposition, one can deduce that $J_{0}$ is self-adjoint and $\sigma(J_{0})$ is discrete and the Green function is a band matrix function so that its entries decrease faster than any nonfinite sequence. If, instead of (5.1), one assumes $A_{n}=A_{n}^{(\epsilon)}$, where

[TABLE]

with $\epsilon_{n}>0$ for all $n\in\mathbb{N}$ and $\epsilon_{n}\xrightarrow[n\to\infty]{}0$, then the corresponding block Jacobi operator given by Definition 1 and denoted by $J$, is also self-adjoint and has discrete spectrum. (Note that $J$ and $J_{0}$ could be examples of self-adjoint block Jacobi operators not satisfying the Carleman criterion when the sequences $\{\lambda_{n}\}_{n=1}^{\infty}$ and $\{\epsilon_{n}\}_{n=1}^{\infty}$ are chosen appropriately). The Green function corresponding to $J$ is not anymore a band matrix. Nevertheless, the entries of the Green matrices also decay as fast as the sequence $\{\epsilon_{n}\}_{n=1}^{\infty}$ permits. This is shown by the following argument. By the Hilbert second resolvent identity, one has, for $\zeta\in\rho(J_{0})\cap\rho(J)$,

[TABLE]

where $J_{\epsilon}$ is the block Jacobi operator given by Definition 1 with $B_{n}=0$ and $A_{n}=\boldsymbol{\epsilon}_{n}$, where

[TABLE]

for any $n\in\mathbb{N}$. Note that $J_{\epsilon}$ is compact and, moreover, the rank of $J_{\epsilon}(J_{0}-\zeta I)^{-1}P_{j}$ is finite since all the nonzero entries of its matrix representation are in a finite vicinity of the blocks indexed by the value of $j$ (denoted $V(j)$). Therefore

[TABLE]

Taking into account that $\norm{A_{n}}=\absolutevalue{\lambda_{n}}(1+o(1))$ as $n\to\infty$, one obtains from Theorem 3.1 that

[TABLE]

where $\widetilde{\gamma}(\zeta)$ is given in (3.41). If $m$ or $j$ tend to $\infty$ and the corresponding series is divergent, then the Green matrix elements $G_{mj}(\zeta)$ could decrease faster than any power when one let either one of the indices $m,j$ grow and the other is kept fixed.

Note that the estimate in (5.2) does not contain any information on the sequence $\{\epsilon_{n}\}_{n=1}^{\infty}$. At the same time, the previous considerations show that the optimal estimates depend on $\max_{i\in V(j)}\{\epsilon_{i}\}$. Hence, for some choices of the sequence $\{\lambda_{n}\}_{n=1}^{\infty}$ the estimate in (5.2) may be close to optimal, however, in most cases, it is far from the real estimates. Example 1 illustrates a case when our results could not be optimal.

Example 2. Let

[TABLE]

Define the sequences of $2\times 2$ matrices $\{B_{n}\}_{n\in\mathbb{N}}$ and $\{A_{n}\}_{n\in\mathbb{N}}$ so that $B_{n}=0$ and $A_{n}=A$. Let $J$ be the operator in $l_{2}(\mathbb{N},\mathbb{C}^{2})$ given by Definition 1. Since the elements of the sequences $\{B_{n}\}_{n\in\mathbb{N}}$ and $\{A_{n}\}_{n\in\mathbb{N}}$ are all equal to constant matrices, the matrix (2.3) is periodic. Let us find conditions on $x$ for the operator $J$ to have a finite gap in the essential spectrum.

Proposition 5.1**.**

For the operator $J$ given in this example, $\sigma_{ess}(J)=[-2+x,2+x]\cup[-2-x,2-x]$. Thus, if $\absolutevalue{x}>2$, then the interval $(2-\absolutevalue{x},-2+\absolutevalue{x})$ is a gap in the essential spectrum of $J$.

Proof.

Denote by $\mathbb{T}$ the unit circle in the complex plane and by $\mu$ the Lebesgue measure on the unit circle normalized so that $\mu(\mathbb{T})=1$. We also consider the Hilbert space $l_{2}(\mathbb{Z},\mathbb{C}^{d})$ and denote any of its elements, *i.e.*the sequence $w=\{w_{k}\}_{k\in\mathbb{Z}}$ (($w_{k}\in\mathbb{C}^{d}$ for all $k\in\mathbb{Z}$) by an infinite column vector $(\dots,w_{-1},w_{0},w_{1},\dots)^{\intercal}$ (*cf.*Section 2.1).

Define in $l_{2}(\mathbb{Z},\mathbb{C}^{d})$, the map $V$ by

[TABLE]

where $z=e^{{\rm i}\theta}$, $\theta\in[0,2\pi)$. It is known that this is a unitary map from $l_{2}(\mathbb{Z},\mathbb{C}^{d})$ onto $L_{2}(\mathbb{T},\mathbb{C}^{d},\mu)$. Moreover (cf.[5, Chap. 2]) any “double-infinite” Jacobi operator $J_{\mathbb{Z}}$ with constant block entries acting in $l_{2}(\mathbb{Z},\mathbb{C}^{d})$ is transformed under this map into the multiplication operator by certain matrix function in $L_{2}(\mathbb{T},\mathbb{C}^{d},\mu)$, namely,

[TABLE]

Here the multiplication operator $M_{\phi}$ is defined by

[TABLE]

where $\phi$ is a $d\times d$-matrix function which will be determined some lines below.

Let $J_{\mathbb{Z}}$ be the operator corresponding to the matrix

[TABLE]

If $w$ is such that $w_{k}=0$ for all $k\neq j$, then

[TABLE]

Thus, $J_{\mathbb{Z}}$ is transformed by $V$ into the operator of multiplication $M_{\phi}$ with $\phi(z)=zA+\overline{z}A^{*}$.

For each $z\in\mathbb{T}$, the spectrum of $\phi(z)$ is

[TABLE]

According to [4, Chap. 8 Sec. 4], the spectrum of $M_{\phi}$ is given by the union of the essential range of the eigenvalues as functions of $z$. Therefore

[TABLE]

Now, if $K$ is given by the matrix

[TABLE]

then $J_{\mathbb{Z}}=J\oplus J_{\rm up}+K$, where $J_{\rm up}$ is the operator associated with the matrix

[TABLE]

in the subspace $l_{2}(\mathbb{Z}_{-},\mathbb{C}^{2})$ ($\mathbb{Z}_{-}=\{\dots,-1,0\}$) of the Hilbert space of $l_{2}(\mathbb{Z},\mathbb{C}^{2})$. Since $K$ is a finite rank operator, it follows from Weyl theorem (see [4, Thm. 3 Sec. 1 Chap. 9]) that

[TABLE]

Hence, if one shows that $\sigma_{ess}(J_{\mathbb{Z}})\subset\sigma_{ess}(J)$, then $\sigma_{ess}(J_{\mathbb{Z}})=\sigma_{ess}(J)$ and the assertion of the proposition follows.

The fact that $\lambda\in\sigma_{ess}(J_{\mathbb{Z}})$ is equivalent to the existence of a Weyl sequence (also known as singular sequence) at $\lambda$ [4, Thm. 2 Sec. 1 Chap. 9], *i.e.*there is $\{v(m)\}_{m=1}^{\infty}$ such that

1. (1)

$\inf_{m\in\mathbb{N}}\norm{v(m)}>0$ 2. (2)

$v(m)\xrightarrow[m\to\infty]{w}0$ 3. (3)

$(J_{\mathbb{Z}}-\lambda I)v(m)\xrightarrow[m\to\infty]{}0$ .

It follows from (1) that there is a $p\in\mathbb{Z}$ such that for all $m\in\mathbb{N}$,

[TABLE]

The sequence $\{v_{j+k}(m)\}_{j\in\mathbb{Z}}$ in $l_{2}(\mathbb{Z},\mathbb{C}^{2})$ satisfies (1)–(3) due to the fact that the $J_{\mathbb{Z}}$ have constant block coefficients. Therefore, if $P_{\mathbb{N}}$ is the projection in $l_{2}(\mathbb{Z},\mathbb{C}^{2})$ onto the subspace $l_{2}(\mathbb{N},\mathbb{C}^{2})$, then

[TABLE]

is a Weyl sequence at $\lambda$ for the operator $J$, which in turn means that $\lambda\in\sigma_{ess}(J)$. ∎

In view of Proposition 5.1, operator $J$ satisfies Hypothesis 1. Let us find the estimates given by Theorems 3.1 and 3.2 for this operator. To this end, one first computes the norm of the matrix $A$ by calculating the largest eigenvalue of $\absolutevalue{A}$. One has

[TABLE]

Inserting this expression into the formula given in Theorem 3.1, one arrives at the following estimate. If $\zeta$ is an eigenvalue and $u$ is the corresponding eigenvector, then

[TABLE]

where the function $\gamma$ is given by (3.5). If $\zeta$ is not an eigenvalue, then

[TABLE]

This result can be compared with the straightforward computation of the generalized eigenvectors by the so-called transfer matrices. By defining

[TABLE]

the recurrence equation $\Upsilon u=zu$ (see (2.2)) can be written as $\widetilde{u}_{n+1}=M_{n}\widetilde{u}_{n}$ for $n>1$, where

[TABLE]

Thus, estimates of the products of the transfer matrices yield decay estimates for generalized eigenvectors. Since in this case $A_{n}=A$ for any $n\in\mathbb{N}$, the matrix $M_{n}(\zeta)$ does not depend on $n$ and will be denoted by $M(\zeta)$. The eigenvalues of $M(\zeta)$ for a fixed $\zeta$ are the solutions with respect to $\mu$ of the equation

[TABLE]

Therefore the four eigenvalues of $M(\zeta)$ are

[TABLE]

By a straightforward computation, the minimal decay of an eigenvector in the gap $(2-\absolutevalue{x},-2+\absolutevalue{x})$, $\absolutevalue{x}>2$ is

[TABLE]

Now assume that $\zeta$ is placed near the edge of the gap given in Proposition 5.1, say $\zeta=2-\absolutevalue{x}+\epsilon$, ($0<\epsilon<<1$). In this case, the minimal decay of an eigenvector corresponding to this $\zeta$ is

[TABLE]

Thus, as $\epsilon\to 0$, the minimal decay of the corresponding eigenvector is

[TABLE]

Compare this with (5.5), where in this case $\gamma(\zeta)\simeq\sqrt{\epsilon}(2\absolutevalue{x}-4)$. Note that in both cases the coefficient determining the decay rate is determined by $\sqrt{\epsilon}$, *i.e.*by the square root of the distance to the edge of the gap in the essential spectrum when this distance is small.

Example 3. Let $J$ be the operator given in Definition 1 with $B_{n}=0$ for any $n\in\mathbb{N}$. Consider the constant matrix $A$ given in (5.3) and define

[TABLE]

where $\alpha\in(1/2,1)$, $c_{2n-1}=c_{1}$, and $c_{2n}=c_{2}$ ($c_{1},c_{2}\in\mathbb{R}$). For this example, it is assumed that $\absolutevalue{x}<2$. This assumption, as seen below, guarantees the existence of a bounded gap in the essential spectrum of $J$.

The block Jacobi operator $J$ exhibits a gap in the essential spectrum. This operator does not reduce to “scalar” Jacobi operators and its spectral analysis requires, as shown in [19], generalizing some of the techniques used for studying Jacobi operators.

Due to the 2-periodic character of the matrix weights, we first construct block of transfer matrices (see for instance [10], and [9] for the block version). Using the matrices given in (5.6), define the monodromy matrix

[TABLE]

for each $n\in\mathbb{N}$. Then

[TABLE]

where $\widetilde{u}_{n}$ is given by (5.7). Thus, the generalized eigenvectors of $J$ at the spectral parameter $\zeta$ are solutions to the discrete linear system (5.10) and, by the same token, the spectral properties of $J$ are determined by this system. We use here an approach to the analysis of (5.10) which has a heuristic component and refer the reader to [19] for the complete proof.

Substituting (5.6) into (5.9), one obtains

[TABLE]

Taking into account (5.8), one verifies by straightforward calculations that the monodromy matrix can be written as follows

[TABLE]

where $\Gamma_{n}$ is such that the sequence $\{\norm{\Gamma_{n}}\}_{n=1}^{\infty}$ is summable. In the asymptotic analysis of (5.10), the sequence $\{\Gamma_{n}\}_{n\in\mathbb{N}}$ is not relevant and the factor $1-\alpha/(2n)$ can be easily deal with at the end of the computation.

Fix arbitrary complex numbers $\omega$ and $\zeta$, and put $\epsilon:=(2n)^{-\alpha}$ with fixed $n\in\mathbb{N}$. Let us compute the determinant of the matrix $\mathcal{M}-\omega I$, where

[TABLE]

(compare this expression with (5.11)). To this end, consider the $2\times 2$ auxiliary matrix

[TABLE]

By using the Schur complement for computing $\det(\mathcal{M}-\omega I)$ (see [22]), one obtains

[TABLE]

where $\widetilde{\gamma}_{-}$ is the adjugate of the matrix $\gamma_{-}$. Substituting (5.3) into the last expression and performing all necessary elementary (though lengthy) computations, one arrives at

[TABLE]

where the scalar $C$ depends only on $x,c_{1},c_{2},\epsilon$, and its value, being a polynomial in $\epsilon$, has no effect in the remaining computations. Recall that $\det\gamma_{\pm}$ are polynomials of $\omega$ and $\epsilon$ in both variables of degree 2.

Let $\mu_{\pm}$ be the eigenvalues of

[TABLE]

Note that each eigenvalue $\mu_{\pm}$ is a multiplicity two eigenvalue of the $4\times 4$ matrix $\mathcal{M}$ when $\epsilon=0$. Now, on the basis of [15, Chap. 2 Sec. 2], since the algebraic and geometric multiplicities of the eigenvalues of $\mathcal{M}$ coincide, the following asymptotic ansatz

[TABLE]

can be substituted into (5.12) to find $\rho$ from the equation $\det(\mathcal{M}-\omega I)=0$ by equating coefficients of the powers of $\epsilon$. From the equation corresponding to $\epsilon^{2}$, one obtains

[TABLE]

Hence, depending on the value of $\zeta$, both $\rho_{\pm}$ are either real or pure imaginary. Note that, in the leading approximation, the system is always in the elliptic regime ($\absolutevalue{\mu_{\pm}}=1$) whereas, in the second approximation, which corresponds to the values of $\rho_{\pm}$, its character depends on the spectral parameter $\zeta$. Indeed, if $\zeta$ is in

[TABLE]

then the system is secondary hyperbolic ($\rho_{\pm}$ are real). If $\zeta$ is not in the closure of the interval given in (5.13), then the system is secondary elliptic ($\rho_{\pm}$ are purely imaginary). The fact that the character of the leading approximation is elliptic means that the asymptotic behaviour of the solutions, namely their growth or decay, is actually determined by the sub-leading coefficients of the approximation, i.e., the values of $\rho$. This is related to the divergence of the series defined by the sequence $\{1/(2n)^{\alpha}\}_{n\in\mathbb{N}}$ (recall that $1/2<\alpha<1$). Note that the two degenerated eigenvalues of $\mathcal{M}$ when $\epsilon=0$ split into two pairs of complex conjugate simple eigenvalues. In the secondary elliptic case, the splitting goes tangentially to the unit circle, while in the secondary hyperbolic case the splitting goes along the radius (perpendicular to the circle).

Now, one performs the asymptotic analysis of (5.10) à la Levinson (see for instance [11]). This allows one to conclude that the interval (5.13) is a gap in the essential spectrum of $J$. Indeed, the idea of the Levinson approach is to replace the nontrivial structure of the solutions to (5.10) by the product of the eigenvalues of $\mathcal{M}$ and then multiply by

[TABLE]

It is worth remarking that the formal application of the Levinson approach is the heuristic part of our analysis. The reason of this is that the product of eigenvalues of the monodromy matrix usually gives the correct main exponential term. However, the power in $n$ factor is not always correct.

The product of the eigenvalues of $\mathcal{M}$ yields (up to a constant factor) the following four expressions

[TABLE]

Thus the heuristic application of the Levinson approach allows one to conclude that there are two linearly independent solutions $u^{\pm}$ such that

[TABLE]

Since $\rho_{\pm}$ are purely imaginary when the spectral parameter $\zeta$ is outside the interval in (5.13), the exponential factor in the last formula has modulus 1. In this case, all the generalized eigenvectors decay as $n^{-\alpha/2}$ and the role of the perturbation $\epsilon$ is bounded by a purely oscillating factor. This behaviuor corresponds to the essential spectrum in the region given by (5.13). A rigorous proof of this fact can be obtained using Weyl sequences constructed on the basis of the asymptotic properties of (5.14) (see [14]). On the other hand, in the secondary hyperbolic case (i.e inside the interval given by (5.13)), $\rho_{\pm}$ are real with opposite signs which implies that one solution of (5.10) grows while the other decays. This case corresponds, at least at the physical level, to the absence of the essential spectrum (for a rigorous proof see the methods used in [18]).

A detailed asymptotic analysis of (5.10) and the proof of the above assertions concerning the spectral properties of $J$ in this example are done in [19] in which a further development of the techniques used in [20] is carried out.

The estimates for the decay of generalized eigenvalues inside the gap are thus given by

[TABLE]

In particular, this implies that

[TABLE]

for some positive constant $C_{0}$.

Let us calculate the bound of the decay of the generalized eigenvectors given by Theorem 3.1 for this case. To this end, we first compute the norm of $A_{n}$. We almost have already done this, since we have (5.4) and therefore

[TABLE]

Also the expression of $\gamma(\zeta)$ can be simplified according to Corollary 3.1. Assuming that $\zeta$ is in the gap (5.13), one thus has

[TABLE]

Hence, for a generalized eigenvector $u$, there are constants $C,\,\widetilde{C}>0$ such that

[TABLE]

Compare this inequality with (5.15). When $x\to 0$ (recall that $\absolutevalue{x}<2$), the expression in the square brackets in (5.16) goes to $1$ and therefore the arguments of the exponential function in (5.16) and (5.15) exactly coincide up to $\epsilon^{\prime}$ which is arbitrarily small.

6 Discrete version of the method

In this section, a different realization of the method used in the previous sections is presented. Here we use an alternative expression for the auxiliary operators (3.7). The result of this change is an estimate that could be more precise in various cases, in particular when the sequence of operators $\{A_{k}\}_{k=1}^{\infty}$ has multiple occurrences of $\norm{A_{k}}\ll 1$.

Redefine the sequence of operators $\{\Phi_{m}\}_{m=1}^{\infty}$ (see the proof of Theorem 3.1) as follows:

[TABLE]

where $\gamma>0$. As in the proof of Theorem 3.1, one establishes the inequalities (3.19) and (3.29). From these inequalities, one obtains

[TABLE]

If $\norm{A_{m}}\xrightarrow[m\to\infty]{}+\infty$, then $\gamma/\norm{A_{m}}$ is arbitrarily small for $m$ sufficiently large. Let us assume, without loss of generality, that $\gamma/\norm{A_{m}}<1$ for all $m\in\mathbb{N}$.

The sequences from the right-hand sides of the inequalities have the form

[TABLE]

The rational functions $\psi_{d}$ and $\widetilde{\psi}_{d}$ replace the transcendental functions $\psi$ and $\widetilde{\psi}$ of the proof of Theorem 3.1. By a reasoning similar to the one used in the proof of Theorem 3.1 to obtain the estimates of the norm of the real and imaginary part of the operator $F$, one concludes that assigning

[TABLE]

one obtains the necessary estimates guaranteeing that the existing of a constant $C=C(\zeta,J,\epsilon)>0$ such that

[TABLE]

for $N$ sufficiently large. Thus we have given the sketch of the proof of the following result.

Theorem 6.1**.**

Assume that Hypothesis 1 holds true and $\norm{A_{m}}\to\infty$ as $m\to\infty$. Fix an arbitrary $\delta>0$, $\epsilon\in(0,1/2)$, and $\eta\in(0,1)$. If $\zeta\not\in\sigma(J)$ with $\real\zeta\in(r,s)$ and $n_{0}$ is so large that $\gamma/\norm{A_{k}}<1$ for $k\geq n_{0}$, then

[TABLE]

where

[TABLE]

when $\absolutevalue{\imaginary\zeta}\leq w(\real\zeta)\frac{\epsilon}{2}$, and

[TABLE]

otherwise. The constant $C$ depends neither on $m$ nor on $j$.

Remark 5**.**

Note that the function $\psi$ given in (3.1) satisfies

[TABLE]

while

[TABLE]

Therefore the first two terms of the expansion of $\psi$ and $\psi_{d}$ coincide as $x\to 0$. On the other hand, $\widetilde{\psi}$ (see (3.1)) obeys

[TABLE]

and

[TABLE]

These asymptotic expansions show that there is an advantage in using $\widetilde{\psi}_{d}$ instead of $\widetilde{\psi}$. It is possible to obtain better estimates of the Green matrix entries by using the discrete version of the method (see [12, Thm. 5]). Although the continuous version of the method, which was presented in the previous sections, gives a more convenient form for the estimates of the Green matrix entries, the discrete version introduced in this section yields a slightly better and more subtle form of the estimates. In general, the discrete version of the method is more natural since the problem has itself a discrete character. Moreover, even in the case of noncommuting entries of the block Jacobi matrix, the operator $\Phi_{m}$ can be chosen as in (6.1), but with $\absolutevalue{A_{k}}$ instead of $\norm{A_{k}}$, $k\in\mathbb{N}$. Of course in this case the products should be taken in chronological order. The exponential form of $\Phi_{m}$ in the proof of Theorem 3.1 has no sense in that case. If we want to obtain a more precise estimate we should choose this discrete version with nonscalar operators. It seems, however, that the continuous version of the method provides estimates accurate enough for most of the applications.

Acknowledgements

The authors thank the anonymous reviewer for pertinent and useful comments and remarks which led to an improved version of the manuscript.

S N was supported by grant RFBR 19-01-00657 A (Sections 1–4) and grant RScF 20-11-20032 (Sections 5–6). He expresses his gratitute to the Institut Mittag-Leffler, where part of this work has been done, for their kind hospitality and to the Knut and Alice Wallenberg Foundation for the support given. L O S has been supported by UNAM-DGAPA-PAPIIT IN110818 and SEP-CONACYT CB-2015 254062. Part of this work was carried out while L O S was on sabbatical leave at the University of Bath from UNAM with the support of PASPA-DGAPA-UNAM.