On the relationship between Weyl functions of Jacobi matrices and response vectors for special dynamical systems with discrete time

TL;DR

This paper establishes a novel connection between Weyl functions of Jacobi matrices and response vectors of specific discrete-time dynamical systems, providing new insights into their spectral properties.

Contribution

It introduces a special representation for Weyl functions of finite and semi-infinite Jacobi matrices based on their relationship with dynamical systems.

Findings

Derived a new representation for Weyl functions.

Linked spectral problems with initial-boundary value problems.

Applicable to bounded Jacobi matrices.

Abstract

We derive special representation for Weyl functions for finite and semi-infinite Jacobi matrices with bounded entries based on a relationship between spectral problem for Jacobi matrices and initial-boundary value problem for auxiliary dynamical systems with the discrete time for Jacobi matrices.

Equations

$$(Hu{)}_n$$

$$(Hu{)}_1$$

$$H^N=\left(\begin{array}{cccccc}{b}_1& a_1& 0& \cdots & 0& 0\\ a_1& b_2& a_2& \cdots & 0& 0\\ 0& a_2& b_3& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & b_{N-1}& a_{N-1}\\ 0& 0& 0& \cdots & a_{N-1}& b_N\end{array}\right).$$

$$m(\lambda )=\left((H-\lambda {)}^{-1}{e}_1,e_1\right),$$

$$m^N(\lambda )=\left((H^N-\lambda {)}^{-1}{e}_1,e_1\right),$$

$$m(\lambda )={\int }_{\mathbb{R}}\frac{d\rho (z)}{z-\lambda },$$

$$\left\{\begin{array}[]{l}u_{n,t+1}+u_{n,t-1}-a_{n}u_{n+1,t}-a_{n-1}u_{n-1,t}-b_{n}u_{n,t}=0,\quad n,t\in\mathbb{N},\\ u_{n,-1}=u_{n,0}=0,\quad n\in\mathbb{N},\\ u_{0,t}=f_{t},\quad t\in\mathbb{N}_{0},\end{array}\right.$$

$${\left(Rf\right)}_t:=u_{1,t}^f,t=1,2,\dots .$$

$$\left\{\begin{array}[]{l}v_{n,t+1}+v_{n,t-1}-a_{n}v_{n+1,t}-a_{n-1}v_{n-1,t}-b_{n}v_{n,t}=0,\,t\in\mathbb{N},n\in 1\ldots N,\\ v_{n,-1}=v_{n,0}=0,\quad n=1,2,\ldots,N+1,\\ v_{0,t}=g_{t},\quad v_{N+1,t}=0,\quad t\in\mathbb{N}_{0},\end{array}\right.$$

$${\left(R_{N}f\right)}_t:=v_{1,t}^g,t=1,2,\dots$$

$$m(\lambda )=-\sum _{t=0}^{\infty }{z}^t{r}_t,$$

$$m^N(\lambda )=-\sum _{t=0}^{\infty }{z}^t{r}_t^N$$

$$\lambda =z+\frac{1}{z},z=\frac{\lambda -\sqrt{{\lambda }^2-4}}{2}=\frac{\lambda -i\sqrt{4-{\lambda }^2}}{2}.$$

$$\displaystyle D:=\left\{x+iy\,\Big{|}\,x>\left(R+\frac{1}{R}\right)\cos{\phi},\right.$$

$$y>0,y>\left(\frac{1}{R}-R\right)\sin \varphi ,\varphi \in (\pi ,2\pi )\}.$$

$$u^{f}_{n,t}=\left\{\begin{array}[]{l}\prod\limits_{k=0}^{n-1}a_{k}f_{t-n}+\sum\limits_{s=n}^{t-1}w_{n,s}f_{t-s-1},\quad n\leqslant t,\,\,n,t\in\mathbb{N},\\ 0,\quad n>t,\,\,n,t\in\mathbb{N},\end{array}\right.$$

$$\left\{\begin{array}[]{l}w_{n,s+1}+w_{n,s-1}-a_{n}w_{n+1,s}-a_{n-1}w_{n-1,s}-b_{n}w_{n,s}=\\ =-\delta_{s,n}(1-a_{n}^{2})\prod\limits_{k=0}^{n-1}a_{k},\,n,s\in\mathbb{N},\,\,s>n,\\ w_{n,n}-b_{n}\prod\limits_{k=0}^{n-1}a_{k}-a_{n-1}w_{n-1,n-1}=0,\quad n\in\mathbb{N},\\ w_{0,t}=0,\quad t\in\mathbb{N}_{0}.\end{array}\right.$$

$$c_t=\sum _{s=0}^t{f}_s{g}_{t-s},t\in {\mathbb{N}}_0.$$

$${\left(Rf\right)}_t=u_{1,t}^f=f_{t-1}+\sum _{s=1}^{t-1}{w}_{1,s}{f}_{t-1-s}t=1,2,\dots ,$$

$$\left(R^{T}f\right)=r\ast f_{\cdot -1},$$

$$u_{n,t}^f=u_{n,\cdot }^{\delta }\ast f_{\cdot -1},$$

$$r_{t-1}={\left(R\delta \right)}_t=u_{1,t}^{\delta }t\in \mathbb{N}.$$

$$v_{n,t}^g=v_{n,\cdot }^{\delta }\ast g_{\cdot -1},$$

$$r_{t-1}^N={\left(R_N\delta \right)}_t=v_{1,t}^{\delta },t\in \mathbb{N}.$$

$$a_{n-1}{\psi }_{n-1}+b_n{\psi }_n+a_n{\psi }_{n+1}=\lambda {\psi }_n,n⩾2,$$

$$P_1(\lambda )=1,P_2(\lambda )=\frac{\lambda -b_1}{a_1},Q_1(\lambda )=0,Q_2(\lambda )=\frac{1}{a_1}.$$

$${\Phi }_N^{+}(\lambda )=1,{\Phi }_{N+1}^{+}(\lambda )=0.$$

$${\Phi }_0^{+}(\lambda )=1,\sum _{n=0}^{\infty }|{\Phi }_n^{+}(\lambda )|<\infty .$$

$$Q_n(\lambda )+m(\lambda )P_n(\lambda )=-{\Phi }_n^{+}(\lambda )=-\frac{{\Phi }_n^{+}(\lambda )}{{\Phi }_0^{+}(\lambda )}.$$

$$m^N(\lambda )=-\frac{Q_{N+1}(\lambda )}{P_{N+1}(\lambda )}=-\frac{{\Phi }_1^{+}(\lambda )}{{\Phi }_0^{+}(\lambda )},$$

$$m(\lambda )=-{\Phi }_1^{+}(\lambda ).$$

$${\psi }_{n-1}+{\psi }_{n+1}=\lambda {\psi }_n,$$

$$P_0=0,P_1=1,Q_0=-1,Q_1=0.$$

Full text

1. Introduction.

Given a sequence of real numbers $(b_{1},b_{2},\ldots)$ and a sequence of positive numbers $(a_{1},a_{2},\ldots)$ such that $\sup_{n\geqslant 1}\{a_{n},|b_{n}|\}<\infty$, we consider an operator $H$ corresponding to a semi-infinite Jacobi matrix, defined on $l^{2}(\mathbb{N})\ni\psi=(\psi_{1},\psi_{2},\ldots)$, given by

[TABLE]

For a fixed $N\in\mathbb{N}$ we consider a finite Jacobi matrix:

[TABLE]

The Weyl $m-$functions for $H$ and $H^{N}$ are defined by

[TABLE]

$e_{1}=(1,0,0,\ldots)^{T}$. By the spectral theorem, this definition is equivalent to the following:

[TABLE]

where $d\rho=d\left(E_{H}e_{1},e_{1}\right)$ and $E_{H}$ is the (projection-valued) spectral measure of the self-adjoint operator $H$. The latter makes Weyl function an important object in spectral and inverse spectral theory of discrete and continuous one-dimensional systems [5, 13, 21, 20, 4].

Let us denote $\mathbb{N}_{0}:=\{0\}\cup\mathbb{N}$. For the same sequences $(b_{1},b_{2},\ldots)$, $(a_{1},a_{2},\ldots)$, and an additional parameter $a_{0}$, which for convenience we set equal to one, we consider two dynamical systems with the discrete time. The first system corresponds to semi-infinite case:

[TABLE]

and is a natural discrete analog of an initial-boundary value problem for the wave equation on a half-line. The dynamic inverse problem for such systems was considered in [3, 10]. In (1.1) the real sequence $f=(f_{0},f_{1},\ldots)$ is interpreted as a boundary control. We denote the solution to (1.1) by $u^{f}_{n,t}$. With (1.1) we associate the response operator $R:\mathbb{R}^{\infty}(\mathbb{N}_{0})\mapsto\mathbb{R}^{\infty}(\mathbb{N})$, acting by the rule:

[TABLE]

The operator $R$ is a discrete analog of a dynamic Dirichlet-to-Neumann operator, which is a classical object in the theory of inverse dynamic problems [7, 9, 14, 12].

The second system which corresponds to a finite case,

[TABLE]

is a discrete analog of an initial-boundary value problem for dynamical systems governed by the wave equation on a finite interval. The solution to (1.3) is denoted by $v^{g}_{n,t}$. The real sequence $g=(g_{0},g_{1},\ldots)$ is a boundary control. With (1.3) we associate the response operators $R_{N}:\mathbb{R}^{\infty}(\mathbb{N}_{0})\mapsto\mathbb{R}^{\infty}(\mathbb{N})$ acting by the rule:

[TABLE]

It was shown in [15, 16] that operators $R,\,R^{N}$ are determined by their kernels, so-called response vectors: $\left(r_{0},r_{1},r_{2},\ldots\right)$, $\left(r_{0}^{N},r_{1}^{n},r_{2}^{N},\ldots\right)$. We will establish the relationship between Weyl functions $m(\lambda)$, $m^{N}(\lambda)$ and response vectors. Such a relationship gives a new way of calculation of Weyl functions, and at the same time it is important in the inverse spectral theory: see [20, 4, 3, 17, 18] for the case of the Schrödinger operator on a half-line. The main result of the paper is the following.

Theorem 1**.**

If the coefficients in the semi-infinite $H$ or finite $H^{N}$ Jacobi matrix operators satisfy $\sup_{n\geqslant 1}\{a_{n},|b_{n}|\}\leqslant B$, then the Weyl functions $m,$ $m^{N}$ admit the representations

[TABLE]

in terms of response vectors (kernels of dynamic response operators associated to (1.1), (1.3)): $\left(r_{0},r_{1},r_{2},\ldots\right)$, $\left(r_{0}^{N},r_{1}^{n},r_{2}^{N},\ldots\right)$, where the variables $\lambda$ and $z$ are related by

[TABLE]

These representations hold for $\lambda\in D$, where $D\subset\mathbb{C}_{+}$ is defined as follows. Let $R:=3B+1$, then

[TABLE]

Our approach to Weyl function was stimulated by works on the Boundary Control method [7, 8, 9] for dynamic inverse problems. For the first time the relationship between Weyl functions and auxiliary dynamical systems was established in [4]. The inverse problem of recovering a Jacobi matrix from the response operator (i.e., from the response vector) was solved in [15, 16] by the Boundary Control method. The inverse problem of recovering a Jacobi matrix from the Weyl function was studied in [13]. Now, using Theorem 1 one can solve the second inverse problem by the following procedure: first one needs to recover the response vector from $m(\lambda)$ or $m^{N}(\lambda)$, which can be done by observing that in the variable $z$ formulas (1.5), (1.6) are Taylor expansions at zero, and then to use the dynamic method described in [16] to recover $a_{i},\,b_{i}$, $i\geqslant 1$. We note that for the first time the Boundary Control method was applied to discrete dynamical systems in [1, 2] in connection with spectral estimation problem.

The paper is organized as follows. In the second section we provide all necessary information on dynamical systems with discrete time corresponding to Jacobi matrices. In the third section we give an (equivalent) definition of the Weyl function which is more relevant to our approach, introduce the “Fourier transform” associated with the discrete Schrödinger operator with zero potential, derive representation formulas for Weyl functions $m^{N}$, $m$, and prove Theorem 1.

2. Dynamical systems with discrete time corresponding to finite and semi-infinite Jacobi matrices.

In this section we provide some results obtained in [15, 16]. We denote by $\mathcal{F}^{\infty}$ the outer space of the system (1.1), the space of controls: $\mathcal{F}^{\infty}:=\mathbb{R}^{\infty}$, with $f=(f_{0},f_{1},f_{2},\ldots)\in\mathcal{F}^{\infty}$.

Lemma 1**.**

The solution to (1.1) admits the representation

[TABLE]

where $w_{n,s}$ is the solution of the following Goursat problem

[TABLE]

This representation formula is an analog of the Duhamel formula for the wave equation with potential on a half-line [3].

Definition 1**.**

For $f,g\in\mathcal{F}^{\infty}$ we define the convolution $c=f*g\in\mathcal{F}^{\infty}$ by the formula

[TABLE]

In accordance with (2.1) and the definition of the response operator (1.2) (we assume here that $a_{0}=1$):

[TABLE]

where $r=(r_{0},r_{1},\ldots,r_{T-1})=(1,w_{1,1},w_{1,2},\ldots)$, is the response vector, the convolution kernel of the response operator. Note that choosing the special control $f=\delta=(1,0,0,\ldots)$, we can write the solution (2.1) as

[TABLE]

and the response vector can be determined as

[TABLE]

Similarly, for the system (1.3) the solution $v^{g}$ admits the representation in terms of convolution with a special solution corresponding to control $\delta$:

[TABLE]

and the response vector is given by

[TABLE]

Remark 1**.**

Formulas (2.4), (2.6) are discrete analog of classical Duhamel representation formula. They can be checked by the direct calculations (see also [15, 16]).

3. Weyl function representation

In this section we provide necessary information about the Weyl function and orthogonal polynomials associated with the Jacobi matrix, discrete Fourier transform related to the unperturbed discrete Schrödinger operator that we use for the variable $t$ of systems (1.1) and (1.3). Then we proceed to prove Theorem 1.

3.1. Additional details about the Weyl function

Consider two solutions $P,$ $Q$ of the equation

[TABLE]

satisfying the initial data

[TABLE]

Thus $P_{n}(\lambda)$ is a polynomial of degree $n-1$, and $Q_{n}(\lambda)$ is a polynomial of degree $n-2$.

In the case of finite Jacobi matrix $H^{N}$ we introduce $\Phi^{+}(\lambda)$ to be a solution to (3.1) fixed by conditions “at the right end”:

[TABLE]

In the case of semi-infinite $H$, in order to introduce the solution $\Phi^{+}(\lambda)$ for $\lambda\notin\mathbb{R}$, we extend the equation (3.1) to $n=1$ using the same additional parameter $a_{0}=1$ as before, and define

[TABLE]

Since entries of the matrix are bounded, it is in the limit point case [5, Chapter 1], and hence there exists only one such solution for $\lambda\notin\mathbb{R}$.

The solution $\Phi^{+}$ is expressed in terms of $P$ and $Q$ as (cf. [13], [11, Chapter 7], [5, Chapters 1,2])

[TABLE]

Then for $N<\infty$

[TABLE]

and for $N=\infty$

[TABLE]

Since the limit point case holds, we have $m^{N}(\lambda)\underset{N\to\infty}{\longrightarrow}m(\lambda)$ uniformly in all bounded domains in the upper and the lower half-planes.

3.2. Discrete Fourier transformation.

Consider two solutions of the equation

[TABLE]

satisfying the initial conditions:

[TABLE]

Clearly, one has:

[TABLE]

On introducing the notation $U_{n}(\lambda)=P_{n+1}(2\lambda)$, we see that $U$ satisfies

[TABLE]

So $U_{n}$ are Chebyshev polynomials of the second kind, and for them the following representation holds:

[TABLE]

We are looking for the unique $l^{2}$-solution $S$ (see (3.2), (3.3)):

[TABLE]

where $m_{0}$ is the Weyl function of $H_{0}$ (the special case of $H$ with $a_{n}\equiv 1$, $b_{n}\equiv 0$). This function can be obtained as the limit of Weyl functions $m_{0}^{N}$ for the problem (3.5) with the Dirichlet condition at $n=N+1$:

[TABLE]

Consider the new variable $z$ related to $\lambda$ by the equalities

[TABLE]

$z\in\mathbb{D}\cap\mathbb{C}_{-}$ for $\lambda\in\mathbb{C}_{+}$, where $\mathbb{D}=\{z\,|\,|z|\leqslant 1\}$. Using the formula for Chebyshev polynomials we can pass to the limit as $N\rightarrow+\infty$ and obtain

[TABLE]

Equation (3.5) has two solutions, $z^{n}$ and $z^{-n}$. Since $S\in l^{2}(\mathbb{N})$ and $S_{1}(\lambda)=m_{0}(\lambda)=-z$, we get

[TABLE]

Let $\chi_{(a,b)}(\lambda)$ be the characteristic function of the interval $(a,b)$. The spectral measure of the unperturbed operator $H_{0}$ corresponding to (3.5) is

[TABLE]

Then the Fourier transformation $F:l^{2}(\mathbb{N})\mapsto L_{2}((-2,2),\rho)$ acts by the rule: for $(v_{n})\in l^{2}(\mathbb{N})$:

[TABLE]

The inverse transform is given by:

[TABLE]

3.3. Relationship between response vectors and Weyl functions

Now we have everything that is necessary to prove Theorem 1.

Proof of Theorem 1..

In (1.1), (1.3) we take the special controls: $f=\delta$, $g=\delta$ and go over the Fourier transform in the variable $t$: for a fixed $n$ we evaluate the sum using the conditions at $t=0$:

[TABLE]

Changing the order of summation yields:

[TABLE]

Introducing the notation

[TABLE]

and assuming that $S$ satisfies (3.6), we deduce from (3.9) that $\widehat{u}$ satisfies

[TABLE]

Similarly, introducing

[TABLE]

one can check that $\widehat{v}$ satisfies

[TABLE]

Then by (3.4) we immediately obtain the representation for the Weyl function in the finite case:

[TABLE]

As a consequence of finiteness of the signal propagation speed in the systems (1.1), (1.3) one has local dependance of the response operator (response vector) on the coefficients $\{a_{n},b_{n}\}$ (see [15, 16]). The latter, in particular, yields that $r_{t}=r^{N}_{t}$ for $t=0,1,\ldots,2N-1$, which, in turn, leads to the convergence

[TABLE]

which takes place in the region which will be specified below. Since $H$ is in the limit point case, Weyl functions $m^{N}$ of finite-dimensional operators $H^{N}$ converge to $m$ uniformly in bounded domains in $\mathbb{C}_{+}$ and $\mathbb{C}_{-}$ as $N\to\infty$, so we deduce that

[TABLE]

Passing from dynamical systems with discrete time (1.1), (1.3) to systems (3.11), (3.13) with the parameter $\lambda$ will be justified as soon as we show that sums in (3.10) and (3.12) converge. To this end we need estimates on $|u^{\delta}_{n,t}|$, $|v^{\delta}_{n,t}|$. We treat the semi-infinite case, the arguments for the estimate for $|v^{\delta}_{n,t}|$ are the same. Introducing the notation

[TABLE]

from the difference equation (the first line) of the system (1.1) we have the following estimate:

[TABLE]

From this relation we obtain that

[TABLE]

Thus $M_{t}$ is bounded by the following:

[TABLE]

The above estimate implies that the sum in (3.10) converges, provided

[TABLE]

or

[TABLE]

Denote $R:=3B+1$. Then (3.17) is valid in the region $D\subset\mathbb{C}_{+}$ specified in (1.7), which finishes the proof of Theorem 1. ∎

Remark 2**.**

In $z$ variable the formulas (1.5), (1.6) gives a Taylor expansions of Weyl functions at zero, these expansions hold in the region $\mathbb{C}_{-}\cap\{\left|z\right|<\frac{1}{3B+1}\}$.

Acknowledgments

The research of Victor Mikhaylov was supported in part by RFBR 17-01-00529. Alexandr Mikhaylov was supported by RFBR 17-01-00099; A. S. Mikhaylov and V. S. Mikhaylov were partly supported by RFBR 18-01-00269 and by VW Foundation program “Modeling, Analysis, and Approximation Theory toward application in tomography and inverse problems.” S. A. Simonov was supported by grants RFBR 17-01-00529, RFBR 16-01-00443A and RFBR 16-01-00635A.