Dynamical inverse problem for Jacobi matrices

TL;DR

This paper addresses the inverse dynamical problem for semi-infinite Jacobi matrices, providing a solution and characterization of inverse data, along with conditions for spectral measures of discrete Schrödinger operators.

Contribution

It offers a novel solution to the inverse problem for Jacobi matrices and characterizes inverse data and spectral measures.

Findings

Solved the inverse problem for the dynamical system with Jacobi matrices.

Provided necessary and sufficient conditions for spectral measures.

Characterized inverse data for semi-infinite Jacobi matrices.

Abstract

We consider the inverse dynamical problem for the dynamical system with discrete time associated with the semi-infinite Jacobi matrix. We solve the inverse problem for such a system and answer a question on the characterization of the inverse data. As a by-product we give a necessary and sufficient condition for the measure on the real line line to be the spectral measure of semi-infinite discrete Schrodinger operator.

Equations

$$(H\varphi {)}_n$$

$$(H\varphi {)}_0$$

$$\left\{\begin{array}[]{l}\mathrm{T}_{t+1}+\mathrm{T}_{t-1}-\lambda\mathrm{T}_{t}=0,\\ \mathrm{T}_{0}=0,\,\,\mathrm{T}_{1}=1.\end{array}\right.$$

$$C_{i,j}^T={\int }_{-\infty }^{\infty }{T}_{T-i+1}(\lambda )T_{T-j+1}(\lambda )d\rho (\lambda ),i,j=1,\dots ,T.$$

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

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

$$u_{n,t}^f=\prod _{k=0}^{n-1}{a}_k{f}_{t-n}+\sum _{s=n}^{t-1}{w}_{n,s}{f}_{t-s-1},n,t\in \mathbb{N}.$$

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

$$0=\prod _{k=0}^{n-1}{a}_k{f}_{t+1-n}+\prod _{k=0}^{n-1}{a}_k{f}_{t-1-n}-a_n\prod _{k=0}^n{a}_k{f}_{t-n-1}-a_{n-1}\prod _{k=0}^{n-2}{a}_k{f}_{t-n+1}$$

$$-b_n\prod _{k=0}^{n-1}{a}_k{f}_{t-n}+\sum _{s=n}^t{w}_{n,s}{f}_{t-s}+\sum _{s=n}^{t-2}{w}_{n,s}{f}_{t-s-2}$$

$$-a_n\sum _{s=n+1}^{t-1}{w}_{n+1,s}{f}_{t-s-1}-a_{n-1}\sum _{s=n-1}^{t-1}{w}_{n-1,s}{f}_{t-s-1}-\sum _{s=n}^{t-1}{b}_n{w}_{n,s}{f}_{t-s-1}.$$

$$0=\left(1-a_n^2\right)\prod _{k=0}^{n-1}{a}_k{f}_{t-n-1}-b_n\prod _{k=0}^{n-1}{a}_k{f}_{t-n}$$

$$-\sum _{s=n}^{t-1}{f}_{t-s-1}\left(b_n{w}_{n,s}+a_n{w}_{n+1,s}+a_{n-1}{w}_{n-1,s}\right)+a_n{w}_{n+1,n}{f}_{t-n-1}$$

$$-a_{n-1}{w}_{n-1,n-1}{f}_{t-n}+\sum _{s=n-1}^{t-1}{w}_{n,s+1}{f}_{t-s-1}+\sum _{s=n+1}^{t-1}{w}_{n,s-1}{f}_{t-s-1}$$

$$=\sum _{s=n}^{t-1}{f}_{t-s-1}\left(w_{n,s+1}+w_{n,s-1}-a_n{w}_{n+1,s}-a_{n-1}{w}_{n-1,s}-b_n{w}_{n,s}\right)$$

$$-b_n\prod _{k=0}^{n-1}{a}_k{f}_{t-n}+\left(1-a_n^2\right)\prod _{k=0}^{n-1}{a}_k{f}_{t-n-1}+a_n{w}_{n+1,n}{f}_{t-n-1}$$

$$-a_{n-1}{w}_{n-1,n-1}{f}_{t-n}+w_{n,n}{f}_{t-n}-w_{n,n-1}{f}_{t-n-1}.$$

$$\sum _{s=n}^{t-1}{f}_{t-s-1}\left(w_{n,s+1}+w_{n,s-1}-a_n{w}_{n+1,s}-a_{n-1}{w}_{n-1,s}-b_n{w}_{n,s}+\left(1-a_n^2\right)\prod _{k=0}^{n-1}{a}_k{\delta }_{sn}\right)$$

$$+f_{t-n}\left(w_{n,n}-a_{n-1}{w}_{n-1,n-1}-b_n\prod _{k=0}^{n-1}{a}_k\right)=0.$$

$$c_t=\sum _{s=0}^t{a}_s{b}_{t-s},t\in \mathbb{N}\cup \{0\}.$$

$${\left(R^{T}f\right)}_t=u_{1,t}^f,t=1,\dots ,T.$$

$${\left(R^{T}f\right)}_t=u_{1,t}^f=a_0{f}_{t-1}+\sum _{s=1}^{t-1}{w}_{1,s}{f}_{t-1-s}t=1,\dots ,T.$$

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

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

$$W^{T}f:=u_{n,T}^f,n=1,\dots ,T.$$

$${\left(W^{T}f\right)}_n=u_{n,T}^f=\prod _{k=0}^{n-1}{a}_k{f}_{T-n}+\sum _{s=n}^{T-1}{w}_{n,s}{f}_{T-s-1},n=1,\dots ,T.$$

$$W^{T}f=\left(\begin{array}{c}{u}_{1,T}\\ u_{2,T}\\ \cdot \\ u_{k,T}\\ \cdot \\ u_{T,T}\end{array}\right)=\left(\begin{array}{cccccc}{a}_0& w_{1,1}& w_{1,2}& \dots & \dots & w_{1,T-1}\\ 0& a_0{a}_1& w_{2,2}& \dots & \dots & w_{2,T-1}\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ 0& \dots & {\prod }_{j=0}^{k-1}{a}_j& w_{k,k}& \dots & w_{k,T-1}\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ 0& 0& 0& 0& \dots & {\prod }_{k=0}^{T-1}{a}_{T-1}\end{array}\right)\left(\begin{array}{c}{f}_{T-1}\\ f_{T-2}\\ \cdot \\ f_{T-k-1}\\ \cdot \\ f_0\end{array}\right)$$

$$J_T:{\mathcal{F}}^T\mapsto {\mathcal{F}}^T,{\left(J_{T}f\right)}_n=f_{T-1-n},n=0,\dots ,T-1,$$

$$A\in {\mathbb{R}}^{T\times T},a_{ii}=\prod _{k=0}^{i-1}{a}_i,a_{ij}=0,i\ne j,$$

$$K\in {\mathbb{R}}^{T\times T},k_{ij}=0,i⩾j,k_{ij}=w_{ij-1},i<j.$$

$$W^T=\left(A+K\right)J^T$$

$${\left(C^{T}f,g\right)}_{{\mathcal{F}}^T}={\left(u_{\cdot ,T}^f,u_{\cdot ,T}^g\right)}_{{\mathcal{H}}^T}={\left(W^{T}f,W^{T}g\right)}_{{\mathcal{H}}^T}.$$

$$C^T=a_0{C}_{ij}^T,C_{ij}^T=\sum _{k=0}^{T-\max i,j}{r}_{|i-j|+2k},r_0=a_0.$$

$$C^T=\left(\begin{array}{ccccc}{r}_0+r_2+\dots +r_{2T-2}& r_1+r_3+\dots +r_{2T-3}& \dots & r_T+r_{T-2}& r_{T-1}\\ r_1+r_3+\dots +r_{2T-3}& r_0+r_2+\dots +r_{2T-4}& \dots & \dots & r_{T-2}\\ \cdot & \cdot & \cdot & \cdot & \cdot \\ r_{T-3}+r_{T-1}+r_{T+1}& \dots & r_0+r_2+r_4& r_1+r_3& r_2\\ r_T+r_{T-2}& \dots & r_1+r_3& r_0+r_2& r_1\\ r_{T-1}& r_{T-2}& \dots & r_1& r_0\end{array}\right)$$

$${\psi }_{n,t}:={\left(u_{\cdot ,n}^f,u_{\cdot ,t}^g\right)}_{{\mathcal{H}}^T}=\sum _{k=1}^T{u}_{k,n}^f{u}_{k,t}^g$$

$${\psi }_{n,t+1}+{\psi }_{n,t-1}-{\psi }_{n+1,t}-{\psi }_{n-1,t}=\sum _{k=1}^T{u}_{k,n}^f\left(u_{k,t+1}^g+u_{k,t-1}^g\right)$$

Full text

Abstract. We consider the inverse dynamical problem for the dynamical system with discrete time associated with the semi-infinite Jacobi matrix. We solve the inverse problem for such a system and answer a question on the characterization of the inverse data. As a by-product we give a necessary and sufficient condition for the measure on the real line line to be the spectral measure of semi-infinite discrete Schrodinger operator.

1. Introduction.

Given a positive sequence $\{a_{0},a_{1},\ldots\}$ and real $\{b_{1},b_{2},\ldots\}$ we consider the operator $H$ corresponding to semi-infinite Jacobi matrix, defined on $l^{2}\ni\phi=(\phi_{0},\phi_{1},\ldots)$, given by

[TABLE]

adding the Dirichlet boundary condition $\phi_{0}=0$ give rise to the spectral measure $d\rho$ (in the case when $H$ is in the limit circle case at infinity, see [2, 7], this measure in non unique and is paramertized by a point on a unit circle). When all $a_{n}=1$, $n=\{0,1,\ldots\}$ the operator $H$ is called the discrete Schrödinger operator. In [5] the authors set up a question of the characterization of the spectral measure for the discrete Schrodinger operator. The goal of the paper is to answer this question. Below we formulate the main result: let $\mathrm{T}_{k}(2\lambda)$ be the Chebyshev polynomials of the second kind: i.e. they satisfy

[TABLE]

Theorem 1**.**

The measure $d\rho$ is a spectral measure of discrete Schrödinger operator if and only if for every $T\geqslant 1$ the matrix $C^{T}$ with the entries

[TABLE]

is positive definite and $\det{C^{T}}=1.$

We use the dynamical approach: we consider the dynamical system with discrete time associated with the Jacobi marix, which is a natural analog of dynamical systems governed by the wave equation on a semi-axis:

[TABLE]

By analogy with continuous problems [3], we treat the real sequence $f=(f_{0},f_{1},\ldots)$ as a boundary control. The solution to (1.1) we denote by $u^{f}_{n,t}$. Having fixed $\tau\in\mathbb{N}$, with (1.1) we associate the response operators, which maps the control $f=(f_{0},\ldots f_{\tau-1})$ to $u^{f}_{1,t}$:

[TABLE]

The inverse problem we will be dealing with is to recover from $R^{\tau}$ the sequences $\{b_{1},b_{2},\ldots,b_{n}\}$, $\{a_{0},a_{1},\ldots,a_{n}\}$ for some $n$. This problems is a natural discrete analog of the inverse problem for the wave equation where the inverse data is the dynamical Dirichlet-to-Neumann map, see [3].

To treat the inverse dynamical problem we will use the Boundary Control method [3] which was initially developed to treat multidimensional dynamical inverse problems, but since then was applied to multy- and one- dimensional inverse dynamical, spectral and scattering problems, problems of signal processing and identification problems.

In the second section we study the forward problem: for (1.1) we prove the analog of d’Alembert integral representation formula, we also introduce and prove the representation formulaes for the main operators of the BC method: response operator, control and connecting operators. In the third section we derive the equations for the inverse problem and give a characterization of the dynamical inverse data for the case of Jacobi matrix and for the case of discrete Schrödinger operator. In the last section we derive the spectral representation formulaes for response and connecting operators and use the results obtained to prove Theorem 1.

2. Forward problem, operators of the Boundary Control method.

We fix some positive integer $T$. By $\mathcal{F}^{T}$ we denote the outer space of the system (1.1), the space of controls: $\mathcal{F}^{T}:=\mathbb{R}^{T}$, $f\in\mathcal{F}^{T}$, $f=(f_{0},\ldots,f_{T-1})$.

First, we derive the representation formulas for the solution to (1.1) which could be considered as analogs of known formulas for the wave equation [1].

Lemma 1**.**

The solution to (1.1) admits the representation

[TABLE]

where $w_{n,s}$ satisfies the Goursat problem

[TABLE]

Proof.

We assume that $u^{f}_{n,t}$ has a form (2.1) with unknown $w_{n,s}$ and plug it to equation in (1.1):

[TABLE]

Evaluating and changing the order of summation we get

[TABLE]

Finally we arrive at

[TABLE]

Counting that $w_{n,s}=0$ when $n>s$ and arbitrariness of $f\in\mathcal{F}^{T}$, we arrive at (2.2). ∎

Definition 1**.**

For $a,b\in l^{\infty}$ we define the convolution $c=a*b\in l^{\infty}$ by the formula

[TABLE]

As an inverse data for (1.1) we use the analog of the dynamical response operator (dynamical Dirichlet-to-Neumann map) [3].

Definition 2**.**

For (1.1) the response operator $R^{T}:\mathcal{F}^{T}\mapsto\mathbb{R}^{T}$ is defined by the rule

[TABLE]

Introduce the notation: the response vector is the convolution kernel of the response operator, $r=(r_{0},r_{1},\ldots,r_{T-1})=(a_{0},w_{1,1},w_{1,2},\ldots w_{1,T-1})$. Then in accordance with (2.1)

[TABLE]

If we take special control $f=\delta=(1,0,0,\ldots)$, then the kernel of response operator becomes

[TABLE]

We introduce the inner space of dynamical system (1.1) $\mathcal{H}^{T}:=\mathbb{R}^{T}$, $h\in\mathcal{H}^{T}$, $h=(h_{1},\ldots,h_{T})$. The control operator $W^{T}:\mathcal{F}^{T}\mapsto\mathcal{H}^{T}$ is defined by the rule

[TABLE]

Directly from (2.1) we deduce that

[TABLE]

The following statement is equivalent to the controllability of (1.1).

Theorem 2**.**

The operator $W^{T}$ is an isomorphism between $\mathcal{F}^{T}$ and $\mathcal{H}^{T}$.

Proof.

We fix some $a\in\mathcal{H}^{T}$ and look for a control $f\in\mathcal{F}^{T}$ such that $W^{T}f=a$. To this aim we write down the operator as

[TABLE]

We introduce the notations

[TABLE]

Then

[TABLE]

Obviously, this operator is invertible, which proves the statement of the theorem. ∎

For the system (1.1) we introduce the connecting operator $C^{T}:\mathcal{F}^{T}\mapsto\mathcal{F}^{T}$ by the quadratic form: for arbitrary $f,g\in\mathcal{F}^{T}$ we define

[TABLE]

We observe that $C^{T}=\left(W^{T}\right)^{*}W^{T}$, so due to Theorem 2, $C^{T}$ is an isomorphism in $\mathcal{F}^{T}$. The fact that $C^{T}$ can be expressed in terms of response $R^{2T-1}$ is crucial in BC-method.

Theorem 3**.**

Connecting operator admits the representation in terms of inverse data:

[TABLE]

[TABLE]

Proof.

For fixed $f,g\in\mathcal{F}^{T}$ we introduce the Blagoveshchenskii function by the rule

[TABLE]

Then we show that $\psi_{n,t}$ satisfies some difference equation. Indeed, we can evaluate:

[TABLE]

So we arrive at the following difference equation on $\psi_{n,t}$:

[TABLE]

We introduce the set

[TABLE]

The solution to (2.12) is given by (see [6])

[TABLE]

We observe that $\psi_{T,T}=\left(C^{T}f,g\right)$, so

[TABLE]

Notice that in the r.h.s. of (2.13) the argument $k$ runs from $1$ to $2T-1.$ We extend $f\in\mathcal{F}^{T}$, $f=(f_{0},\ldots,f_{T-1})$ to $f\in\mathcal{F}^{2T}$ by:

[TABLE]

Due to this odd extension, $\sum_{k,\tau\in K(T,T-1)}f_{k}(R^{T}g)_{\tau}=0$, so (2.13) gives

[TABLE]

Finally we infer that

[TABLE]

from where the statement of the theorem follows. ∎

3. Inverse problem.

The dependence of the solution (1.1) $u^{f}$ on the coefficients $a_{n},b_{n}$ resemble one of the wave equation with the potential. From the very system one can see that for $M\in\mathbb{N}$, $u^{f}_{M,M+1}$ depends on $\{a_{0},\ldots,a_{M-1}\}$, $\{b_{1},\ldots,b_{M}\}$, which implies that $u^{f}_{1,2M}$ depends of the same set of parameters. From where follows

Remark 1**.**

The response $R^{2T}$ (or, what is equivalent, the response vector $(r_{0},r_{1},\ldots,r_{2T-1})$) depends on $\{a_{0},\ldots,a_{T-1}\}$, $\{b_{1},\ldots,b_{T}\}$.

This is an analog of the effect of the finite speed of wave propagation in the wave equation. This leads to the following natural set up of the dynamical inverse problem: by the given operator $R^{2T}$ to recover $\{a_{0},\ldots,a_{T-1}\}$ and $\{b_{1},\ldots,b_{T}\}$.

3.1. Krein equations

Let $\alpha,\beta\in\mathbb{R}$ and $y$ be solution to

[TABLE]

We set up the following control problem: to find a control $f^{T}\in\mathcal{F}^{T}$ such that

[TABLE]

Due to Theorem 2, this problem has unique solution. Let $\varkappa^{T}$ be a solution to

[TABLE]

We show that the control $f^{T}$ satisfies the Krein equation:

Theorem 4**.**

The control $f^{T}$, defined by (3.2) satisfies the following equation in $\mathcal{F}^{T}$:

[TABLE]

Proof.

Let us take $f^{T}$ solving (3.2). We observe that for any fixed $g\in\mathcal{F}^{T}$:

[TABLE]

Indeed, changing the order of summation in the r.h.s. of (3.5), we get

[TABLE]

which gives (3.5) due to (3.3). Using this observation, we can evaluate

[TABLE]

From where (3.4) follows. ∎

Having found $f^{\tau}\in W^{\tau}$ for $\tau=1,\ldots,T$, we can recover $b_{n},$ $a_{n}$, $n=1,\ldots,T-1$. We will describe the procedure. From (2.1) and (2.2) we infer that

[TABLE]

Notice that we know $a_{0}=r_{0}$. Let $T=2$, then we have:

[TABLE]

In (3.7) we know $y_{1}=\beta,$ $a_{0},$ $f^{2}_{1},$ $f^{2}_{0}$, so we can recover $b_{1}$. On the other hand, using (3.1), we have a system

[TABLE]

Since $a_{1}>0,$ we can recover $y_{2}$ and $a_{1}.$ Assume that we have already found $y_{k-1},$ $b_{k-2},$ $a_{k-2}$ for $k\leqslant n$, we will find $y_{n},$ $a_{n-1},$ $b_{n-1}$. We have that

[TABLE]

Since we know $y_{n-1},$ $f^{n}_{0}$, $f^{n}_{1}$, and $a_{k},$ $b_{k}$, $k\leqslant n-2,$ from (3.9) we can recover $b_{n-1}$. Then we use (3.1) and (3.8) to write down the system

[TABLE]

From which we recover $a_{n-1}$ and $y_{n}.$

3.2. Factorization method

We make use the fact that matrix $C^{T}$ has a special structure – it is a product of triangular matrix and its conjugate. We rewrite the operator $W^{T}$ as $W^{T}=\overline{W}^{T}J$ where

[TABLE]

Using the definition (2.8) and the invertibility of $W^{T}$ (cf. Theorem 2), we have:

[TABLE]

We can rewrite the latter equation as

[TABLE]

Here the matrix $\overline{C}^{T}$ has the entries:

[TABLE]

and operator $\left(\overline{W}^{T}\right)^{-1}$ has the form

[TABLE]

We multiply the $k-$th row of $\overline{W}^{T}$ by $k-$th column of $\left(\overline{W}^{T}\right)^{-1}$ to get $a_{k,k}a_{0}a_{1}\ldots a_{k-1}=1$, so

[TABLE]

Multiplying the $k-$th row of $\overline{W}^{T}$ by $k+1-$th column of $\left(\overline{W}^{T}\right)^{-1}$, we get

[TABLE]

from where

[TABLE]

Thus we can rewrite (3.10) as

[TABLE]

In the above equation $\overline{c}_{ij}$ are given (see (3.11)), the entries $a_{ij}$ are unknown. As a direct consequence of (3.15) we get

[TABLE]

which yields

[TABLE]

From where we derive that

[TABLE]

Combining the latter equation with (3.13), we deduce

[TABLE]

similarly,

[TABLE]

so we can write

[TABLE]

here we assume that $\det{C^{0}}=1,$ $\det{C^{-1}=1}$.

Now using (3.15) we can write down the equation on the last column of $\left(\overline{W}^{T}\right)^{-1}$:

[TABLE]

Here we know $a_{T,T}$, so $(a_{1,T},\ldots,a_{T-1,T})^{*}$ satisfies

[TABLE]

which is equivalent to

[TABLE]

Introduce the notation:

[TABLE]

that is $\overline{C}^{k-1}_{k}$ is constructed from $C^{k-1}$ by substituting the last column by $(\overline{c}_{1,k},\ldots,\overline{c}_{k-1,k})$. Then by linear algebra, from (3.21) we have:

[TABLE]

here we assume that $\det{C^{-1}_{0}}=0.$ On the other hand, from (3.13), (3.14) we see that

[TABLE]

Equating (3.23) and (3.24), we see that

[TABLE]

from where

[TABLE]

3.3. Characterization of the inverse data.

In the second section we considered the forward problem (1.1), for $(a_{0},\dots,a_{T-1})$, $(b_{1},\dots,b_{T-1})$ we constructed the matrix $W^{T}$ (2.1), (2.2), the response vector $(r_{0},r_{1},\dots,r_{2T-2})$ (see (2.3)) and the connecting operator $\overline{C}^{T}$ defined in (2.9), (3.11). From the theorem 2 we know that $C^{T}$ is positively definite. We have also shown that if coefficients $r_{0},r_{1},\dots,r_{2T-2}$ correspond to $(a_{0},\dots,a_{T-1})$ $(b_{1},\dots,b_{T-1})$ then we can recover those $a^{\prime}$s and $b^{\prime}s$ by (3.19) and (3.26).

Now we set up a question: can one determine whether a vector $(r_{0},r_{1},r_{2},\ldots,r_{2T-2})$ is a response vector for the dynamical system (1.1) with some $(a_{0},\ldots,a_{T-1})$ $(b_{1},\ldots,b_{T-1})$ or not? The answer is the following theorem.

Theorem 5**.**

The vector $(r_{0},r_{1},r_{2},\ldots,r_{2T-2})$ is a response vector for the dynamical system (1.1) if and only if the matrix $C^{T}$ (2.9) is positively definite.

Proof.

First we observe that in the conditions of the theorem we can substitute $C^{T}$ by $\overline{C}^{T}$ (3.11). The necessary part of the theorem is proved in the preceding sections. We are left to prove the sufficiency of these conditions.

Let we have a vector $(r_{0},r_{1},\dots,r_{2T-2})$ such that the matrix $\overline{C}^{T}$ constructed from it using (3.11) satisfies conditions of the theorem. Then we can construct sequences $(a_{0},\dots,a_{T-1})$, $(b_{1},\dots,b_{T-1})$ using (3.19) and (3.26) and consider the dynamical system (1.1) with this coefficients. For this system we construct the response $(r^{new}_{0},r^{new}_{1},\dots,r^{new}_{2T-2})$ and connecting operator $\mathcal{C}^{T}$ and its rotated $\mathcal{\overline{C}}^{T}$ using (2.9) and (3.11). We will show that the response vectors coincide.

First of all we note that we have two matrices constructed by (3.11), one comes from the vector $(r_{0},r_{1},\dots,r_{2T-2})$ and the other comes from $(r_{0}^{new},r^{new}_{1},\dots,r^{new}_{2T-2})$. Also they have a common property that $\overline{C}^{T}>0,$ $\mathcal{\overline{C}}^{T}>0$ (one by theorem’s condition and the other by representation $\mathcal{\overline{C}}^{T}=(\overline{W}^{T}_{new})^{*}\overline{W}^{T}_{new}$). Secondly we note that if we calculate the elements of sequences $(a_{0},\dots,a_{T-1})$, $(b_{1},\dots,b_{T-1})$ using (3.19) and (3.26) from any of $\overline{C}^{T}$ and $\mathcal{\overline{C}}^{T}$ matrices, we get the same answer. If so, we get that

[TABLE]

From these equalities by simple arguments we deduce that

[TABLE]

From these equalities immediately follows that

[TABLE]

which finishes the proof.

∎

3.4. Discrete Schrödinger operator

Here we consider the case of the dynamical Schrödinger operator, i.e. the system (1.1) with $a_{k}=1$, $k\in\mathbb{N}$, see[6]. In this particular case the control operator (2.6) is given by $W^{T}=\left(I+K\right)J^{T}$, so all the diagonal elements of the matrix in (2.6) are equal to one. The latter immediately yields $\det{W^{T}}=1$. Due to this fact, the connecting operator (2.8), (2.9), has a remarkable property that $\det{C^{k}}=1$, $k=1,\ldots,T.$ This fact actually says that not all elements in the response vector are independent: $r_{2m}$ depends on $r_{2l+1}$, $l=0,\ldots,m-1$, moreover, this property characterize the dynamical data of the discrete Schrödinger operators:

Theorem 6**.**

The vector $(1,r_{1},r_{2},\ldots,r_{2T-2})$ is a response vector for the dynamical system (1.1) with $a_{k}=1$ if and only if the matrix $C^{T}$ (2.9) is positive definite and $\det C^{l}=1,$ $l=1,\ldots,T$.

Proof.

As in Theorem 5 we use $\overline{C}^{T}$ instead of $C^{T}.$ The necessity of the conditions was explained. We are left with the sufficiency part.

Notice that $r_{0}=a_{0}=1$. Let a vector $(1,r_{1},\dots,r_{2T-2})$ be such that the matrix $\overline{C}^{T}$ constructed from it using (3.11) satisfies conditions of the theorem. We construct the potential $(b_{1},\dots,b_{T-1})$ using (3.26) and consider the dynamical system (1.1) with these $b_{k}$ and $a_{k}=1$. For this system we construct the response $(1,r_{1}^{new},\ldots,r_{2T-2}^{new})$ and the connecting operator $\mathcal{\overline{C}}^{T}$ using (2.3), (2.9) and (3.11). We will show that responses coincide.

We notice that if we calculate $(b_{1},\dots,b_{T-1})$ using (3.26) with any of $\overline{C}^{T}$ or $\mathcal{\overline{C}}^{T}$ matrices, we get the same answer. The latter implies (we count that $\det{\overline{C}^{k}}=\det{\mathcal{\overline{C}}^{k}}=1$)

[TABLE]

By induction arguments we get

[TABLE]

which yields $r_{k}=r_{k}^{new}$, $k=1,\ldots,2T-2$. That finishes the proof. ∎

4. Spectral representation of $C^{T}$ and $r_{t}$.

We fix $N\in\mathbb{N}$. Along with (1.1) we consider the analog of the wave equation on the interval: we impose the Dirichlet condition at $n=N+1$. Then for a control $f=(f_{0},f_{1},\ldots)$ and $h\in\mathbb{R}$ we consider

[TABLE]

We denote the solution to (4.1) by $v^{f}$.

Let $\phi_{n}(\lambda)$ be the solution to

[TABLE]

Denote by $\{\lambda_{k}\}_{k=1}^{N}$ the roots of the equation $\phi_{N+1}+h\phi_{N}=0$, it is known [2], that they are real. We introduce the vectors $\phi^{n}\in\mathbb{R}^{N}$ by the rule $\phi^{n}_{i}:=\phi_{i}(\lambda_{n})$, $n,i=1,\ldots,N,$ and define the numbers $\rho_{k}$ by

[TABLE]

where $(\cdot,\cdot)$– is a scalar product in $\mathbb{R}^{N}$.

Definition 3**.**

The set

[TABLE]

is called the spectral data.

We take $y\in\mathbb{R}^{N},\,\,y=(y_{1},\ldots,y_{N})$, for each $n$ we multiply the equation in (4.1) by $y_{n}$, sum up and evaluate the following expression, changing the order of summation

[TABLE]

Now we choose $y=\phi^{l}$, $l=1\ldots,N$. On counting that $\phi^{l}_{0}=0$, $\phi^{l}_{N+1}=-h\phi_{N},$ $\phi^{l}_{1}=1,$ $v_{0,t}=f_{t}$, $v_{N+1,t}+hv_{N,t}=0$ we evaluate (4.5) arriving at:

[TABLE]

We assume that the solution to (4.1) has a form

[TABLE]

Proposition 1**.**

The coefficients $c^{k}$ admits the representation:

[TABLE]

where $T(2\lambda)=(T_{1}(2\lambda),T_{2}(2\lambda),T_{3}(2\lambda),\ldots)$ are Chebyshev polynomials of the second kind.

Proof.

We plug (4.7) into (4.6) and evaluate, counting that $a_{n-1}\phi^{l}_{n-1}+a_{n}\phi^{l}_{n+1}+b_{n}\phi^{l}_{n}=\lambda_{l}\phi^{l}_{n}$:

[TABLE]

Changing the order of summation and using (4.3) we finally arrive at the following equation on $c^{k}_{t}$, $k=1,\ldots,N$:

[TABLE]

We assume that solution to (4.9) has a form $c^{k}=\frac{a_{0}}{\rho_{k}}T*f,$ or

[TABLE]

Plugging it into (4.9), we get

[TABLE]

We see that (4.10) holds if $T$ solves

[TABLE]

Thus $T_{k}(2\lambda)$ are Chebyshev polynomials of the second kind. ∎

For the system (4.1) the control operator $W^{T}_{N,h}:\mathcal{F}^{T}\mapsto\mathcal{H}^{N}$ is defined by the rule

[TABLE]

The representation for this operator immediately follows from (4.7), (4.8). Because of the dependence of the solution on the coefficients, which was discussed in the third section, we see that $v^{f}_{N,N}$ does not ”feel” the boundary condition at $n=N$, so

[TABLE]

We introduce the response operator $R^{T}_{N,h}:\mathcal{F}^{T}\mapsto\mathbb{R}^{T}$ by the rule

[TABLE]

The connecting operator $C^{T}_{N,h}:\mathcal{F}^{T}\mapsto\mathcal{F}^{T}$ is introduced in the similar way: for arbitrary $f,g\in\mathcal{F}^{T}$ we define

[TABLE]

The dependence of the solution (1.1) $u^{f}$ on $a_{n},b_{n}$ is discussed in the beginning of the section three (see Remark 1). This dependence in particular implies that for $M\in\mathbb{N}$,

[TABLE]

i.e. $v^{f}_{1,2M}$ does not ”feel” the boundary condition at $n=M$. We introduce the special control $\delta=(1,0,0,\ldots)$, then the kernel of response operator (4.12) is

[TABLE]

on the other hand, we can use (4.7), (4.8) to obtain:

[TABLE]

So on introducing the spectral function

[TABLE]

from (4.15), (4.16) we deduce that

[TABLE]

Due to (4.14), we get

[TABLE]

Taking in (4.18) $N$ to infinity, and varying $h$, we come to the

[TABLE]

where $d\rho$ is a spectral measure of the operator $H$ (non-unique when $H$ is in the limit-circle case at infinity).

Let us evaluate $(C^{T}_{N}f,g)$ for $f,g\in\mathcal{F}^{T}$, using the expansion (4.7):

[TABLE]

from the equality above it is evident that (cf. (2.9))

[TABLE]

Taking into account (4.11), we obtain that $C^{T}=C^{T}_{N,h}$ with $N\geqslant T$, so (4.19) yields for $N\geqslant T:$

[TABLE]

and passing to the limit as $N\to\infty$ yields

[TABLE]

where $d\rho$ is a spectral measure of $H$.

Is is known [5] that any probability measure with finite moments on $\mathbb{R}$ give rise to the Jacobi operator, i.e. is a spectral measure of this operator. In [5] the authors posed the question on the characterization of the spectral measure for the semi-infinite discrete Schrödinger operator. The following theorem answers this question

Theorem 7**.**

The measure $d\rho$ is a spectral measure of discrete Schrödinger operator if and only if for every $T$ the matrix $C^{T}$ is positive definite and $\det{C^{T}}=1,$ where

[TABLE]

Proof.

We consider the system (1.1) with $a_{k}=1$. Let $d\rho$ be a spectral measure of $H$. For every $T\in\mathbb{N}$ we construct the connecting operator $C^{T}$ (see (2.8)) using the representation (4.22). According to Theorem 6, such $C^{T}$ is positive definite and $\det{C^{T}}=1$.

On the other hand, if given measure $d\rho$ satisfies conditions of the theorem, for every $T$ we can construct $C^{T}$ by (4.22) and by Theorem 6 recover coefficients $b_{n}$ by (3.26). ∎

Acknowledgments

The research of Victor Mikhaylov was supported in part by NIR SPbGU 11.38.263.2014. A. S. Mikhaylov and V. S. Mikhaylov were partly supported by VW Foundation program ”Modeling, Analysis, and Approximation Theory toward application in tomography and inverse problems.”