Dynamical inverse problem for Jacobi matrices

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.

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: (1.1)    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, n, t ∈ N, u n,−1 = u n,0 = 0, n ∈ N, u 0,t = f t , t ∈ N ∪ {0}.
By analogy with continuous problems [3], we treat the real sequence f = (f 0 , f 1 , . . .) as a boundary control. The solution to (1.1) we denote by u f n,t . Having fixed τ ∈ N, with (1.1) we associate the response operators, which maps the control f = (f 0 , . . . f τ −1 ) to u f 1,t : (R τ f ) t := u f 1,t , t = 1, . . . , τ. The inverse problem we will be dealing with is to recover from R τ the sequences {b 1 , b 2 , . . . , b n }, {a 0 , a 1 , . . . , 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.

Forward problem, operators of the Boundary Control method.
We fix some positive integer T . By F T we denote the outer space of the system (1.1), the space of controls: F T := R T , f ∈ F T , f = (f 0 , . . . , 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 where w n,s satisfies the Goursat problem (2.2)    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 = −δ s,n (1 − a 2 n ) n−1 k=0 a k , n, s ∈ N, s > n, w n,n − b n n−1 k=0 a k − a n−1 w n−1,n−1 = 0, n ∈ N, w 0,t = 0, t ∈ N 0 .
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): Evaluating and changing the order of summation we get s + a n w n+1,s + a n−1 w n−1,s ) + a n w n+1,n f t−n−1 −a n−1 w n−1,n−1 f t−n + t−1 k=0 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 .
Counting that w n,s = 0 when n > s and arbitrariness of f ∈ F T , we arrive at (2.2).
As an inverse data for (1.1) we use the analog of the dynamical response operator (dynamical Dirichlet-to-Neumann map) [3].
Theorem 2. The operator W T is an isomorphism between F T and H T .
Proof. We fix some a ∈ H T and look for a control f ∈ F T such that W T f = a. To this aim we write down the operator as (2.6) We introduce the notations Obviously, this operator is invertible, which proves the statement of the theorem.
For the system (1.1) we introduce the connecting operator C T : F T → F T by the quadratic form: for arbitrary f, g ∈ F T we define We observe that C T = W T * W T , so due to Theorem 2, C T is an isomorphism in 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: Proof. For fixed f, g ∈ F T we introduce the Blagoveshchenskii function by the rule Then we show that ψ n,t satisfies some difference equation. Indeed, we can evaluate: So we arrive at the following difference equation on ψ n,t : We introduce the set The solution to (2.10) is given by (see [6]) We observe that ψ T,T = C T f, g , so Notice that in the r.h.s. of (2.11) the argument k runs from 1 to 2T − 1. We extend Finally we infer that where the statement of the theorem follows.
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. 3.1. Krein equations. Let α, β ∈ R and y be solution to We set up the following control problem: to find a control f T ∈ F T such that . . , T. Due to Theorem 2, this problem has unique solution. Let κ T be a solution to We show that the control f T satisfies the Krein equation: Proof. Let us take f T solving (3.2). We observe that for any fixed g ∈ F T : Indeed, changing the order of summation in the r.h.s. of (3.5), we get which gives (3.5) due to (3.3). Using this observation, we can evaluate From where (3.4) follows.
Having found f τ ∈ W τ for τ = 1, . . . , T , we can recover b n , a n , n = 1, . . . , T −1. We will describe the procedure. From (2.1) and (2.2) we infer that Notice that we know a 0 = r 0 . Let T = 2, then we have: In (3.7) we know y 1 = β, 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 y 2 = a 0 a 1 f 2 0 , a 1 y 2 + a 0 α + b 1 β = 0 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 n, we will find y n , a n−1 , b n−1 . We have that a k a n−1 f 2 0 , (3.8) Since we know y n−1 , f n 0 , f n 1 , and a k , b k , k n − 2, from (3.9) we can recover b n−1 . Then we use (3.1) and (3.8) to write down the system y n = n−2 k=0 a k a n−1 f 2 0 , a n−1 y n + a n−2 y n−2 + b n−1 y n−1 = 0.
From which we recover a n−1 and y n .

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 Using the definition (2.8) and the invertibility of W T (cf. Theorem 2), we have: We can rewrite the latter equation as Here the matrix C T has the entries: to get a k,k a 0 a 1 . . . a k−1 = 1, so Multiplying the k−th row of W T by k + 1−th column of W Thus we can rewrite (3.10) as In the above equation c ij are given (see (3.11)), the entries a ij are unknown. As a direct consequence of (3.15) we get From where we derive that Combining the latter equation with (3.13), we deduce , so we can write 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
Here we know a T,T , so (a 1,T , . . . , a T −1,T ) * satisfies Introduce the notation: is constructed from C k−1 by substituting the last column by (c 1,k , . . . , c k−1,k ). Then by linear algebra, from (3.21) we have: here we assume that det C −1 0 = 0. On the other hand, from (3.13), (3.14) we see that Equating (3.23) and (3.24), we see that

Discrete Schrödinger operator.
Here we consider the case of the dynamical Schrödinger operator, i.e. the system (1.1) with a k = 1, k ∈ N, see [6]. In this particular case the control operator (2.6) is given by W T = (I + K) 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, . . . , T. This fact actually says that not all elements in the response vector are independent: r 2m depends on r 2l+1 , l = 0, . . . , m − 1, moreover, this property characterize the dynamical data of the discrete Schrödinger operators: Theorem 6. The vector (1, r 1 , r 2 , . . . , 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, . . . , T .
Proof. As in Theorem 5 we use 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 , . . . , r 2T −2 ) be such that the matrix C T constructed from it using (3.11) satisfies conditions of the theorem.
We will show that responses coincide. We notice that if we calculate (b 1 , . . . , b T −1 ) using (3.26) with any of C T or C T matrices, we get the same answer. The latter implies (we count that det which yields r k = r new k , k = 1, . . . , 2T − 2. That finishes the proof.
Denote by {λ k } N k=1 the roots of the equation φ N +1 + hφ N = 0, it is known [2], that they are real. We introduce the vectors φ n ∈ R N by the rule φ n i := φ i (λ n ), n, i = 1, . . . , N, and define the numbers ρ k by where (·, ·)-is a scalar product in R N .
Definition 3. The set is called the spectral data.
Proof. We plug (4.7) into (4.6) and evaluate, counting that a n−1 φ l n−1 + a n φ l n+1 + b n φ l n = λ l φ l n : Changing the order of summation and using (4.3) we finally arrive at the following equation on c k t , k = 1, . . . , N : We assume that solution to (4.9) has a form c k = a0 ρ k T * f, or (4.10) Plugging it into (4.9), we get We see that (4.10) holds if T solves Thus T k (2λ) are Chebyshev polynomials of the second kind.
For the system (4.1) the control operator W T N,h : F T → H N is defined by the rule W T N,h f := v f n,T , n = 1, . . . , N. 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 (4.11) u f n,t = v f n,t , n t N, and W N = W N N,h . We introduce the response operator R T N,h : F T → R T by the rule (4.12) R T N,h f t = v f 1,t , t = 1, . . . , T. The connecting operator C T N,h : F T → F T is introduced in the similar way: for arbitrary f, g ∈ F T we define 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 ∈ N, 1,2M does not "feel" the boundary condition at n = M . We introduce the special control δ = (1, 0, 0, . . .), then the kernel of response operator (4.12) is on the other hand, we can use (4.7), (4.8) to obtain: So on introducing the spectral function Due to (4.14), we get Taking in (4.18) N to infinity, and varying h, we come to the where dρ 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 ∈ F T , using the expansion (4. Taking into account (4.11), we obtain that C T = C T N,h with N T , so (4.19) yields for N T :  where dρ is a spectral measure of H. Is is known [5] that any probability measure with finite moments on 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 Proof. We consider the system (1.1) with a k = 1. Let dρ be a spectral measure of H. For every T ∈ 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 dρ 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).