DYNAMIC INVERSE PROBLEM for JACOBI MATRICES Alexandr Mikhaylov and Victor Mikhaylov

Inverse Problems and Imaging doi:10.3934/ipi.2019021 Volume 13, No. 3, 2019, 431–447

DYNAMIC INVERSE PROBLEM FOR JACOBI MATRICES

Alexandr Mikhaylov∗ and Victor Mikhaylov St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, 27, Fontanka, 191023 St. Petersburg, Russia and Saint Petersburg State University, St.Petersburg State University 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia

(Communicated by Matti Lassas)

Abstract. We consider the inverse dynamic problem for a dynamical system with discrete time associated with a semi-infinite Jacobi matrix. We derive discrete analogs of Krein equations and answer a question on the characterization of dynamic inverse data. As a consequence we obtain a necessary and sufficient condition for a measure on a real line to be a spectral measure of a semi-infinite discrete Schrödingeroperator.

1. Introduction. For a given sequence of positive numbers {a0, a1,...} and real numbers {b1, b2,...}, we consider a dynamical system with discrete time associated with a Jacobi marix:   un, t+1 + un, t−1 − anun+1, t − an−1un−1, t − bnun, t = 0, n, t ∈ N, (1) un, −1 = un, 0 = 0, n ∈ N,  u0, t = ft, t ∈ N ∪ {0}, which is a natural analog of a dynamical systems governed by a wave equation on a semi-axis. By an analogy with continuous problems [6], we treat the real sequence f f = (f0, f1,...) as a boundary control. The solution to (1) is denoted by un, t. Having ﬁxed τ ∈ N, we associate the response operators with (1), which maps the f control f = (f0, . . . fτ−1) to u1, t:

τ f (R f)t := u1, t, t = 1, . . . , τ. The inverse problem we will be dealing with consists in recovering the sequences τ {b1, b2, . . . , bn}, {a0, a1, . . . , an} for some n from R . This problems is a natural discrete analog of an inverse problem for a wave equation on a half-line, in which the dynamic Dirichlet-to-Neumann map is used as inverse data [8,4, 10]. The area of dynamic inverse problems is wide and well-developed, to mention reviews and books [6,7, 16, 15, 14] and literature cited therein, at the same time the authors are not familiar with any literature on inverse dynamic problems for discrete systems with discrete time. In our approach we use the Boundary Control (BC) method [6,7], For the ﬁrst time the BC method was applied to discrete dynamical systems in [1,3] in connection with the spectral estimation problem.

2010 Mathematics Subject Classiﬁcation. Primary: 35R30, 93B30; Secondary: 39A12, 35C15, 35P99. Key words and phrases. Inverse problem, Jacobi matrices, discrete Schr¨odingeroperator, boundary control method, characterization of inverse data.

431 c 2019 American Institute of Mathematical Sciences 432 Alexandr Mikhaylov and Victor Mikhaylov

For the same sequences {a1, a2,...}, {b1, b2,...} (we note that a0 is an additional parameter used in system (1)) we consider the operator H corresponding to a semi- 2 inﬁnite Jacobi matrix, deﬁned on l 3 φ = (φ1, φ2,...), given by

(2) (Hφ)n = anφn+1 + an−1φn−1 + bnφn, n > 2, (3) (Hφ)1 = b1φ1 + a1φ2, n = 1.

Let us ﬁx N ∈ N and h ∈ R, the operator corresponding to a ﬁnite Jacobi matrix is denoted by HN , it is given by (2), (3) and condition at the “right end”:

(4) φN+1 + hφN = 0.

When all an = 1, n = 0, 1,... the operator H is called a discrete Schrödinger operator. In [13] the authors posed a question of the characterization of a spectral measure for a discrete Schrödingeroperator. We give the answer on this question as a consequence to a theorem on characterization of dynamic inverse data. In the second section we study the forward problem: for (1) we prove the analog of d’Alembert integral representation formula, we also introduce and prove representation formulaes for the main operators of the BC method: response operator, control and connecting operators. The continuous analogs of dynamic inverse problems for one-dimensional systems were considered in [4, 10,8], where the authors deal with wave equations on a half-line, in [11] the Dirac system is considered and in [9, 12] the two-velocity system is studied. In the third section we derive Krein equations of the inverse problem and answer a question on the characterization of dynamic inverse data in the case of Jacobi matrices and in the case of discrete Schrödingeroperators. More on dynamic approach to Krein equations one can find in [10, 18]. The application of the BC-method to the problem of characterization of dynamic inverse data (different from one in the present paper) is considered in [9, 12, 11]. In the last section we derive spectral representations for response and connecting operators, that gives a possibility to use methods developed in this paper for solving inverse spectral problems for Jacobi matrices in finite and semi-infinite cases. Then we use the results obtained to prove the theorem on the characterization of a spectral measure for a discrete Schrödingeroperator. In [5, 19, 20] relationships between dynamic and spectral inverse data were established for some one-dimensional dynamical systems. In [18] the authors established links between BC-method and the method of de Branges on the example of several continuous and discrete one-dimensional systems.

2. Forward problem, operators of the Boundary Control method. We ﬁx some positive integer T and denote by F T the outer space of the system (1), the T T T space of controls (inputs): F := R , f ∈ F , f = (f0, . . . , fT −1), we use the ∞ ∞ notation F = R when control acts for all t > 1. The representation formula in the following lemma can be considered as a discrete analog of a d’Alembert integral representation formula for a solution of an initial-boundary value problem for a wave equation with a potential on a half-line. [4].

Lemma 2.1. A solution to (1) admits the representation

n−1 t−1 f Y X (5) un, t = akft−n + wn, sft−s−1, n, t ∈ N, k=0 s=n

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 Dynamic inverse problem for Jacobi matrices 433 where wn,s satisﬁes the Goursat problem  w + w − a w − a w − b w =  n, s+1 n, s−1 n n+1, s n−1 n−1, s n n, s  2 Qn−1 = −δs,n(1 − an) k=0 ak, n, s ∈ N, s > n, (6) Qn−1  wn, n − bn k=0 ak − an−1wn−1, n−1 = 0, n ∈ N,  w0, t = 0, t ∈ N0. f Proof. We assume that un,t has a form (5) with unknown wn,s and plug it into (1): n−1 n−1 n n−2 Y Y Y Y 0 = akft+1−n + akft−1−n − an akft−n−1 − an−1 akft−n+1 k=0 k=0 k=0 k=0 n−1 t t−2 Y X X −bn akft−n + wn, sft−s + wn, sft−s−2 k=0 s=n s=n t−1 t−1 t−1 X X X −an wn+1, sft−s−1 − an−1 wn−1, sft−s−1 − bnwn, sft−s−1. s=n+1 s=n−1 s=n Changing the order of summation and evaluating we get n−1 n−1 2 Y Y 0 = 1 − an akft−n−1 − bn akft−n k=0 k=0 t−1 X − ft−s−1 (bnwn, s + anwn+1,s + an−1wn−1, s) + anwn+1, nft−n−1 s=n t−1 t−1 X X −an−1wn−1, n−1ft−n + wn, s+1ft−s−1 + wn, s−1ft−s−1 s=n−1 s=n+1 t−1 X = ft−s−1 (wn, s+1 + wn, s−1 − anwn+1, s − an−1wn−1, s − bnwn, s) s=n n−1 n−1 Y 2 Y −bn akft−n + 1 − an akft−n−1 + anwn+1, nft−n−1 k=0 k=0 −an−1wn−1, n−1ft−n + wn, nft−n − wn, n−1ft−n−1. Finally we arrive at t−1 X ft−s−1 (wn, s+1 + wn, s−1 − anwn+1, s − an−1wn−1, s − bnwn, s s=n n−1 ! n−1 ! 2 Y Y + 1 − an akδsn + ft−n wn, n − an−1wn−1, n−1 − bn ak = 0. k=0 k=0

Counting that wn, s = 0 when n > s and the arbitrariness of f, we arrive at (6). Deﬁnition 2.2. For f, g ∈ F ∞ we deﬁne the convolution c = f ∗ g ∈ F ∞ by the formula t X ct = fsgt−s, t ∈ N ∪ {0}. s=0 The input 7−→ output correspondence in the system (1) is realized by a response operator:

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 434 Alexandr Mikhaylov and Victor Mikhaylov

Deﬁnition 2.3. For (1) the response operator RT : F T 7→ RT is deﬁned by the rule T f R f t = u1, t, t = 1,...,T.

This operator plays the role of inverse data. The quantity u1, t −u0, t is an analog of ux(0, t) (derivative in x direction at x = 0) in a continuous case, so such introduced response RT f can be considered as a discrete dynamic Dirichlet-to-Neumann map. On the relationship between RT and the Weyl function for corresponding Jacobi matrix see [20]. The response vector is a convolution kernel of a response operator, r = (r0, r1, . . . , rT −1) = (a0, w1,1, w1,2, . . . w1,T −1), in accordance with (5):

t−1 T f X (7) R f t = u1, t = a0ft−1 + w1, sft−1−s t = 1,...,T. s=1 T R f = r ∗ f·−1. By choosing the special control f = δ = (1, 0, 0,...), the kernel of a response operator can be determined as

T δ R δ t = u1, t = rt−1. The inverse problem we will be dealing with consists of recovering a Jacobi matrix (i.e. the sequences {a0, a1, a2,...}, {b1, b2,...}) from the response operator. For a ﬁxed T ∈ N we introduce the inner space of dynamical system (1) HT := T T f R , h ∈ H , h = (h1, . . . , hT ), the space of states. The wave u·,T is considered as a f T state of the system (1) at the moment t = T . By (5) we have that u·,T ∈ H . We note that in our situation, the space of controls and the space of states coincide: F T = HT = RT , although it is not usually the case [6,7, 11]. The input 7−→ state correspondence of the system (1) is realized by a control operator W T : F T 7→ HT , deﬁned by the rule

T f W f n := un, T , n = 1,...,T. From (5) we deduce the representation for W T :

n−1 T −1 T f Y X W f n = un, T = akfT −n + wn, sfT −s−1, n = 1,...,T. k=0 s=n The following statement is equivalent to a boundary controllability of (1). Lemma 2.4. The operator W T is an isomorphism between F T and HT . Proof. We ﬁx some a ∈ HT and look for a control f ∈ F T such that W T f = a. We write down the action of the operator W T as (8)       u1,T a0 w1,1 w1,2 ...... w1,T −1 fT −1 u2,T   0 a0a1 w2,2 ...... w2,T −1   fT −2        T  ·   ······   ·  W f =   =  Qk−1    . uk, T   0 ... aj wk,k . . . wk,T −1  fT −k−1    j=0     ·   ······   ·  u QT −1 f T,T 0 0 0 0 ... k=0 aT −1 0

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 Dynamic inverse problem for Jacobi matrices 435

On introducing the notations

T T JT : F 7→ F , (JT f)n = fT −1−n, n = 0,...,T − 1, i−1 T ×T Y A ∈ R , aii = ak, aij = 0, i 6= j, k=0

T ×T K ∈ R , kij = 0, i > j, kij = wij−1, i < j, we have that W T = (A + K) J T . Obviously, this operator is invertible, which proves the statement of the lemma.

For the system (1) we introduce the connecting operator CT : F T 7→ F T by the quadratic form: for arbitrary f, g ∈ F T we deﬁne

T f g T T (9) C f, g T = u·,T , u·,T = W f, W g T . F HT H ∗ That is CT = W T W T , and by (9), CT connects the metrics of the outer and inner spaces. The fact that the connecting operator can be represented in terms of inverse data is crucial in the BC method.

Theorem 2.5. The connecting operator CT is an isomorphism in F T , it admits the representation in terms of inverse data:

T −max i,j T T T X (10) C = a0Cij,Cij = r|i−j|+2k, r0 = a0. k=0   r0 + r2 + ... + r2T −2 r1 + ... + r2T −3 . . . rT + rT −2 rT −1 r1 + r3 + ... + r2T −3 r0 + ... + r2T −4 ...... rT −2   T  ·····  C =    rT −3 + rT −1 + rT +1 . . . r0 + r2 + r4 r1 + r3 r2     rT + rT −2 . . . r1 + r3 r0 + r2 r1  rT −1 rT −2 . . . r1 r0

∗ Proof. We observe that CT = W T W T , so due to Lemma 2.4, CT is an isomorphism in F T . For ﬁxed f, g ∈ F T we introduce the Blagoveshchenskii function by the rule T X ψ := uf , ug = uf ug . n, t ·, n ·, t HT k, n k, t k=1

We show that ψn, t satisﬁes certain diﬀerence equation. Indeed, we can evaluate:

T X f g g ψn, t+1 + ψn, t−1 − ψn+1, t − ψn−1, t = uk, n uk, t+1 + uk, t−1 k=1 T T X f f g X f g g g − uk, n+1 + uk, n−1 uk, t = uk, n akuk+1, t + ak−1uk−1, t + bkuk, t k=1 k=1

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 436 Alexandr Mikhaylov and Victor Mikhaylov

T T X f f f g X g f f − akuk+1, n + ak−1uk−1, n + bkuk, t uk, t = uk, t akuk+1, n + ak−1uk−1, n k=1 k=1 g f f g g f f g +a0u0, tu1, n − a0u0, nu1, t + aT uT +1, tuT, n − aT uT +1, nuT, t T X g f f − uk, t akuk+1, n + ak−1uk−1, n = a0 [gt(Rf)n − fn(Rg)t] . k=1

Thus we arrived at the following diﬀerence equation on ψn,t: ψ + ψ − ψ − ψ = h , n, t ∈ , (11) n, t+1 n, t−1 n+1, t n−1, t n, t N0 ψ0, t = 0, ψn, 0 = 0,

hn, t = a0 [gt(Rf)n − fn(Rg)t] . We introduce the set K(n, t) := {(n, t) ∪ {(n − 1, t − 1), (n + 1, t − 1)} ∪ {(n − 2, t − 2), (n, t − 2), (n + 2, t − 2)} ∪ ... ∪ {(n − t, 0), (n − t + 2, 0),..., (n + t − 2, 0), (n + t, 0)}} t τ [ [ = (n − τ + 2k, t − τ) . τ=0 k=0 The solution to (11) is given by (see [17]) X ψn, t = h(k, τ). (k,τ)∈K(n,t−1) T We observe that ψT,T = C f, g , so X (12) CT f, g = h(k, τ). (k,τ)∈K(T,T −1) We note that in the right hand side of (12) the argument k runs from 1 to 2T − 1. T 2T Extending f ∈ F , f = (f0, . . . , fT −1) to f ∈ F by:

fT = 0, fT +k = −fT −k, k = 1, 2,...,T − 1, P T we deduce that k,τ∈K(T,T −1) fk(R g)τ = 0, so (12) leads to T X 2T 2T 2T 2T C f, g = gτ R f = g0 R f + R f + ... + R f k 1 3 2T −1 k,τ∈K(T,T −1) 2T 2T 2T 2T +g1 R f + R f + ... + R f + ... + gT −1 R f . 2 4 2T −2 T The latter relation yields CT f = R2T f + ... + R2T f , R2T f + ... + R2T f ,..., R2T f 1 2T −1 2 2T −2 T from where the statement of the theorem follows.

3. Inverse problem. The speed of a wave propagation in (1) is ﬁnite, which implies the following dependence of inverse data on coeﬃcients {an, bn}: for M ∈ N f the element u1,2M−1 depends on {a0, a1, . . . , aM−1} , {b1, . . . , bM−1}, on observing this we can make the following Remark 1. The response operator R2T (or, what is equivalent, the response vector (r0, r1, . . . , r2T −2)) depends on {a0, . . . , aT −1}, {b1, . . . , bT −1}.

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 Dynamic inverse problem for Jacobi matrices 437

This observation leads to the natural set up of the dynamic inverse problem 2T for the system (1): by the given operator R to recover {a0, . . . , aT −1} and {b1, . . . , bT −1}.

3.1. Krein equations. Let α, β ∈ R and y be a solution to a y + a y + b y = 0, (13) k k+1 k−1 k−1 k k y0 = α, y1 = β. We set up the following special control problem: to ﬁnd a control f T ∈ F T that T T drives the system (1) to prescribed statey ¯ := (y1, . . . , yT ) ∈ H : T T T T T (14) W f =y ¯ , W f k = yk, k = 1,...,T. −1 Due to Lemma 2.4, this control problem has a unique solution f T = W T y¯T . Let κT be a solution to T T κt+1 + κt−1 = 0, t = 0,...,T, (15) T T κT = 0, κT −1 = 1. It is important fact that the control f T can be found as a solution to the Krein equation: −1 Theorem 3.1. The control f T = W T y¯T solving the special control problem (14), satisﬁes the following equation in F T :

T T h T T ∗ T i (16) C f = a0 βκ − α R κ .

Proof. Let f T be a solution to (14). We observe that for any ﬁxed g ∈ F T :

T −1 g X g g T (17) uk, T = uk, t+1 + uk, t−1 κt . t=1 Indeed, changing the order of a summation in the right hand side of (17) yields

T −1 T −1 X g g T X T T g g T g T uk, t+1 + uk, t−1 κt = κt+1 + κt−1 uk, t + uk, 0κ1 − uk, T κT −1, t=1 t=1 which gives (17) due to (15). Using this observation, we can evaluate

T T T −1 T T X g X X g g T C f , g = ykuk, T = uk, t+1 + uk, t−1 κt yk k=1 k=1 t=0 T −1 T ! X T X g g g = κt akuk+1, tyk + ak−1uk−1, tyk + bkuk, tyk t=0 k=1 T −1 T X T X g g = κt uk, t(akyk+1 + ak−1yk−1 + bkyk + a0u0, ty1 t=0 k=1 T −1 g g g X T T +aT uT +1, tyT − a0u1, ty0 − aT uT, tyT +1 = κt a0βgt − a0α R g t t=0 T T h T T ∗ T i = κ , a0 βg − α R g = a0 βκ − α R κ , g . From where (16) follows.

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 438 Alexandr Mikhaylov and Victor Mikhaylov

Now we describe the procedure of the recovering a0, an, bn, n = 1,...,T − 1 from the solutions of Krein (16) equations f τ ∈ F τ for τ = 1,...,T . From (5) and (6) we infer that

T −1 f T Y T uT,T = akf0 , k=0 T −2 T −2 f T Y T Y uT −1,T = akf1 + (b1 + b2 + ... + bT −1). k=0 k=0

Notice that we know a0 = r0. Let T = 2, then we have:

f 2 2 y2 = u2, 2 = a0a1f0 , f 2 2 2 (18) y1 = u1, 2 = a0f1 + a0b1f0 , 2 2 In (18) we know y1 = β, a0, f1 , f0 , so we can recover b1. On the other hand, using (13), we have a system 2 y2 = a0a1f0 , a1y2 + a0α + b1β = 0

Since a1 > 0, we can recover y2 and a1. We proceed by the induction: assuming that we have already recovered yk−1, bk−2, ak−2 for k 6 n, we recover yn, an−1, bn−1. Bearing in mind that

n−2 f n Y 2 (19) yn = un, n = akan−1f0 , k=0 n−2 n−2 f n Y n Y n (20) yn−1 = un−1, n = akf1 + ak(b1 + ... + bn−2 + bn−1)f0 , k=0 k=0 n n and that we know yn−1, f0 , f1 , and ak, bk, k 6 n − 2, we can recover bn−1 from (20). Using (13) and (19) leads to the following relations

Qn−2 2 yn = k=0 akan−1f0 , an−1yn + an−2yn−2 + bn−1yn−1 = 0, from which we can recover an−1 and yn.

3.2. Factorization method. We make use of the fact that the matrix CT has a special structure: it is a product of a triangular matrix and its conjugate. We T rewrite the operator W T as W T = W J where       a0 w1,1 w1,2 . . . w1,T −1 0 0 0 ... 1 f0  0 a0a1 w2,2 . . . w2,T −1  0 0 0 ... 0  f2   ·····      T   ·····  ·  W f =  Qk−1       0 ... aj . . . wk,T −1  0 ... 1 0 0 fT −k−1  j=1       ·····  ·····  ·  QT −1 1 0 0 0 0 f 0 0 0 ... j=1 aj T −1 Using the deﬁnition (9) and the invertibility of W T (cf. Lemma 2.4) gives that

∗ −1∗ −1 CT = W T W T , or W T CT W T = I.

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 Dynamic inverse problem for Jacobi matrices 439

The latter equation can be rewritten as ∗ T −1 T T −1 T (21) W C W = I, C = JCT J,

T where the matrix C is deﬁned by   r0 r1 r2 . . . rT −1 T r1 r0 + r2 r1 + r3 . . . ..  (22) Cij = CT +1−j,T +1−i, C = a0   , r2 r1 + r3 r0 + r2 + r4 . . . ..  ·····

T −1 and W has a form   a1, 1 a1, 2 a1, 3 . . . a1,T −1 T  0 a2, 2 a2, 3 . . . ..  (23) W =   .  ··· aT −1,T −1 aT −1,T  0 ...... 0 aT,T

T T −1 We multiply the k−th row of W by k−th column of W to get ak, ka0a1 ... ak−1 = 1, so diagonal elements of (23) satisfy the relation: −1 k−1  Y (24) ak, k =  aj . j=0

T T −1 Multiplying the k−th row of W by k + 1−th column of W leads to the relation

ak, k+1a0a1 . . . ak−1 + ak+1, k+1wk, k = 0, from where we deduce that −2  k  Y (25) ak, k+1 = −  aj akwk, k. j=0 All aforesaid leads to the equivalent form of (21): (26)       a1, 1 0 . 0 c11 .. .. c1T a1,1 a1,2 .. a1,T a1, 2 a2, 2 0 .   ......   0 a2,2 .. a2,T        = I  ····   ····   ····  a1,T . . aT,T cT 1 .. cTT 0 ...... aT,T We note that in the case of a discrete Schr¨odingeroperator, (26) is equivalent to Gelfand-Levitan type equations (see [17]). In the above equality cij are given (see (22)), the entries aij are unknown. A direct consequence of (26) is an equality for determinants: ∗ T −1 T T −1 det W det C det W = 1, which yields 1 T − 2 a1, 1 ∗ ... ∗ aT,T = det C .

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 440 Alexandr Mikhaylov and Victor Mikhaylov

From the above equality we derive that

− 1 − 1 1 2 ! 2 k ! 2 1− 2 det C det C a1, 1 = det C , a2, 2 = 1 , ak, k = k−1 . det C det C Combining latter relations with (24), we deduce that

1 k−1 k ! 2 Y det C ai = k−1 , i=0 det C similarly, for k + 1: 1 k k+1 ! 2 Y det C ai = k . i=0 det C These two relations lead to 1 1 k+1 2 k−1 2 det C det C (27) ak = k , k = 1,...,T − 1. det C Here we set det C0 = 1, det C−1 = 1. T −1 Now using (26) we write down the equation on the last column of W :         a1, 1 0 . 0 c1, 1 .. .. c1,T a1,T 0  a1, 2 a2, 2 0 .   ......  a2,T  0       =    ····   ····   ·  · a1,T −1 . . aT −1,T −1 cT −1, 1 .. cT −1,T −1 aT,T 0

Note that we know aT,T , so we rewrite the above equality in the form of equation ∗ on (a1,T , . . . , aT −1,T ) :       a1, 1 0 . 0 c1, 1 .. .. c1,T a1,T  a1, 2 a2, 2 0 .   ......   a2,T         ····   ····   ·  a1,T −1 . . aT −1,T −1 cT −1, 1 .. cT −1,T −1 aT −1,T       a1, 1 0 . 0 a1,T 0  a1, 2 a2, 2 0 .   a2,T  0 +aT,T     =   ,  ····   ·  · a1,T −1 . . aT −1,T −1 aT −1,T 0 which is equivalent to the equation       c1, 1 .. .. c1,T a1,T a1,T  ......   a2,T   a2,T  (28)     = −aT,T   .  ····   ·   ·  cT −1, 1 .. cT −1,T −1 aT −1,T aT −1,T Introduce the notation:   c1, 1 .. .. c1, k−2 c1, k k−1  ......  C :=   , k  ····  ck−1, 1 .. ck−1, k−2 ck−1, k

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 Dynamic inverse problem for Jacobi matrices 441

k−1 k−1 that is Ck is constructed from the matrix C by substituting the last column by (c1, k,..., ck−1, k). Then from (28) we deduce that: T −1 det CT (29) aT −1,T = −aT,T T −1 , det C −1 where we assumed that det C0 = 0. On the other hand, from (24), (25) we see that −1 T −1  T −1 Y X (30) aT −1,T =  aj bk. j=0 k=1 Equating (29) and (30) gives that

T −1 T −1 T T X det CT X det CT +1 bk = − T −1 , bk = − T , k=1 det C k=1 det C from where k k−1 det Ck+1 det Ck (31) bk = − k + k−1 , k = 1,...,T − 1. det C det C 3.3. Characterization of inverse data. In the second section the forward problem for (1) was considered: for (a0, . . . , aT −1), (b1, . . . , bT −1) we constructed the ma- T trix W (5), (6), the response vector (r0, r1, . . . , r2T −2) (see (7)) and the connecting T T operator C defined in (10), (22). From Lemma 2.4 we know that CT (and C ) is positively definite. We have also shown that if the vector (r0, r1, . . . , r2T −2) is a response vector corresponding to (1) with the coefficients a0, . . . , aT −1, b1, . . . , bT −1, 0 0 then one can recover those a s and b s by a0 = r0 and formulas (27) and (31). Now we set up the following question: can one determine whether a vector (r0, r1, r2, . . . , r2T −2) is a response vector for dynamical system (1) with some coefficients (a0, . . . , aT −1)(b1, . . . , bT −1)? The answer is the following theorem.

Theorem 3.2. The vector (r0, r1, r2, . . . , r2T −2) is a response vector for the dynamical system (1) if and only if the matrix CT defined by (10) is positive definite. Proof. Since the necessary part of the theorem is proved in the preceding sections we are left to prove the sufficiency part. We also observe that in the conditions of T the theorem we can substitute CT by C (22). T Let a vector (r0, r1, . . . , r2T −2) be such that the matrix C constructed from it with the help of (22) is positive definite. Then we can construct the sequences (a0, . . . , aT −1), (b1, . . . , bT −1) using a0 = r0 and formulas (27) and (31) and consider dynamical system (1) with this coefficients. For this system we construct the new new new T T response (r0 , r1 , . . . , r2T −2), the connecting operator C and corresponding C new new new using (10) and (22). We show that the response vector (r0 , r1 , . . . , r2T −2) co- incides with the given one (r0, r1, . . . , r2T −2). We have two matrices constructed by (22), one comes from the response vector new new new (r0, r1, . . . , r2T −2) and the other comes from (r0 , r1 , . . . , r2T −2). Both of them T T T are positive definite C > 0, C > 0 (C by the conditions of a theorem, and T C since it comes from a connecting operator CT ). We note that if we calculate the elements of sequences (a1, . . . , aT −1), (b1, . . . , bT −1) using (27) and (31) from

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 442 Alexandr Mikhaylov and Victor Mikhaylov

T T any of C and C matrices, we get the same answer. If so, we obtain that for k = 1,...,T − 1 the following relations hold:

k−1 k k−1 k det Ck det Ck+1 det Ck det Ck+1 k−1 − k = k−1 − k , det C det C det C det C 1 1 1 1 k+1 2 k−1 2 k+1 2 k−1 2 det C det C det C det C k = k , det C det C 0 0 −1 −1 det C = det C = 1, det C = det C = 1, −1 −1 det C0 = det C0 = 0. From these equalities we deduce that

k k det C = det C , k k det Ck+1 = det Ck+1. The above relations immediately yield that

new rk = rk , k = 1,..., 2T − 2. which ﬁnishes the proof.

3.4. Discrete Schrödingeroperator. We consider the special case of dynamical system (1): we assume that Jacobi matrix corresponds to a discrete Schrödinger operator, i.e. ak = 1, k ∈ N ∪ {0}, see also [17]. In this particular case the control operator (8) is given by W T = (I + K) J T , so all the diagonal elements of the matrix in (8) are equal to one. The latter immediately implies that det W T = 1. Due to this fact, the connecting operator (9), (10), has a remarkable property that det Ck = 1, k = 1,...,T. This implies that not all elements in the response vector are independent: r2m depends on r2l+1, l = 0, . . . , m − 1, moreover, this property characterize the dynamic data of discrete Schrödingeroperators:

Theorem 3.3. The vector (1, r1, r2, . . . , r2T −2) is a response vector for the dynam- T ical system (1) with ak = 1 if and only if the matrix C (10) is positive deﬁnite and det Cl = 1, l = 1,...,T .

T Proof. As in Theorem 3.2 we use C instead of CT . The necessity of the conditions was explained. We are left with the suﬃciency part. Note that r0 = a0 = 1. Let a vector (1, r1, . . . , r2T −2) be such that the ma- T trix C constructed from it using (22) satisﬁes conditions of the theorem. We calculate b1, . . . , bT −1 using (31) and consider the dynamical system (1) with these bk and ak = 1, k = 0,...,T − 1. For this system we construct the response new new T (1, r1 , . . . , r2T −2) and the connecting operator C using (7), (10) and (22). We new new will show that the response (1, r1 , . . . , r2T −2) coincide with the given one. T T We note that if we calculate b1, . . . , bT −1 using (31) with any of C or C matri- k k ces, we get the same answer. The latter implies (we count that det C = det C = 1)

k−1 k k−1 k det Ck − det Ck+1 = det Ck − det Ck+1, −1 −1 det C0 = det C0 = 0.

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 Dynamic inverse problem for Jacobi matrices 443

By the induction arguments we obtain that k k det Ck+1 = det Ck+1, k k det C = det C = 1, new which yields rk = rk , k = 1,..., 2T − 2. That ﬁnishes the proof.

T 4. Spectral representation of C and rt. We fix N ∈ N and h ∈ R. Along with (1) we consider the dynamical system with discrete time which is an analog of the wave equation with a potential on an interval: imposing a boundary condition at n = N + 1 we come to (32)   vn,t+1 + vn,t−1 − anvn+1,t − an−1vn−1,t − bnvn,t = 0, t ∈ N ∪ {0}, n ∈ 1,...,N, vn, −1 = vn, 0 = 0, n = 1, 2,...,N + 1,  v0, t = ft, vN+1, t + hvN, t = 0, t ∈ N0, f where f = (f0, f1,...) is a boundary control. The solution to (32) is denoted by v . Let φn(λ) be a solution to anφn+1 + an−1φn−1 + bnφn = λφn, φ0 = 0, φ1 = 1. N Denote by {λk}k=1 the roots of the equation φN+1(λ) + hφN (λ) = 0, it is known [2, 21] that they are real and distinct. We introduce the vectors φn ∈ RN by the n rule φi := φi(λn), n, i = 1,...,N, and define the numbers ρk by k l ρk (33) (φ , φ ) = δkl , a0 where (·, ·) is a scalar product in RN . Definition 4.1. The set of pairs N {λk, ρk}k=1 is called spectral data of operator HN (see (2)–(4)).

N Taking arbitrary y = (y1, . . . , yN ) ∈ R , for each n we multiply the ﬁrst equation in (32) by yn, sum up from n = 1 to n = N and evaluate, changing the order of summation: N X 0 = (vn, t+1yn + vn, t−1yn − anvn+1, tyn − an−1vn−1, tyn − bnvn, tyn) n=1 N X = (vn, t+1yn + vn, t−1yn − vn, t(an−1yn−1 + anyn+1) − bnvn, tyn) n=1

(34) −aN vN+1, tyN − a0v0, ty1 + a0v1, ty0 + aN vN, tyN+1. l l l Now we choose y = φ , l = 1 ...,N. On counting that φ0 = 0, φN+1 = −hφN , l φ1 = 1, v0, t = ft, vN+1, t + hvN, t = 0 we evaluate (34) arriving at N X l l l l l (35) vn,t+1φn + vn,t−1φn − vn,t an−1φn−1 + anφn+1 + bnφn − a0ft = 0. n=1 Let us look for a solution to (32) in a form PN k k f k=1 ct φn, n = 1,...,N, (36) vn,t = ft, n = 0.

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 444 Alexandr Mikhaylov and Victor Mikhaylov

Proposition 1. Coeﬃcients ck admit the representation: k a0 (37) c = T (λk) ∗ f, ρk where T (2λ) = (T1(2λ),T2(2λ),T3(2λ),...) are Chebyshev polynomials of the second kind. l l Proof. We plug (36) into (35) and evaluate, counting that an−1φn−1 + anφn+1 + l l bnφn = λlφn: N X l (vn, t+1 + vn, t−1 − λlvn, t) φn = a0ft, n=1 N N X X k k k k k k l ct+1φn + ct−1φn − λlct φn φn = a0ft. n=1 k=1 k Changing the order of summation and using (33) yields that ct for k = 1,...,N satisfy the following problem:

k k k a0 c + c − λkc = ft, t+1 t−1 t ρk (38) k k c−1 = c0 = 0. We look for the solution to (38) in a form ck = a0 T ∗ f : ρk t k a0 X (39) ct = Tlft−l. ρk l=0 Plugging this expression into (38) yields: t+1 t−1 t ! a0 X X X a0 flTt+1−l + flTt−1−l − λk flTt−l = ft, ρk ρk l=0 l=0 l=0 t X fl (Tt+1−l + Tt−1−l − λkTt−l) + ftT1 − ft−1T0 = ft. l=0 The above equality implies that (39) holds in the case if T solves Tt+1 + Tt−1 − λkTt = 0, T0 = 0,T1 = 1.

Thus Tk(2λ) are Chebyshev polynomials of the second kind and the formula (37) is proved.

T T N For the system (32) the control operator WN, h : F 7→ H is deﬁned by the rule T f WN, hf := vn, T , n = 1,...,N. The representation for this operator immediately follows from (36), (37). Due to the dependence of the solution on the coeﬃcients, which was discussed in the third f section (see Remark1), we see that vN,N does not “feel” the boundary condition at n = N, that is why f f N N (40) un, t = vn, t, n 6 t 6 N, and W = WN, h. T T T For the system (32) the response operator RN, h : F 7→ R is introduced by the rule (41) RT f = vf , t = 1,...,T. N, h t 1, t

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 Dynamic inverse problem for Jacobi matrices 445

T T T The connecting operator CN,h : F 7→ F is defined via the quadratic form: for arbitrary f, g ∈ F T we have that T f g T T CN, hf, g T = v·,T , v·,T = WN, hf, WN, hg N . F HN H f The dependence of the solution (1) u on an, bn (see Remark1) in particular implies that for M ∈ N, 2M 2M (42) R = RM, h. f i.e. v1, 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 (41) is (43) rN,h = RT δ = vδ , t = 1,...,T. t−1 N, h t 1, t on the other hand, we can use (36), (37) to obtain: N δ X a0 (44) v1, t = Tt(λk). ρk k=1 So on introducing the spectral function X a0 ρN, h(λ) = , ρk {k | λk<λ} from (43), (44) we deduce that Z ∞ N,h N,h rt−1 = Tt(λ) dρ (λ), t ∈ N. −∞ Due to (42), we have that Z ∞ N,h N,h (45) rt−1 = rt−1 = Tt(λ) dρ (λ), t ∈ 1,..., 2N. −∞ Taking in (45) N to infinity, we come to the spectral representation of the response vector: Z ∞ rt−1 = Tt(λ) dρ(λ), t ∈ N, −∞ where dρ is a spectral measure of the operator H (2), (3) (non-unique when H is in the limit-circle case at infinity). For f, g ∈ F T we can evaluate, using the expansion (36): N N N N T X f g X X a0 k X a0 l (CN, hf, g) = vn, T vn, T = TT (λk) ∗ fϕn TT (λl) ∗ gϕn ρk ρl n=1 n=1 k=1 l=1 N Z ∞ T −1 T −1 X a0 X X N, h = TT (λk) ∗ fTT (λk) ∗ g = TT −l(λ)fl TT −m(λ)gm dρ (λ) ρk k=1 −∞ l=0 m=0 from the above equality the spectral representation of CT (cf. (10)) follows: Z ∞ T N, h (46) {CN, h}l+1, m+1 = TT −l(λ)TT −m(λ) dρ (λ), l, m = 0,...,T − 1. −∞ T T Taking into account (40), we obtain that C = CN, h with N > T , so (46) yields for N > T : Z ∞ T N,h {C }l+1, m+1 = TT −l(λ)TT −m(λ) dρ (λ), l, m = 0,...,T − 1, −∞

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 446 Alexandr Mikhaylov and Victor Mikhaylov and passing to the limit as N → ∞ yields Z ∞ T {C }l+1,m+1 = TT −l(λ)TT −m(λ) dρ(λ), l, m = 0,...,T − 1, −∞ where dρ is a spectral measure of H. Is is known [13] that any probability measure on R with finite moments give rise to the Jacobi operator, i.e. is a spectral measure of this operator. In [13] the authors posed the question on the characterization of a spectral measure for semi- infinite discrete Schrödingeroperator. The following theorem gives an answers to this question Theorem 4.2. The measure dρ is a spectral measure of a semi-infinite discrete Schrödingeroperator if and only if for every T the matrix CT is positive definite and det CT = 1, where Z ∞ T (47) Ci,j = TT −i+1(λ)TT −j+1(λ) dρ(λ), i, j = 1,...,T. −∞

Proof. We consider the system (1) with ak = 1. Let dρ be a spectral measure of H (note that this measure is unique since H is in limit point at infinity case). For every T ∈ N we construct the connecting operator CT (see (9)) using the representation (47). According to Theorem 3.3, such CT is positive definite and det CT = 1. On the other hand, if given measure dρ satisfies conditions of the theorem, for T every T we can construct C by (47) and by Theorem 3.3 recover coefficients bn using (31).

Acknowledgments. The research of Victor Mikhaylov was supported by RFBR 17-01-00529 and by the Ministry of Education and Science of Republic of Kaza- khstan under grant AP05136197. Alexandr Mikhaylov was supported by RFBR 17-01-00099; A. S. Mikhaylov and V. S. Mikhaylov were partly supported by and by VW Foundation program “Modeling, Analysis, and Approximation Theory to- ward application in tomography and inverse problems.”

REFERENCES

[1] S. Avdonin and A. Bulanova, Boundary control approach to the spectral estimation problem. The case of multiple poles, Math. Contr. Sign. Syst., 22 (2011), 245–265. [2] F. V. Atkinson, Discrete and Continuous Boundary Problems, Acad. Press, 1964. [3] S. Avdonin, A. Bulanova and D. Nicolsky, Boundary control approach to the spectral estimation problem. The case of simple poles, Sampling Theory in Signal and Image Processing, 8 (2009), 225–248. [4] S. Avdonin and V. S. Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 045009, 19 pp. [5] S. Avdonin, V. Mikhaylov and A. Rybkin, The boundary control approach to the Titchmarsh- Weyl m−function, Comm. Math. Phys., 275 (20017), 791–803. [6] M. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1–R67. [7] M. Belishev, Boundary control and tomography of Riemannian manifolds (the BC-method), Uspekhi Matem. Nauk., 72 (2017), 3–66, (in Russian). [8] M. Belishev, Boundary control and inverse problems: A one-dimensional version of the boundary control method, J. Math. Sci. (N.Y.), 155 (2008), 343–378. [9] M. Belishev and S. Ivanov, Characterization of data of dynamical inverse problem for two- velocity system, J. Math. Sci. (N.Y.), 109 (2002), 1814–1834. [10] M. Belishev and V. Mikhaylov, Uniﬁed approach to classical equations of inverse problem theory, Journal of Inverse and Ill-Posed Problems, 20 (2012), 461–488.

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447 Dynamic inverse problem for Jacobi matrices 447

[11] M. Belishev and V. Mikhaylov, Inverse problem for one-dimensional dynamical Dirac system (BC-method), Inverse Problems, 30 (2014), 125013, 26 pp. [12] M. Belishev and A. Pestov, Characterization of the inverse problem data for one-dimensional two-velocity dynamical system, St. Petersburg Mathematical Journal, 26 (2015), 411–440. [13] F. Gesztesy and B. Simon, m−functions and inverse spectral analisys for finite and semi- infinite Jacobi matrices, J. d’Analyse Math., 73 (1997), 267–297. [14] V. Isakov, Inverse Problems for Partial Differential Equations, Appl. Math. Studies, Springer, v. 127, 1998. [15] S. I. Kabanikhin, Inverse and Ill-Posed Problems. Theory and Applications, Inverse and Ill- posed Problems Series, 55. Walter de Gruyter GmbH & Co. KG, Berlin, 2012. [16] A. Kachalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chap- man&Hall, 2001. [17] A. Mikhaylov and V. Mikhaylov, Dynamical inverse problem for the discrete Schrödinger operator, Nanosystems: Physics, Chemistry, Mathematics, 7 (2016), 842–854. [18] A. Mikhaylov and V. Mikhaylov, Boundary Control method and de Branges spaces. Schrödingeroperator, Dirac system, discrete Schrödingeroperator, Journal of Mathemati- cal Analysis and Applications, 460 (2018), 927–953. [19] A. Mikhaylov and V. Mikhaylov, Relationship between different types of inverse data for the one-dimensional Schrödingeroperator on a half-line, J. Math. Sci. (N.Y.), 226 (2017), 779–794. [20] A. Mikhaylov, V. Mikhaylov and S. Simonov, On the relationship between Weyl functions of Jacobi matrices and response vectors for special dynamical systems with discrete time, Mathematical Methods in the Applied Sciences, 41 (2018), 6401–6408. [21] B. Simon, The classical moment problem as a self-adjoint finite difference operator, Advances in Math., 137 (198), 82–203. Received October 2017; revised October 2018. E-mail address: [email protected] E-mail address: [email protected]

Inverse Problems and Imaging Volume 13, No. 3 (2019), 431–447