RETRACTED On construction of a complex finite Jacobi matrix from two spectra

Electronic Journal of Linear Algebra Volume 26 Volume 26 (203) Article 8 203 On construction of a complex finite Jacobi matrix from two spectra Gusein Sh. Guseinov guseinov@ati.edu.tr Follow this and additional wors at: http://repository.uwyo.edu/ela Recommended Citation Guseinov, Gusein Sh.. (203), " On construction of a complex finite Jacobi matrix from two spectra", Electronic Journal of Linear Algebra, Volume 26. DOI: https://doi.org/0.300/08-380.642 This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact scholcom@uwyo.edu.

ON CONSTRUCTION OF A COMPLEX FINITE JACOBI MATRIX FROM TWO SPECTRA GUSEIN SH. GUSEINOV Abstract. This paper concerns with the inverse spectral problem for two spectra of finite order complex Jacobi matrices (tri-diagonal symmetric matrices with complex entries). The problem is to reconstruct the matrix using two sets of eigenvalues, one for the original Jacobi matrix and one for the matrix obtained by replacing the last diagonal element of the Jacobi matrix by some other number. The uniqueness and existence results for solution of the inverse problem are established and an explicit procedure of reconstruction of the matrix from the two spectra is given. Key words. Jacobi matrix, Difference equation, Eigenvalue, Normalizing numbers, Inverse spectral problem. AMS subject classifications. 65F8.. Introduction. Let J be an N N Jacobi matrix of the form b 0 a 0 0 0 0 0 a 0 b a 0 0 0 0 a b 2 0 0 0 J =. (.)........, 0 0 0... b N 3 a N 3 0 0 0 0 a N 3 b N 2 a N 2 0 0 0 0 a N 2 b N where for each n, a n and b n are arbitrary complex numbers such that a n is different from zero: (.2) a n,b n C, a n 0. A distinguishing feature of the Jacobi matrix (.) from other matrices is that the eigenvalue problem Jy = λy for a column vector y = {y n } N n=0 is equivalent to the second order linear difference equation (.3) a n y n +b n y n +a n y n+ = λy n, Received by the editors on September 3, 202. Accepted for publication on January 2, 203. Handling Editor: Raphael Loewy. Department of Mathematics, Ati University, 06836 Ince, Anara, Turey (guseinov@ati.edu.tr). 0

02 G.Sh. Guseinov n {0,,...,N }, a = a N =, for {y n } N n=, with the boundary conditions (.4) y = y N = 0. This allows, using techniques from the theory of three-term linear difference equations [4], to develop a thorough analysis of the eigenvalue problem Jy = λy. If J has distinct eigenvalues λ,...,λ p with multiplicities m,...,m p and for some (.5) the matrix (.6) bn b N, b 0 a 0 0 0 0 0 a 0 b a 0 0 0 0 a b 2 0 0 0 J =............... 0 0 0... b N 3 a N 3 0 0 0 0 a N 3 b N 2 a N 2 0 0 0 0 a N 2 bn (in which all a n and b n are the same as in J, except b N which is replaced by b N ) has distinct eigenvalues λ,..., λ q with multiplicities n,...,n q, then the collections (.7) {λ,m ( =,...,p)} and { λ i,n i (i =,...,q)} are called the two spectra (or the two-spectra) of J. The inverse problem about two spectra consists in reconstruction of the matrix J from its two spectra. This problem consists of the following parts: (i) To elucidate the uniqueness problem consisting of whether the matrix J and the number b N are determined uniquely by the two spectra of J. (ii) To find necessary and sufficient conditions for two collections of numbers in (.7) to be the two spectra for some matrix of the form (.) with entries from class (.2). (iii) To indicate an algorithm for the construction of the matrix J and the number bn from the two spectra of J. For real finite Jacobi matrices, this problem was completely solved by the author in [2]. Note that in case of real entries the finite Jacobi matrix is selfadjoint and its eigenvalues are real and distinct. In the complex case, however, the Jacobi matrix is, in general, no longer selfadjoint and its eigenvalues may be complex and multiple.

On Construction of a Complex Finite Jacobi Matrix from Two Spectra 03 In the present paper we show, by reducing the inverse problem about two spectra to the inverse problem about spectral data consisting of the eigenvalues and normalizing numbers of the matrix, that the complex Jacobi matrix and the number b N are determined from two spectra given in (.7) uniquely up to signs of the off-diagonal elements of the matrix. We indicate also necessary and sufficient conditions for two collections of numbers of the form given in (.7) to be two spectra of a Jacobi matrix J of the form (.) with entries belonging to the class (.2), as well as a procedure of reconstruction of J and b N from the two spectra. For given collections in (.7), assuming that λ λ i for all possible values of and i, and that q (.8) m λ n i λi, (.9) (m )! we define the number a by (.0) a = (m )! and construct the numbers (.) Then we set (.2) a β j = (m j)! s l = d m dλ m l=,l d m dλ m d m j dλ m j (λ λ l ) m l l=,l l=,l 0, q (λ λ i ) ni (λ λ l ) m l (λ λ l ) m l (j =,...,m ; =,...,p). q (λ λ i ) ni q (λ λ i ) ni m ( ) l β j λ l j+, l = 0,,2,..., j j= where ( ) ( l j is a binomial coefficient and we put l j ) = 0 if j > l. Using these numbers we introduce the determinants s 0 s s n s s 2 s n+ (.3) D n =.., n = 0,,2,....... s n s n+ s 2n

04 G.Sh. Guseinov The main result of this paper is the following theorem. Theorem.. Let two collections of numbers in (.7) be given, where λ,...,λ p are distinct complex numbers with p {,...,N} and m,...,m p are positive integers such that m + + m p = N; the λ,..., λ q are distinct complex numbers with q {,...,N} and n,...,n q are positive integers such that n + + n q = N. In order for these collections to be two spectra for a Jacobi matrix J of the form (.) with entries belonging to the class (.2), it is necessary and sufficient that the following two conditions are satisfied: (i) λ λ i for all {,...,p},i {,...,q}, and (.8), (.9) hold; (ii) D n 0, for n {,2,...,N }, where D n is the determinant defined by (.3), (.2), (.), (.0). Under the conditions (i) and (ii) the entries a n and b n of the matrix J for which the collections in (.7) are two spectra, are recovered by the formulae (.4) (.5) a n = ± D n D n+, n {0,,...,N 2}, D =, D n b n = n n, n {0,,...,N }, = 0, 0 = s, D n D n where D n is defined by (.3), (.2), (.), (.0), and n is the determinant obtained from the determinant D n by replacing in D n the last column by the column with the components s n+,s n+2,...,s 2n+. Further, the element b N of the matrix J corresponding to the matrix J and defined by (.6) is determined by the formula (.6) bn = b N + q n i λi m λ. It follows from the above solution of the inverse problem about two spectra that the matrix (.) is not uniquely restored from the two spectra. This is lined with the fact that the a n are determined from (.4) uniquely up to a sign. To ensure that the inverse problem is uniquely solvable, we have to specify additionally a sequence of signs + and. Namely, let {σ 0,σ,...,σ N 2 } be a given finite sequence, where for each n {0,,...,N 2}the σ n is+or. We have2 N such differentsequences. Nowto determine a n uniquely from (.4) for n {0,,...,N 2} we can choose the sign σ n when extracting the square root. In this way, we get precisely 2 N distinct Jacobi matrices possessing the same two spectra. The inverse problem is solved uniquely from the data consisting of the two spectra and a sequence {σ 0,σ,...,σ N 2 } of signs + and. Thus, we can say that the inverse problem with respect to the two

On Construction of a Complex Finite Jacobi Matrix from Two Spectra 05 spectra is solved uniquely up to signs of the off-diagonal elements of the recovered Jacobi matrix. The paper is organized as follows. Section 2 is auxiliary and presents a solution of the inverse spectral problem for complex finite Jacobi matrices in terms of the eigenvalues and normalizing numbers. Lastly, Section 3 presents the solution of the inverse problem for complex finite Jacobi matrices in terms of the two spectra. 2. Inverse problem with respect to eigenvalues and normalizing numbers. In the sequel, we will use the following well-nown useful lemma. We bring it here for easy reference. Lemma 2.. Let A(λ) and B(λ) be polynomials with complex coefficients and dega < degb = N. Suppose that B(λ) = b(λ z ) m (λ z p ) mp, where z,...,z p are distinct complex numbers, b is a nonzero complex number, and m,...,m p are positive integers such that m + +m p = N. Then there exist uniquely determined complex numbers a j (j =,...,m ; =,...,p) such that (2.) A(λ) m B(λ) = a j (λ z ) j j= for all values of λ different from z,...,z p. The numbers a j are given by the equation d m j [ a j = (m j)! λ z dλ m (λ z ) m A(λ) ] (2.2), j B(λ) (2.3) j =,...,m ; =,...,p. Proof. For each {,...,p}, we have A(λ) B(λ) = C (λ) (λ z ) m, where the function C (λ) = (λ z ) m A(λ) B(λ) A(λ) = b(λ z ) m (λ z ) m (λ z+ ) m + (λ zp ) mp is regular (analytic) at z. We can expand C (λ) into a Taylor series about the point z, (2.4) C (λ) = d s (λ z ) s, s=0

06 G.Sh. Guseinov where d s = C(s) (z ), s = 0,,2,... s! Substituting (2.4) in (2.3) we get that near z, (2.5) Consider the function (2.6) m A(λ) B(λ) = d s (λ z ) m s +(a Taylor series about z ). s=0 By (2.5) and (2.6), near z l, Φ(λ) = A(λ) m B(λ) d s (λ z ) m s. s=0 Φ(λ) = (a Taylor series about z l ), l m s=0 d s (λ z ) m s and this is analytic at z l (l =,...,p). Therefore, the function Φ(λ) is analytic everywhere, that is, Φ(λ) is an entire function. Next, since dega < degb, Φ(λ) 0 as λ. Thus, the entire function Φ(λ) is bounded and tends to zero as λ. By the well-nown Liouville theorem, we conclude that Φ(λ) 0. Thus, we have and m d s (λ z s=0 ) m s = m d,m j j= A(λ) B(λ) = d,m j = C(m j) (z ) = (m j)! (m j)! d m j λ z dλ m j (λ z ) j [ (λ z ) m A(λ) ]. B(λ) These prove (2.) and (2.2). Note that decomposition (2.) is unique as for the a j in this decomposition Eq. (2.2) necessarily holds. Now it is easy to get the following important for us consequence of this lemma. Lemma 2.2. Let A(λ) be a polynomial in λ with complex coefficients and dega N, where N is a positive integer. Next, let for some p {,...,N},z,...,z p be distinct complex numbers and m,...,m p be positive integers such that m + +m p = N. Further, suppose that (2.7) A (j) (z ) = 0 for j = 0,,...,m ; =,...,p.

On Construction of a Complex Finite Jacobi Matrix from Two Spectra 07 Then there is a constant c such that (2.8) A(λ) = c(λ z ) m (λ z p ) mp, where c = 0 if dega < N and c 0 if dega = N. (2.9) Proof. Let us set B(λ) = (λ z ) m (λ z p ) mp. If dega < N, then applying Lemma 2. to A(λ)/B(λ) we can write (2.), (2.2). From condition (2.7) and Eq. (2.2) it follows that a j = 0 (j =,...,m ; =,...,p). Then (2.) gives A(λ)/B(λ) = 0 for all values of λ different from z,...,z p, that is, A(λ) 0. Thus, (2.8) holds with c = 0 if dega < N. Let now dega = N. Then dividing A(λ) by B(λ) with a reminder term, we can write (2.0) A(λ) = cb(λ)+r(λ), where c is a nonzero complex number and R(λ) is a polynomial of degr < N. It follows from (2.7), (2.9), and (2.0) that R (j) (z ) = 0 for j = 0,,...,m ; =,...,p. Then by the first part of the lemma we get that R(λ) 0 and (2.0) taes the form (2.8). Next in this section, we follow the author s paper []. Given a Jacobi matrix J of the form (.) with the entries (.2), consider the eigenvalue problem Jy = λy for a column vector y = {y n } N n=0, that is equivalent to the problem (.3), (.4). Denote by {P n (λ)} N n= and {Q n(λ)} N n= the solutions of Eq. (.3) satisfying the initial conditions (2.) (2.2) P (λ) = 0, P 0 (λ) = ; Q (λ) =, Q 0 (λ) = 0. For each n 0, P n (λ) is a polynomial of degree n and is called a polynomial of first ind and Q n (λ) is a polynomial of degree n and is nown as a polynomial of second ind. These polynomials can be found recurrently from Eq. (.3) using initial

08 G.Sh. Guseinov conditions (2.) and (2.2). The leading terms of the polynomials P n (λ) and Q n (λ) have the forms (2.3) The equality λ n λ n P n (λ) = +, Q n (λ) = +. a 0 a a n a 0 a a n det(j λi) = ( ) N a 0 a a N 2 P N (λ) holds (see [, 2]) so that the eigenvalues of the matrix J coincide with the zeros of the polynomial P N (λ). (2.4) Further, the identity holds (see [, Lemma 4]). P N (λ)q N (λ) P N (λ)q N (λ) = Let R(λ) = (J λi) be the resolvent of the matrix J (by I we denote the identity matrix of needed dimension) and e 0 be the N-dimensional column vector with the components,0,...,0. The rational function (2.5) w(λ) = R(λ)e 0,e 0 = (λi J) e 0,e 0, introduced earlier in [3], we call the resolvent function of the matrix J, where, denotes the standard inner product in C N. This function is nown also as the Weyl- Titchmarsh function of J. The entries R nm (λ) of the matrix R(λ) = (J λi) (resolvent of J) are of the form { Pn (λ)[q m (λ)+m(λ)p m (λ)], 0 n m N, (2.6) R nm (λ) = P m (λ)[q n (λ)+m(λ)p n (λ)], 0 m n N, (see [, 2]) where (2.7) M(λ) = Q N(λ) P N (λ). According to (2.5), (2.6), (2.7) and using initial conditions (2.), (2.2), we get (2.8) w(λ) = R 00 (λ) = M(λ) = Q N(λ) P N (λ). By (2.3), we have P N (λ) = c(λ λ ) m (λ λ p ) mp,

On Construction of a Complex Finite Jacobi Matrix from Two Spectra 09 where c is a nonzero constant and λ,...,λ p are distinct eigenvalues of J with multiplicities m,...,m p. Therefore, we can decompose the rational function w(λ) expressed by (2.8) into partial fractions (Lemma 2.) to get (2.9) where (2.20) w(λ) = β j = (m j)! m j= d m j dλ m j β j (λ λ ) j, (j =,...,m ; =,...,p) are called the normalizing numbers of the matrix J. (2.2) [ (λ λ ) m Q ] N(λ) P N (λ) The collection of the eigenvalues and normalizing numbers {λ,β j(j =,...,m ; =,...,p)}, of the matrix J of the form (.), (.2) is called the spectral data of this matrix. Thus, the spectral data consist of the eigenvalues and associated normalizing numbers derived by decomposing the resolvent function (Weyl-Titchmarsh function) w(λ) into partial fractions using the eigenvalues. Determination of the spectral data of a given Jacobi matrix is called the direct spectral problem for this matrix. The inverse spectral problem consists in reconstruction of the matrix J from its spectral data. This problem was solved by the author in [] and we will present here the final result. (2.22) Let us set s l = m ( ) l j j= β jλ l j+, l = 0,,2,..., where ( ) ( l j is a binomial coefficient and we put l j ) = 0 if j > l. Next, using these numbers s l we introduce the determinants s 0 s s n D n s s 2 s n+ (2.23) =.., n = 0,,2,....... s n s n+ s 2n

0 G.Sh. Guseinov Let us bring an important property of the determinants D n in the form of a lemma. Lemma 2.3. Given any collection (2.2), for the determinants D n defined by (2.23), (2.22), we have D n = 0 for n N, where N = m + +m p. Proof. Given a collection (2.2), define a linear functional Ω on the linear space of all polynomials in λ with complex coefficients as follows: if G(λ) is a polynomial then the value Ω,G(λ) of the functional Ω on the element (polynomial) G is (2.24) Ω,G(λ) = Let m 0 be a fixed integer and set (2.25) Then, due to (2.24), (2.26) m G (j ) (λ ) β j. (j )! j= T(λ) = λ m (λ λ ) m (λ λ p ) mp = t m λ m +t m+ λ m+ + +t m+n λ m+n +λ m+n. Ω,λ l T(λ) = 0, l = 0,,2,... Consider (2.26) for l = 0,,2,...,N +m, and substitute (2.25) in it for T(λ). Taing into account that where s l is defined by (2.22), we get Ω,λ l = s l, l = 0,,2,..., t m s l+m +t m+ s l+m+ + +t m+n s l+m+n +s l+m+n = 0, l = 0,,2,...,N +m. Therefore, (0,...,0,t m,t m+,...,t m+n,) is a nontrivial solution of the homogeneous system of linear algebraic equations x 0 s l +x s l+ + +x m s l+m +x m+ s l+m+ + +x m+n s l+m+n +x m+n s l+m+n = 0, l = 0,,2,...,N +m, with the unnowns x 0,x,...,x m,x m+,...,x m+n,x m+n. Therefore, the determinant of this system, which coincides with D N+m, must be zero.

On Construction of a Complex Finite Jacobi Matrix from Two Spectra The solution of the above inverse problem with respect to eigenvalues and normalizing numbers is given by the following theorem (see Theorem 6 in []). Theorem 2.4. Let an arbitrary collection (2.2) of numbers be given, where p N,m,...,m p are positive integers with m + + m p = N,λ,...,λ p are distinct complex numbers. In order for this collection to be the spectral data for a Jacobi matrix J of the form (.) with entries belonging to the class (.2), it is necessary and sufficient that the following two conditions are satisfied: (i) p β = ; (ii) D n 0, for n {,2,...,N }, where D n is the determinant defined by (2.23), (2.22). Under the conditions (i) and (ii) the entries a n and b n of the matrix J for which the collection (2.2) is spectral data, are recovered by the formulae (2.27) (2.28) a n = ± D n D n+, n {0,,...,N 2}, D =, b n = n D n D n n D n, n {0,,...,N }, = 0, 0 = s, where D n is defined by (2.23), (2.22), and n is the determinant obtained from the determinant D n by replacing in D n the last column by the column with the components s n+,s n+2,...,s 2n+. Note that the condition (ii) of Theorem 6 in [] contains an extra condition which requires that D N = 0. However, by Lemma 2.3 of the present paper, that condition is fulfilled automatically and is therefore redundant. Concerning the formulas (2.27) and (2.28) for a n and b n, respectively, see formulas (3.35) and (3.36) in []. It follows from (2.27) and (2.28) that the inverse problem with respect to the spectral data is solved uniquely up to signs of the off-diagonal elements of the recovered Jacobi matrix. Let us remar that in the case of arbitrary real distinct numbers λ,...,λ N and positivenumbersβ,...,β N thecondition(ii)oftheorem2.4issatisfiedautomatically and in this case, we have D n > 0, for n {,2,...,N }; see [2, Lemma 7]. However, in the complex case the condition (ii) of Theorem 2.4 need not be satisfied automatically. Indeed, let N = 3 and as the collection (2.2) we tae {λ,λ 2,λ 3,β,β 2,β 3}, where λ,λ 2,λ 3,β,β 2,β 3 are arbitrary complex numbers such that λ λ 2, λ λ 3, λ 2 λ 3,

2 G.Sh. Guseinov We have β 0, β 2 0, β 3 0, β +β 2 +β 3 =. s l = β λ l +β 2λ l 2 +β 3λ l 3, l = 0,,2,..., and it is not difficult to show that D = s 0 s s s 2 = β β 2(λ λ 2 ) 2 +β β 3(λ λ 3 ) 2 +β 2β 3(λ 2 λ 3 ) 2, s D 2 0 s s 2 = s s 2 s 3 s 2 s 3 s = β β 2β 3(λ λ 2 ) 2 (λ λ 3 ) 2 (λ 2 λ 3 ) 2, 4 0 = s = β λ +β 2 λ 2 +β 3 λ 3, = s 0 s 2 s s = β β 2(λ +λ 2 )(λ λ 2 ) 2 3 +β β 3 (λ +λ 3 )(λ λ 3 ) 2 +β 2 β 3 (λ 2 +λ 3 )(λ 2 λ 3 ) 2, s 0 s s 3 2 = s s 2 s 4 s 2 s 3 s = β β 2β 3 5 λ λ 2 λ 3 λ 2 λ 2 2 λ 2 3 λ λ 2 λ 3 λ 3 λ 3 2 λ 3. 3 We see that the condition D 0 is not satisfied automatically, and therefore, one must require D 0 as a condition. For example, if then we get D = 0. β = β 2 = β 3 = 3, λ = ±i 3, λ 2 =, λ 3 = 0, 2 3. Construction from two spectra. Let J be an N N Jacobi matrix of the form (.) with entries satisfying (.2). Define J to be the Jacobi matrix given by (.6), where the number b N satisfies (.5). Denote by λ,...,λ p all the distinct eigenvalues of the matrix J and by m,...,m p their multiplicities, respectively, as the

On Construction of a Complex Finite Jacobi Matrix from Two Spectra 3 roots of the characteristic polynomial det(j λi) so that p N,m + +m p = N, and (3.) det(λi J) = (λ λ ) m (λ λ p ) mp. Further, denote by λ,..., λ q all the distinct eigenvalues of the matrix J and by n,...,n q their multiplicities, respectively, as the roots of the characteristic polynomial det( J λi) so that q N,n + +n q = N, and (3.2) det(λi J) = (λ λ ) n (λ λ q ) nq. Recallthatwecallthecollections{λ,m ( =,...,p)}and{ λ i,n i (i =,...,q)} the two spectra of the matrix J. Our goal in this section is to prove Theorem. which solves the inverse problem for two spectra, consisting in the reconstructionof the matrix J and the number b N from the two spectra of J. We will reduce the inverse problem for two spectra to the inverse problem for eigenvalues and normalizing numbers solved above in Section 2. First let us study some necessary properties of the two spectra of the Jacobi matrix J. Let P n (λ) and Q n (λ) be the polynomials of the first and second ind for the matrix J. The similar polynomials for the matrix J we denote by P n (λ) and Q n (λ). By (2.3), we have (3.3) (3.4) det(j λi) = ( ) N a 0 a a N 2 P N (λ), ) det ( J λi = ( ) N a 0 a a N 2 PN (λ), so that the eigenvalues λ,...,λ p and λ,..., λ q of the matrices J and J and their multiplicities coincide with the zerosand their multiplicities ofthe polynomials P N (λ) and P N (λ), respectively. Lemma 3.. For the resolvent function w(λ) of the matrix J, defined by (2.5), the following formula holds: (3.5) w(λ) = Q N (λ) P N (λ) + b N b N P N (λ) P N (λ). Proof. Comparing the matrices J and J defined by (.) and (.6), respectively, we can see that P n (λ) = P n (λ), n {,0,,...,N },

4 G.Sh. Guseinov (3.6) (3.7) and (3.8) (3.9) a N 2 P N 2 (λ)+b N P N (λ)+p N (λ) = λp N (λ), a N 2 P N 2 (λ)+ b N P N (λ)+ P N (λ) = λp N (λ), Q n (λ) = Q n (λ), n {,0,,...,N }, a N 2 Q N 2 (λ)+b N Q N (λ)+q N (λ) = λq N (λ), a N 2 Q N 2 (λ)+ b N Q N (λ)+ Q N (λ) = λq N (λ). Subtracting (3.6) and (3.7), and also (3.8) and (3.9), we get Hence (3.0) P N (λ) P N (λ) = (b N b N )P N (λ), Q N (λ) Q N (λ) = (b N b N )Q N (λ). P N (λ) = P N (λ) P N (λ), Q N (λ) = Q N (λ) Q N (λ). b N b N b N b N Replacing P N (λ) and Q N (λ) in the identity (2.4) by their expressions from (3.0), we get (3.) P N (λ)q N (λ) P N (λ) Q N (λ) = b N b N. Dividing both sides by P N (λ) P N (λ) gives Q N (λ) P N (λ) Q N (λ) P N (λ) = b N b N P N (λ) P N (λ). Therefore, by formula (2.8) for the resolvent function w(λ), we obtain (3.5). Lemma 3.2. The matrices J and J have no common eigenvalues, that is, λ λ i for all values of and i. Proof. Suppose that λ is an eigenvalue of the matrices J and J. Then by (3.3) and (3.4), we have P N (λ) = P N (λ) = 0. But this is impossible by (3.) and the condition b N b N.

On Construction of a Complex Finite Jacobi Matrix from Two Spectra 5 The following lemma allows us to calculate the difference b N b N in terms of the two spectra. (3.2) holds. Lemma 3.3. The equality (trace formula) m λ q n i λi = b N b N Proof. For any matrix A = [a j ] N j, the spectral trace of A coincides with the matrix trace of A: if µ,...,µ p are the distinct eigenvalues of A of multiplicities m,...,m p as the roots of the characteristic polynomial det(a λi), then Indeed, this follows from m µ = N a. det(λi A) = (λ µ ) m (λ µ p ) mp by comparison of the coefficients of λ N on the two sides. Therefore, we can write m λ = b 0 +b + +b N 2 +b N, q n i λi = b 0 +b + +b N 2 + b N. Subtracting the last two equalities side by side we arrive at (3.2). The following statement follows from Lemma 3.3. Corollary 3.4. Under the condition (.5), m λ q n i λi 0. The following lemma gives a formula for calculating the normalizing numbers β j (j =,...,m ; =,...,p) in terms of the two spectra. Lemma 3.5. Under the condition (.5), the statement in (.9) is valid and if we define the number a by (.0), then for each {,...,p} and j {,...,m } the

6 G.Sh. Guseinov formula (3.3) holds. β j = a (m j)! d m j dλ m j l=,l (λ λ l ) m l q (λ λ i ) ni Proof. Substituting (2.9) in the left-hand side of (3.5) established in Lemma 3., we can write m j= β j Hence, we get, using (3.3), (3.4), β j = (3.4) (m j)! where (λ λ ) j = Q N (λ) P N (λ) + b N b N P N (λ) P N (λ). d m j = b N b N (m j)! = c(b N b N ) (m j)! dλ m j d m j d m j dλ m j { [ ]} QN (λ λ ) m (λ) P N (λ) + b N b N P N (λ) P N (λ) dλ m j [ ] (λ λ ) m P N (λ) P N (λ) (λ λ ) m ( det(λi J)det λi J ), c = (a 0 a a N 2 ) 2. We have used the fact that since P N (λ ) 0 by Lemma 3.2, { } d m j Q dλ m (λ λ ) m N (λ) = 0 (j =,...,m ). j P N (λ) Substituting (3.) and (3.2) in (3.4), we get (3.5) where β j = a (m j)! d m j dλ m j l=,l a = c(b N b N ). (λ λ l ) m l, q (λ λ i ) ni

On Construction of a Complex Finite Jacobi Matrix from Two Spectra 7 Next, putting j = in Eq. (3.5) and then summing this equation over =,...,p and taing into account the condition (i) of Theorem 2.4, we get = a (m )! d m dλ m l=,l (λ λ l ) m l Hence, (.9) and a = a follow and (3.5) coincides with (3.3). The following statement follows from Lemma 3.5. Corollary 3.6. We have. q (λ λ i ) ni β j = β j and hence s l = s l, D n = D n, where β j,s l,d n are defined by (2.20), (2.22), (2.23) and β j,s l,d n by (.), (.2), (.3). Let us now prove Theorem. stated in the Introduction. Proof. The necessity of the conditions (i) and (ii) of Theorem. follows from Lemmas 3.2, 3.5, Corollary 3.4, and Theorem 2.4 by Corollary 3.6. To prove sufficiency suppose that two collections of numbers in (.7) are given which satisfy the conditions of Theorem.. We construct β j (j =,...,m ; =,...,p) by (.), (.0). It follows directly that β =. Thus, the collection {λ,β j (j =,...,m ; =,...,p)} satisfies conditions (i) and (ii) of Theorem 2.4 (with β j replaced by β j), and hence, there exists a Jacobi matrix J of the form (.) with entries from the class (.2) such that λ are the eigenvalues of the multiplicity m and β j are the corresponding normalizing numbers for J. Having the matrix J, in particular, its entry b N, we construct the number b N by (3.6) bn = b N + q n i λi m λ and then the matrix J by (.6) according to the matrix J and (3.6). Note that by the condition (.8), we have (3.7) bn b N.

8 G.Sh. Guseinov Itremainstoshowthat λ i aretheeigenvaluesoftheconstructedmatrix J,ofmultiplicity n i. To do this we denote the eigenvalues of J by µ,..., µ s and their multiplicities by ñ,...,ñ s. We have to show that s = q, µ i = λ i,ñ i = n i (i =,...,q). Let us set f(λ) = (λ λ l ) m l, g(λ) = l= q (λ λ i ) ni, h(λ) = s (λ µ i. )ñi By the direct problem, we have (Lemma 3.5) ã d m j [ ] β j = (m j)! dλ m (λ λ ) m (3.8), j f(λ)h(λ) where ã = (m )! d m dλ m l=,l (λ λ l ) m l. s (λ µ i )ñi On the other hand, by our construction of β j, we have (.), (.0) which can be written in the form a d m j [ ] β j = (m j)! dλ m (λ λ ) m (3.9), j f(λ)g(λ) where a = (m )! d m dλ m l=,l Subtracting (3.8) and (3.9) side by side we get (3.20) where (3.2) (3.22) d m j dλ m j (λ λ l ) m l. q (λ λ i ) ni A(λ) F (λ) = 0 (j =,...,m ; =,...,p), q s A(λ) = ãg(λ) ah(λ) = ã (λ λ i ) ni a (λ µ i, )ñi F (λ) = f(λ)g(λ)h(λ) (λ λ ) m = l=,l (λ λ l ) m l q s (λ λ i ) ni (λ µ i. )ñi Note that by (3.2), A(λ) is a polynomial of degree N because n + +n q = N and ñ + + ñ s = N; besides, by (3.22), F (λ ) 0( =,...,p) observing that µ i s are distinct from λ by Lemma 3.2.

On Construction of a Complex Finite Jacobi Matrix from Two Spectra 9 Taing j = m in (3.20), we get Next, taing j = m in (3.20), we get A(λ ) F (λ ) = 0, that is, A(λ ) = 0. A (λ )F (λ ) A(λ )F (λ ) F 2(λ = 0, ) and hence, A (λ ) = 0. Continuing in this way, we find that A (j) (λ ) = 0 for all j = 0,,...,m ; =,...,p. Therefore, applying Lemma 2.2, we get that that is, (3.23) where c is a constant. Hence and we get (3.24) (3.25) A(λ) = c(λ λ ) m (λ λ p ) mp, q s ã (λ λ i ) ni a (λ µ i = c )ñi (ã a)λ N Next, from (3.6), we have ã ( ã = cλ N q n i λi a ( c q n i λi a (λ λ ) m, s ñ i µ i )λ N + m λ )λ N + ã a = c, s ñ i µ i = c m λ. q n i λi m λ = b N b N, and applying Lemma 3.3 to the matrices J and J we can write s ñ i µ i m λ = b N b N.

20 G.Sh. Guseinov From the last two equations we get (3.26) q n i λi = s ñ i µ i. Taing (3.26) into account in (3.25) we obtain Hence, by (3.24), ( c m λ (ã a) s ñ i µ i = c m λ. ) s ñ i µ i = 0, that is, c(b N b N ) = 0. Therefore, c = 0 by (3.7), and hence, ã = a by (3.24). Now (3.23) gives q (λ λ i ) ni = s (λ µ i )ñi for all values of λ. It follows that q = s, λ i = µ i,n i = ñ i (i =,...,q) with a possible reorder of the µ i s. Formulae (.4), (.5) follow from (2.27), (2.28) by Corollary 3.6 and formula (.6) follows from Lemma 3.3. Formulae (.4), (.5), and (.6) show that the two spectra in (.7) determine the Jacobi matrix J of the form (.) in the class (.2) and the number b N in C in the matrix J defined by (.6) uniquely up to signs of the off-diagonal elements of J. Acnowledgment. The author is very much indebted to the anonymous referee for his/her constructive and valuable comments and suggestions which greatly improved the original manuscript of this paper. REFERENCES [] G.Sh. Guseinov. Inverse spectral problems for tridiagonal N by N complex Hamiltonians. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 5:Paper 08, 28 pages, 2009. [2] G.Sh. Guseinov. On an inverse problem for two spectra of finite Jacobi matrices. Applied Mathematics and Computation, 28:7573 7589, 202. [3] H. Hochstadt. On the construction of a Jacobi matrix from spectral data. Linear Algebra and its Applications, 8:435 446, 974. [4] W.G. Kelley and A. C. Peterson. Difference Equations: An Introduction with Applications, 2nd edition. Academic Press, San Diego, 200.