CONSTRUCTION OF A COMPLEX JACOBI MATRIX FROM TWO-SPECTRA

Transcription

1 Hacettepe Journal of Mathematics and Statistics Volume 40(2) (2011), CONSTRUCTION OF A COMPLEX JACOBI MATRIX FROM TWO-SPECTRA Gusein Sh Guseinov Received 21:06 :2010 : Accepted 25 :04 :2011 Abstract In this paper we study the inverse spectral problem for two-spectra of finite order complex Jacobi matrices (tri-diagonal matrices) 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 deleting the first column and the first row of the Jacobi matrix An explicit procedure of reconstruction of the matrix from the two-spectra is given Keywords: Jacobi matrix, Spectral data, Inverse spectral problem 2010 AMS Classification: 15 A29 1 Introduction Communicated by Cihan Orhan An N N complex Jacobi matrix is a matrix of the form b 0 a a 0 b 1 a a 1 b (11) J =, b N 3 a N a N 3 b N 2 a N a N 2 b N 1 where for each n, a n and b n are arbitrary complex numbers such that a n is different from zero: (12) a n, b n C, a n 0 The general inverse spectral problem is to reconstruct the matrix given some of its spectral characteristics (spectral data) Department of Mathematics, Atilim Universty, Incek, Ankara, Turkey guseinov@atilimedutr

2 298 G Sh Guseinov In the real case (13) a n, b n R, a n 0, many versions of the inverse spectral problem for J have been investigated in the literature, see [1] and references given therein In [9, 10] Hochstadt investigated the following version of the inverse problem for twospectra Let J 1 be the (N 1) (N 1) matrix obtained from J by deleting its first row and first column Let {λ k } and {µ k } be the eigenvalues of J and J 1, respectively The eigenvalues of a Jacobi matrix with entries satisfying (13) are real and distinct, and the eigenvalues of J and J 1 interlace: λ 1 < µ 1 < λ 2 < µ 2 < λ 3 < < µ N 1 < λ N The author showed in [9, 10] that if the {λ k } are the eigenvalues of some Jacobi matrix of the form (11) with the entries a n, b n R, a n > 0, and if {µ k } are the eigenvalues of the corresponding J 1, then there is precisely one such matrix with these {λ k, µ k }, and gave a constructive method for calculating the entries of J in terms of the given eigenvalues The same problem was subsequently considered in [5, 8, 4], where different proofs of the uniqueness and existence of J were offered (for the case of infinite Jacobi matrices see [2, 6, 3]) In the present paper, we consider the above formulated problem about two-spectra for complex Jacobi matrices of the form (11) with entries satisfying (12) In the complex case in general the matrix J is no longer selfadjoint and therefore its eigenvalues may be complex and multiple Recently, the author introduced in [7] the concept of spectral data for complex Jacobi matrices of finite order and solved the inverse problem of recovering the matrix from its spectral data Here we show that the inverse problem for two-spectra of complex Jacobi matrices can be solved by reducing it to the inverse problem for spectral data 2 Spectral data and the inverse problem In this auxiliary section we follow the author s paper [7] Let J be a complex Jacobi matrix of the form (11) with entries satisfying (12) Further, let R(λ) = (J λi) 1 be the resolvent of the matrix J (by I we denote the identity matrix of needed dimension) and let e 0 be the N-dimensional column vector with the components 1, 0,, 0 The rational function (21) w(λ) = ((R(λ)e 0, e 0) = ( (λi J) 1 e 0, e 0 ) we call the resolvent function of the matrix J, where (, ) denotes the standard inner product in C N Denote by λ 1,, λ p all the distinct eigenvalues of the matrix J, and by m 1,, m p their multiplicities, respectively, as the roots of the characteristic polynomial det(j λi), so that 1 p N and m m p = N We can decompose the rational function w(λ) into partial fractions to get (22) w(λ) = m p k k=1 j=1 β kj (λ λ k ) j, where β kj are some complex numbers uniquely determined by the matrix J

3 Construction of a Complex Jacobi Matrix from Two-Spectra Definition The collection of quantities (23) {λ k, β kj (j = 1,, m k ; k = 1,, p)}, we call the spectral data of the matrix J For each k {1,, p} the (finite) sequence {β k1,, β kmk } we call the normalizing chain (of the matrix J) associated with the eigenvalue λ k Determining the spectral data of a given Jacobi matrix is called the direct spectral problem for this matrix Let us indicate a convenient way to compute the spectral data of complex Jacobi matrices For this purpose we should describe the resolvent function w(λ) of the Jacobi matrix Given a Jacobi matrix J of the form (11) with the entries (12), consider the eigenvalue problem Jy = λy for a column vector y = {y n} N 1 n=0, that is equivalent to the second order linear difference equation (24) a n 1y n 1 + b ny n + a ny n+1 = λy n, n {0, 1,, N 1}, a 1 = a N 1 = 1, for {y n} N n= 1, with the boundary conditions y 1 = y N = 0 Denote by {P n(λ)} N n= 1 and {Q n(λ)} N n= 1 the solutions of equation (24) satisfying the initial conditions (25) (26) P 1(λ) = 0, P 0(λ) = 1; Q 1(λ) = 1, Q 0(λ) = 0 For each n 0, P n(λ) is a polynomial of degree n called a polynomial of the first kind, and Q n(λ) is a polynomial of degree n 1 known as a polynomial of second kind The equality (27) det (J λi) = ( 1) N a 0a 1 a N 2P N(λ) holds, so that the eigenvalues of the matrix J coincide with the roots of the polynomial P N(λ) In [7] it is shown that the entries R nm(λ) of the matrix R(λ) = (J λi) 1 (resolvent of J) are of the form { P n(λ)[q m(λ) + M(λ)P m(λ)], 0 n m N 1, R nm(λ) = P m(λ)[q n(λ) + M(λ)P n(λ)], 0 m n N 1, where M(λ) = QN(λ) P N(λ) Therefore according to (21) and using initial conditions (25), (26), we get (28) w(λ) = R 00(λ) = M(λ) = QN(λ) P N(λ) Denote by λ 1,, λ p all the distinct roots of the polynomial P N(λ) (which coincide by (27) with the eigenvalues of the matrix J), and by m 1,, m p their multiplicities, respectively: P N(λ) = c(λ λ 1) m1 (λ λ p) mp, where c is a constant We have 1 p N and m 1+ +m p = N Therefore by (28) the decomposition (22) can be obtained by rewriting the rational function Q N(λ)/P N(λ) as the sum of partial fractions

4 300 G Sh Guseinov We can also get another convenient representation for the resolvent function as follows If we delete the first row and the first column of the matrix J given in (11), then we get the new matrix where J 1 = N N N 4 N 3 N N 3 N 2 n = a n+1, n {0, 1,, N 3}, n = b n+1, n {0, 1,, N 2} The matrix J 1 is called the first truncated matrix (with respect to the matrix J) 22 Theorem The equality holds det(j1 λi) w(λ) = det(j λi) Proof Let us denote the polynomials of the first and the second kinds, corresponding to the matrix J 1, by P n (1) (λ) and Q (1) n (λ), respectively It is easily seen that (29) P (1) n (λ) = a 0Q n+1(λ), n {0, 1,, N 1},, Q (1) n (λ) = 1 a 0 {(λ b 0)Q n+1(λ) P n+1(λ)}, n {0,1,, N 1} Indeed, both sides of each of these equalities are solutions of the same difference equation n 1yn 1 + b(1) n y n + n y n+1 = λy n, n {0, 1,, N 2}, N 2 = 1, and the sides coincide for n = 1 and n = 0 Therefore the equalities hold by the uniqueness theorem for solutions Consequently, taking into account (27) for the matrix J 1 instead of J and using (29), we find (210) det(j 1 λi) = ( 1) N 1 0 a(1) 1 N 3 P (1) N 1 (λ) Comparing this with (27), we get = ( 1) N 1 a 1 a N 2a 0Q N(λ) Q N(λ) det(j(1) P = λi) N(λ) det(j λi) so that the statement of the theorem follows by (28) The inverse spectral problem is stated as follows: (i) To see if it is possible to reconstruct the matrix J given its spectral data (23) If it is possible, to describe the reconstruction procedure (ii) To find the necessary and sufficient conditions for a given collection (23) to be the spectral data for some matrix J of the form (11) with entries belonging to the class (12)

5 Construction of a Complex Jacobi Matrix from Two-Spectra 301 This problem was recently solved by the author in [7] and the following results were established Given the collection (23) define the numbers ( ) m p k l (211) s l = j 1 k=1 j=1 where ( l j 1 s l let us introduce the determinants s 0 s 1 s n β kj λ l j+1 k, l = 0, 1,2,, ) ( is a binomial coefficient and we put l j 1) = 0 if j 1 > l Using the numbers s 1 s 2 s n+1 (212) D n =, n = 0, 1,, N s n s n+1 s 2n 23 Theorem Let an arbitrary collection (23) of complex numbers be given, where λ 1,, λ p, (1 p N) are distinct, 1 m k N, and m m p = N In order for this collection to be the spectral data for some Jacobi matrix J of the form (11) with entries belonging to the class (12), it is necessary and sufficient that the following two conditions be satisfied: (i) p k=1 β k1 = 1; (ii) D n 0 for n {1, 2,, N 1}, and D N = 0, where D n is the determinant defined by (212), (211) Under the conditions of Theorem 23, the entries a n and b n of the matrix J for which the collection (23) is spectral data, are recovered by the formulas (213) (214) a n = ± D n 1D n+1 D n, n {0,1,, N 2}, D 1 = 1, b n = n D n n 1 D n 1, n {0, 1,, N 1}, 1 = 0, 0 = s 1, where D n is defined by (212) and (211), 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+1, s n+2,, s 2n+1 24 Remark It follows from the above solution of the inverse problem that the matrix (11) is not uniquely restored from the spectral data This is linked with the fact that the a n are determined from (213) 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 1, σ 2,, σ N 1} be a given finite sequence, where for n {1, 2,, N 1} each σ n is + or We have 2 N 1 such different sequences Now to determine a n uniquely from (213) for n {0, 1,, N 2} we can choose the sign σ n when extracting the square root In this way we get precisely 2 N 1 distinct Jacobi matrices possessing the same spectral data The inverse problem is solved uniquely from the data consisting of the spectral data and a sequence {σ 1, σ 2,, σ N 1} of signs + and Thus, we can say 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 3 Inverse problem for two-spectra Let J be an N N Jacobi matrix of the form (11) with entries satisfying (12) Denote by λ 1,, λ p all the distinct eigenvalues of the matrix J and by m 1,, m p their multiplicities, respectively, as the roots of the characteristic polynomial det(j λi), so

6 302 G Sh Guseinov that 1 p N and m 1 + +m p = N Further, let J 1 be the (N 1) (N 1) matrix obtained from J by deleting its first row and first column Denote by µ 1,, µ q all the distinct eigenvalues of the matrix J 1, and by n 1,, n q their multiplicities, respectively, as the roots of the characteristic polynomial det(j 1 λi), so that 1 q N 1 and n n q = N 1 The collections (31) {λ k, m k (k = 1,, p)} and {µ k, n k (k = 1,, q)} form the spectra of the matrices J and J 1, respectively We call these collections the two-spectra of the matrix J The inverse problem for two-spectra consists in the reconstruction of the matrix J by its two-spectra We will reduce the inverse problem for two-spectra to the inverse problem for the spectral data solved above in Section 2 First let us study some necessary properties of the two-spectra of the Jacobi matrix J Having the matrix J consider the difference equation (24) and let {P n(λ)} N n= 1 and {Q n(λ)} N n= 1 be the solutions of this equation satisfying the initial conditions (25) and (26), respectively Then by (27) and (210), we have (32) (33) det(j λi) = ( 1) N a 0a 1 a N 2P N(λ), det (J 1 λi) = ( 1) N 1 a 0a 1 a N 2Q N(λ), so that the eigenvalues and their multiplicities of the matrix J coincide with the roots and their multiplicities of the polynomial P N(λ) and the eigenvalues and their multiplicities of the matrix J 1 coincide with the roots and their multiplicities of the polynomial Q N(λ) The equation (34) P N 1(λ)Q N(λ) P N(λ)Q N 1(λ) = 1 holds (see [7, Lemma 4]) 31 Lemma The matrices J and J 1 have no common eigenvalues, that is, λ k µ j for all possible values of k and j Proof Suppose that λ is an eigenvalue of the matrices J and J 1 Then by (32) and (33) we have P N(λ) = Q N(λ) = 0 But this is impossible by (34) In virtue of (22) and Theorem 22, we have Hence, m p k k=1 j=1 β kj = β kj (λ λ k ) j = 1 (m k j)! q (λ µ i) n i p (λ λ l ) m l i=1 l=1 lim λ λ k d mk j dλ m k j (λ λ k) m k q (λ µ i) n i p (λ λ l ) m l i=1 l=1

7 Construction of a Complex Jacobi Matrix from Two-Spectra 303 Therefore (35) β kj = 1 (m k j)! lim λ λ k d mk j dλ m k j (j = 1,, m k ; k = 1,, p) q (λ µ i) n i i=1 p (λ λ l ) m l l=1 l k Formula (35) shows that the normalizing numbers β kj of the matrix J are determined uniquely by the two-spectra {λ k, m k (k = 1,, p)} and {µ k, n k (k = 1,, q)} of this matrix Since the inverse problem for spectral data is solved uniquely up to the signs of the off-diagonal elements of the recovered matrix (see Remark 24 made above in Section 2), we get the following result 32 Theorem The two spectra in (31) determine the matrix J uniquely up to signs of the off-diagonal elements of J The procedure of reconstruction of the matrix J from its two-spectra consists in the following If we are given the two-spectra {λ k, m k (k = 1,, p)} and {µ k, n k (k = 1,, q)}, we find the quantities β kj from (35) and then solve the inverse problem with respect to the spectral data {λ k, β kj (j = 1,, m k ; k = 1,, p)} to recover the matrix J by using formulas (213) and (214) References [1] Chu, M T and Golub, G H Inverse Eigenvalue Problems: Theory, Algorithms, and Applications (Oxford University Press, New York, 2005) [2] Fu, L and Hochstadt, H Inverse theorems for Jacobi matrices, J Math Anal Appl 47, , 1974 [3] Gasymov, M G and Guseinov, G Sh On inverse problems of spectral analysis for infinite Jacobi matrices in the limit-circle case, Dokl Akad Nauk SSSR 309, , 1989; English transl in Soviet Math Dokl 40, , 1990 [4] Gesztesy, F and Simon, B M M-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J Anal Math 73, , 1997 [5] Gray, L J and Wilson, DG Construction of a Jacobi matrix from spectral data, Linear Algebra Appl 14, , 1976 [6] Guseinov, G Sh Determination of the infinite Jacobi matrix with respect to two-spectra, Mat Zametki 23, , 1978; English transl in Math Notes 23, , 1978 [7] Guseinov, G Sh Inverse spectral problems for tridiagonal N by N complex Hamiltonians, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 5, Paper 018, 28 pages, 2009 [8] Hald, OH Inverse eigenvalue problems for Jacobi matrices, Linear Algebra Appl 14, 63 85, 1976 [9] Hochstadt, H On some inverse problems in matrix theory, Arch Math 18, , 1967 [10] Hochstadt, H On construction of a Jacobi matrix from spectral data, Linear Algebra Appl 8, , 1974