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Persymmetric Jacobi Matrices, Isospectral Deformations And PERSYMMETRIC JACOBI MATRICES, ISOSPECTRAL DEFORMATIONS AND ORTHOGONAL POLYNOMIALS VINCENT GENEST, SATOSHI TSUJIMOTO, LUC VINET, AND ALEXEI ZHEDANOV Abstract. Persymmetric Jacobi matrices are invariant under reflection with respect to the anti- diagonal. The associated orthogonal polynomials have distinctive properties that are discussed. They are found in particular to be also orthogonal on the restrictions either to the odd or to the even points of the complete orthogonality lattice. This is exploited to design very efficient inverse problem algorithms for the reconstruction of persymmetric Jacobi matrices from spectral points. Isospectral deformations of such matrices are also considered. Expressions for the associated polynomials and their weights are obtained in terms of the undeformed entities. 1. Introduction Jacobi matrices that are invariant under reflection with respect to the anti-diagonal are said to be persymmetric or mirror-symmetric. These matrices are known to possess tractable inverse spectral problems[2, 3, 4] and arise in a number of situations especially in engineering (see for instance [10]). Interestingly, they have recently appeared in the analysis of perfect state transfer in quantum spin chains[1, 11, 17]. In the latter context the main question is: what should the design of the chain be if it is to act as a quantum wire and transport quantum states from one location to another with probability one. The answer exploits the connection between orthogonal polynomials and Jacobi matrices as well as the features that mirror-symmetry brings. The present paper provides a systematic examination of the properties that the reflection invariance of persymmetric matrices induces on the associated orthogonal polynomials. Isospectral deformations of persymmetric Jacobi matrices have also found applications in the transport of quantum states along spin chains for instance. Such Jacobi matrices have been seen to lead to spin chain models that exhibit the phenomenon of fractional revival [7, 8, 9, 11]. When this occurs , small clones of the initial state are reproduced at the beginning and the end of the wire periodically. Fractional revival also allows to transport quantum information and may serve as a mechanism to generate maximally entangled states. This motivates the study of the properties of the orthogonal polynomials associated to isospectral deformations of persymmetric Jacobi matrices, a task that is also carried out here. arXiv:1605.00708v2 [math.CA] 14 Feb 2017 The outline of the paper is thus as follows. In section 2, we review relevant aspects of the theory of Jacobi matrices and orthogonal polynomials and introduce persymmetric Jacobi matrices and their general features. We obtain in Section 3 the essential properties of the persymmetric orthogonal polynomials. We show in particular that these polynomials are orthogonal on the sublattices made out either of the odd or of the even points of the full orthogonality grid. This is exploited to present two new efficient algorithms for the reconstruction of persymmetric Jacobi matrices from their Key words and phrases. persymmetric matrices, inverse spectral problems, orthogonal polynomials, isospectral deformations. 1 2 VINCENT GENEST, SATOSHITSUJIMOTO, LUC VINET, AND ALEXEI ZHEDANOV spectra. Isospectral deformations are considered in Section 4 where the associated polynomials and weights are given in terms of those of the undeformed persymmetric Jacobi matrix. 2. Persymmetric Jacobi matrices Consider the tri-diagonal or Jacobi Hermitian matrix of size N 1 N 1 + × + b0 a1 0 ⎛a1 b1 a2 0 ⎞ ⎜ ⎟ ⎜ 0 a2 b2 a3 ⎟ J ⎜ ⎟ . ⎜ ⎟ = ⎜ ⎟ ⎜ ...⋱ aN ⋱ 1 bN 1 aN ⎟ ⎜ − − ⎟ ⎜ ... 0 aN bN ⎟ ⎝ ⎠ In what follows we will assume that all entries ai,bi, i 0, 1,... are real. Let ei, i 0, 1,...,N be the canonical basis in the N= 1-dimensional linear space such = + (2.1) Jen an 1en 1 bnen anen 1, n 0, 1, . N. = + + + + − = In order for relations (2.1) to be valid for n 0 and n N we put additionally a0 aN 1 0. Of course, this condition is not a restriction but= rather a convention.= = + = Introduce also the eigenvectors Xs of the matrix J (2.2) JXs xsXs, s 0, 1,...N = = with eigenvalues xs. It is well known that if ai 0 then all the eigenvalues of the matrix J are real and nondegenerate [2], i.e. ≠ (2.3) xs xs′ , if s s′. ≠ ≠ One can expand the vectors Xs in terms of the basis vectors en [2]: N (2.4) Xs √wsχn xs en, = n 0 ( ) Q= where χn x are orthonormal polynomials defined by the initial conditions χ 1 x 0, χ0 x 1 and the 3-term( ) recurrence relations − ( ) = ( ) = (2.5) an 1χn x bnχn x anχn 1 x xχn x . + ( ) + ( ) + − ( ) = ( ) The reciprocal expansion of the vectors en in terms of the eigenvectors is [2]: N (2.6) en √wsχn xs Xs. = s 0 ( ) Q= In view of the orthonormality of the two bases, the polynomials χn x are orthogonal with respect to the weight function ws: ( ) N (2.7) wsχn xs χm xs δnm. s 0 ( ) ( ) = Q= Sometimes it is convenient to introduce the so-called monic orthogonal polynomials (2.8) Pn x a1a2 ...anχn x . ( ) = ( ) These polynomials have the expansion n n 1 (2.9) Pn x x O x − ( ) = + ( ) ISOSPECTRAL DEFORMATIONS 3 and satisfy the 3-term recurrence relation (2.10) Pn 1 x bnPn x unPn 1 x xPn x , + − 2 ( ) + ( ) + ( ) = ( ) where un an 0. The recurrence relation (2.10) corresponds to the Jacobi matrix = > b0 1 0 ⎛u1 b1 1 0 ⎞ ⎜ ⎟ ⎜ 0 u2 b2 1 ⎟ (2.11) K ⎜ ⎟ , ⎜ ⎟ = ⎜ ⎟ ⎜ ...⋱ uN ⋱ 1 bN 1 1 ⎟ ⎜ − − ⎟ ⎜ ... 0 uN bN ⎟ ⎝ ⎠ which is related to the Jacobi matrix J by a similarity transformation 1 (2.12) K SJS− = with some diagonal matrix S. Sometimes the matrix K is preferable because one can change the signs of the elements an of the matrix J without changing the spectral data xs, s 0, 1,...,N . Note that the eigenvalues xs are the roots of the characteristic polynomial{PN 1 =x [2]: } + ( ) (2.13) PN 1 x x x0 x x1 ... x xN , + ( ) = ( − )( − ) ( − ) where the polynomial PN 1 x can be determined from the recurrence relation (2.10) for n N. In what follows we will+ assume( ) that the eigenvalues are ordered as follows from the smallest= to the largest: (2.14) x0 x1 x2 ⋅ ⋅ ⋅ xN . < < < < The weights ws are given by the the formula [6] hN (2.15) ws , s 0, 1, . , N, = PN xs PN′ 1 xs = ( ) + ( ) where (2.16) hn u1u2 ...un = are normalization constants. For Hermitian Jacobi matrices with nonzero entries an, all weights are nonnegative, ws 0, and moreover they are normalized [6] ≥ N (2.17) ws 1. s 0 = Q= Note that for monic orthogonal polynomials the orthogonality relation looks as N (2.18) wsPn xs Pm xs hn δnm. s 0 ( ) ( ) = Q= There is an important interlacing property of the zeros of the polynomials of an orthogonal set [2]: all zeros of the polynomial Pn x , n 1, 2,...,N are distinct and moreover any zero of the polynomial Pn x lies between two neighboring( ) = zeros of the polynomial Pn 1 x . In particular, this + means that the( sequence) PN xs , s 0, 1, 2,... has alternating signs: ( ) ( ) = N s (2.19) PN xs −1 + Ws, s 0, 1, . , N, ( ) = ( ) = where Ws is a strictly positive sequence Ws 0. > 4 VINCENT GENEST, SATOSHITSUJIMOTO, LUC VINET, AND ALEXEI ZHEDANOV Let R be the “reflection” matrix 0 0 ... 0 1 ⎛ 0 0 ... 1 0 ⎞ R ⎜ ⎟ . = ⎜...............⎟ ⎜ 1 0 ... 0 0 ⎟ ⎝ ⎠ Clearly, R is an involution, i.e. (2.20) R2 I, = where I is the identity matrix. Hence the eigenvalues of the matrix R are either 1 or -1. The action of R on the basis vectors en is obviously: (2.21) Ren eN n. = − The Hermitian matrix J is called persymmetric (or mirror-symmetric) [10] if it commutes with the reflection matrix (2.22) JR RJ. = Condition (2.22) means that the entries of the persymmetric matrix satisfy the conditions (2.23) aN 1 i ai, bN i bi, i 0, 1, . N. + − = − = = Hence a persymmetric matrix J takes the form b0 a1 0 ⎛a1 b1 a2 0 ⎞ ⎜ ⎟ ⎜ 0 a2 b2 a3 ⎟ J ⎜ ⎟ . ⎜ ⋱ ⋱ ⎟ = ⎜ ⎟ ⎜ ... a b a ⎟ ⎜ 2 1 1⎟ ⎜ ... 0 a1 b0 ⎟ ⎝ ⎠ For the Jacobi matrix K (2.11) the persymmetric property means that (2.24) KR RKT , = where KT is the transposed matrix. From the commutativity of the matrix J with the matrix R and from the nondegeneracy of the eigenvalues of matrix J, it follows that any eigenvector Xs should be also an eigenvector of the reflection operator R: (2.25) RXs εsXs, = where εs ±1. We will specify the values εs in the next section. = 3. Basic properties of persymmetric orthogonal polynomials In this section we present the basic properties of the orthogonal polynomials χ x (or Pn x ) that are associated to the persymmetric Jacobi matrices J. Some of these properties are( ) well known,( ) some are new. First of all, consider the property (2.25). Expanding the eigenvector Xs in terms of the basis en and using (2.4) we arrive at the relation (3.1) χN n xs εsχn xs . − ( ) = ( ) ISOSPECTRAL DEFORMATIONS 5 Relation (3.1) should be valid for all n 0, 1,...,N. Choose n N. Then χ0 1 and relation (3.1) becomes = = = (3.2) χN xs εs, ( ) = where εs ±1. But by property (2.19) we have that the only choice for εs is = N s (3.3) εs −1 + . = ( ) We thus have the following. Lemma 3.1. The orthonormal polynomials corresponding to the persymmetric matrix J satisfy the relation N s (3.4) χN n xs −1 + χn xs , n 0, 1, . , N. − ( ) = ( ) ( ) = For the monic polynomials we have a similar relation: N s (3.5) PN n xs −1 + νnPn xs , − ( ) = ( ) ( ) where (3.6) νn »hN n hn.
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