A formula for eigenvalues of Jacobi matrices with a reflection symmetry

The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the $2M$-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix { entries} is obtained. In the limit $M\to\infty$ this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, which super- and sub-diagonal matrix elements are equal to -1. We obtain such requirements in the simplest case of the discrete Schr\"odinger operator acting in ${l}^2( \mathbb{N})$, which does not have bound and semi-bound states, and which potential has a compact support.


Introduction
The theory of tridiagonal (Jacobi) matrices has rich applications in different fields of physics and mathematics including the theory of Anderson localization [1,2,3], the thermodynamic Casimir effect [4,5], orthogonal polynomials [6], and nonlinear integrable lattice models [7,8,9]. The spectral properties of Jacobi matrices are to much extent similar to the spectral properties of the continuous one-dimensional Sturm-Liouville operators. For a detailed review of the present stage of the theory of Jacobi operators see the monograph by G. Teschl [8].
In this paper we address the spectral properties of a special class of Jacobi matrices, which are symmetric with respect to both main and secondary diagonals. To be more specific, let us introduce the real Jacobi matrix, the Hamiltonian which acts in the Hilbert space C N . Throughout this paper, the dimension of this space is supposed to be even, The (real) entries of the matrix H will be subject to the following constrains b N +1−j = b j , j = 1, . . . N, a N −j = a j , j = 1, . . . N − 1.
Jacobi matrices satisfying this symmetry property were studied by Hochstadt [10].
Since the Hamiltonian H commutes with the reflection operator where {Ψ(j)} N j=1 ∈ C N , the eigenvectors {Ψ n } N n=1 of the matrix (1) can be classified by their parities κ n = ±1, R Ψ n = κ n Ψ n .
There are exactly M eigenvectors Ψ (ev) m of the matrix H, which are even under the action of the reflection (4), and M eigenvectors Ψ (od) m , which are odd. We shall use different notation µ m and ν n for the corresponding eigenvalues, Proceeding then to the limit M → ∞, we obtain from (8) identities (55), (56) for the scattering data corresponding to the discrete Schrödinger operator (i.e., the Jacobi operator acting in l 2 (N), which is described by the infinite-dimensional matrix (1) with a j = −1 and b j = 2 + v j , j ∈ N) under certain assumptions on the potential {v j } j∈N . These results are described in Section 4 and Theorem 2 therein.
A part of the results described here were given in the preprint [11].
Turning to the eigenvalue problem for a general Jacobi matrix specified in the Theorem, we first rewrite it as a difference equation j = 1, . . . , N, n = 1, . . . , N, subjected to constrains (3) and supplemented with the Dirichlet boundary conditions Ψ n (0) = Ψ n (N + 1) = 0.
Let us consider two associated eigenvalue problems for the even and odd states, which are restricted to the M -dimensional quotient vector space C M . We shall use the lower case ψ to denote vectors in this quotient space, The vectors ψ corresponding to eigenvalues µ m , m = 1, . . . , M . Similarly, the vectors ψ corresponding to eigenvalues ν m , m = 1, . . . , M . Note, that the matrices H (ev) and H (od) are simply related Here we have taken into account, that the matrix H (od) is symmetric. Therefore, and respectively. It follows from the Vieta's theorem, that the symmetric polynomials of eigenvalues {µ m } M m=1 , as well as the symmetric polynomials of eigenvalues {ν m } M m=1 , can be written as polynomial functions of the variables {a j } M j=1 and {b j } M j=1 . This is true, in particular for the function since the product on the left-hand side is a symmetric polynomial of {µ m } M m=1 , and it is also a symmetric polynomial of {ν m } M m=1 . However, the polynomialQ(a, b) in the right-hand side of (22) must not depend on {b j } M j=1 , i.e.Q (a, b) ≡ Q(a).
To prove this, let us fix all a j at some arbitrary nonzero complex values a where we have explicitly indicated the algebraic dependence of the eigenvalues µ m and ν m on the parameters b j . One can easily see from (24) that the polynomial P (b) has no zeros. Really, if P (b) = 0 at some b = b (0) , then at least one eigenvalue µ m 0 (b (0) ) of the matrix H (ev) (b (0) ) must coincide with some eigenvalue ν n 0 (b (0) ) of the matrix H (od) (b (0) ), as it is implied by equality (24). However, this is a contradiction with Lemma 1, which guaranties that matrices H (ev) (b (0) ) and H (od) (b (0) ) have no common eigenvalues. Since the polynomial P (b) has no zeroes, it is just a constant, and we arrive at equation (23). Thus, we have proven that with some polynomial Q(a). In order to find this polynomial explicitly, let us address some of its properties.
where Q 0 (a) is also some polynomial of {a j } M j=1 .
P r o o f. The statement (a) follows immediately from (25). To prove (b), let us, first, note, that Q(a) vanishes, if one of the variables a j takes the zero value. Really, if a j = 0 for some j, the matrices H (ev) and H (od) defined by (17), (18) become block-diagonal containing two blocks of dimensions j and M − j. Furthermore, the j-dimensional blocks in their top-left corners are identical. Therefore, j eigenvalues of the matrix H (ev) merge with j eigenvalues of the matrix H (od) in this case, First, let us choose j = 1, and vary the parameter a 1 near the zero point a 1 → 0 and keeping the rest a j ′ , j ′ = 2, . . . , M at some fixed non-zero values. Then, a simple perturbative analysis of equations yields for small a 1 : Combining this result with (25), we conclude that the polynomial Q(a) has at least the second order zero at a 1 . Second, let us choose j = 2, and tune the parameter a 2 near the origin a 2 → 0 keeping all others a j nonzero. Then, for j = 1, 2, one can easily obtain from (28), (29) Therefore, two brackets (µ 1 − ν 1 ) and (µ 2 − ν 2 ) in the product on the left-hand side of (25) vanish at a 2 → 0. Since the both brackets are of the order O(a 2 2 ), the polynomial Q(a) must have at least the fourth order zero at a 2 .
Combining (26) and (27), one concludes that the polynomial Q 0 (a) on the right-hand side of (27) is uniform, and its degree is zero. Therefore, Q 0 (a) does not depend on parameters a j being just a constant, Q 0 (a) ≡ C M . To fix this constant, we choose parameters {a j , b j } M j=1 according to (9). For such a choice of the Jacobi matrix, the product specified on the left-hand side of (25) is given by equation (13). Comparing the latter with (27), we find and arrive at equality (8).
R e m a r k 0.1. If all a j = 0, and the matrix (1) satisfying (3) is real, its eigenvalues λ n are also real and non-degenerate. In this case, one can order them so that It is possible to show that This follows from the Remark after Theorem 3 in [10] and Lemma 1.6 in [8].
3 Scattering problem for the discrete Schrödinger operator The main issue of this Section is to fix notations for later use. We briefly summarize some well-known facts of the scattering theory for the discrete Schrödinger operator. The scattering theory for general Jacobi operators, which is to some extent parallel to the scattering theory for the continuous Schrödinger equation (see, for example [12]), is described in much details in the monograph [8].
Consider the discrete Schrödinger equation (14) in l 2 (N) with the potential v j ∈ R, supplemented with the Dirichlet boundary condition In order to simplify further analysis, only the potentials V = {v j } j∈N with a compact support will be considered, with some J ∈ N. The latter requirement means, that equation (36) looks as the free equation for j ≥ J + 1.
The Jacobi operator considered in the previous Section reduces to (37)  At a given λ, equations (36), (37) with omitted boundary condition (38) have two linearly independent solutions, and the general solution of (36), (37) can be written as their linear combination. Following [8], we introduce the Wronskian of two sequences {ψ 1 (j)} j∈N , and It does not depend on j, if ψ 1 (j) and ψ 2 (j) are solutions of equations (36), (37). Let us turn now to the scattering problem associated with equations (36)-(39), which corresponds to the case 0 < λ < 4. Instead of the parameter λ ∈ σ cont [H], it is also convenient to use the momentum p and the related complex parameter z = e ip : Three solutions of (36) are important for the scattering problem.
• The fundamental solution ϕ(j, p), which is fixed by the boundary conditions • Two Jost solutions f (j, p), and f (j, −p), which are determined by their behavior at large j, and describe the out-and in-waves, respectively, f (j, ±p) = exp(±ipj) = z ±j , at j ≥ J + 1.
Note, that this formula is exact only in the considered case of the potential V with a compact support. In the general case, it must by replaced by the asymptotical formula f (j, ±p) ∼ exp(±ipj) = z ±j , as j → ∞. This comment also relates to equations (48) and (61) below.
The fundamental solution ϕ(j, p) can be represented as a linear combination of two Jost solutions, The complex coefficient F (p) in the above equation is known as the Jost function. It is determined by (45) for real momenta p in the interval p ∈ (−π, π), where it satisfies the relation and can be written as with real σ(p) and η(p). At large j ≥ J + 1, the fundamental solution behaves as where A(p) = exp[σ(p)] is the scattering amplitude, and η(p) is the scattering phase. The latter can be defined in such a way, that η(−p) = −η(p) for −π < p < π. Note, that the Jost function F (p) can be also represented as the Wronskian of the Jost and fundamental solutions with arbitrary j ∈ N.
The following exact representation holds for the Jost function F (p) in terms of the fundamental solution ϕ(j, p): cf. equation (1.4.4) in [12] in the continuous case. For |z| = 1, denote byF (z) the Jost function F (p) expressed in terms of the complex parameter z: F (p) =F (z = e ip ). The functionF (z) can be analytically continued into the circle |z| < 1, where it has finite number of zeros {α n } N n=1 . These zeroes determine the discrete spectrum {λ n } N n=1 of the problem (36)-(38): Of course, these eigenvalues are real in the boundary problem with a real potential. Besides (39), two further restrictions will be imposed in the sequel on the potential V . Conditions (i) and (ii) imply, that the operator H does not have bound and semi-bound states [12], respectively. Since ϕ(j, p) is a polynomial of the spectral parameter λ = 2 − z − z −1 of the degree j − 1, the Jost function (50) expressed in the parameter z is a polynomial of the degree 2J − 1: where the real coefficients c j (V ) polynomially depend on the potential v j , j = 1, . . . , J. This indicates, that the set of Jost polynomial (52) has the co-dimension J −1 in the 2J-dimensional linear space of polynomials of the degree 2J − 1. Due to the constrains (i), (ii), we get |z n (V )| > 1, for all n = 1, . . . , 2J − 1.
Let us periodically continue the scattering phase η(p) from the interval (−π, π) to the whole real axis p ∈ R. It follows from (53), that for a potential V satisfying (39), (i), (ii), the scattering phase is a smooth odd 2π-periodical function in the whole real axis: η(p) ∈ C ∞ (R/2πZ).
4 Identities on the scattering data for the discrete Schrödinger operator In this Section we apply Theorem 1 to study some spectral properties of the discrete Schrödinger operator acting in l 2 (N), and prove the following where P indicates the principal value integral. Equality (55) can be rewritten as well in terms of the Jost functionF (z): where the integration path is counter-clockwise oriented. Equality (56) can be also written in terms the zeros {z n (V )} 2J−1 n=1 of the Jost functionF (z): As it was explained in the Introduction, the eigenfunctions Ψ l (j) of such a problem are either even, or odd with respect to the reflection (4), see equations (7). In both cases, the equality must hold. On the other hand, the eigenfunction Ψ l (j) corresponding to the eigenvalue λ l obeys the Dirichlet boundary condition at j = 0, and, therefore, is proportional to the fundamental solution ϕ(j, p l ), which determines the spectrum λ l ≡ ω(p l ) in terms of the scattering phase η(p). The equivalent, but more convenient form of equation (61) reads where k l were defined in (10).
For an arbitrary M > J, let us put a j = −1 for all j ∈ [0, M ] in equality (8), and take the logarithm of the ratio of two versions of this equality written for the potential (58), and for the zero potential, respectively. As a result, we get where In order to proceed to the large-M limit on the right-hand side, let us introduce three one-parametric sets of the C ∞ -functions k(p; ǫ), p(k; ǫ), and F (k, k ′ ; ǫ), where ǫ is a small parameter lying in the interval 0 ≤ ǫ ≤ δ, and δ is some fixed positive number such that The function k(p; ǫ) from the first set is defined in the real axis p ∈ R by the equation At a fixed ǫ ∈ [0, δ], the function k(p; ǫ) monotonically increases in p being odd and quasiperiodic, ∂ p k(p; ǫ) > 0, k(−p; ǫ) = −k(p, ǫ), k(p + 2π; ǫ) = k(p; ǫ) + 2π.
For ǫ ∈ [0, δ], the inverse function p(k; ǫ) is a well defined odd quasiperiodic C ∞ -function of k ∈ R, which also smoothly depends on the parameter ǫ. The Taylor expansion of p(k, ǫ) in ǫ reads as where A j (k) are 2π-periodic odd bounded functions of k. The function representing the third set, is 2π-periodic and even in both arguments k, k ′ ∈ R, and F (k, converges at 0 ≤ ǫ ≤ δ uniformly in k, k ′ ∈ R 2 . The two initial partial derivatives in this expansion read as These and all higher derivatives ∂ n F (k,k ′ ;ǫ) ∂ǫ n ǫ=0 , n = 3, . . ., in the Taylor expansion (69) are the 2π-periodic C ∞ [R 2 ]-functions of their arguments k, k ′ ∈ R 2 , which are therefore bounded in R 2 together with all their partial derivatives in k and k ′ . Let us now rewrite the sum (64) as and expand the right-hand side in ǫ to the second order, It follows from the above analysis, that the correction term in the square brackets is uniformly bounded in n and m, with some positive C independent on n, m. Taking this into account, one obtains from (70)-(73), where where and the m-dependent correction term O(M −2 ) is again uniformly bounded in m. Replacing in (78) the Riemann sum of the regular function of the momenta k 2n−1 by the corresponding integral in the limit of large M , one obtains The m-dependent correction term O(M −1 ) is uniformly bounded in m, since Calculation of the large-M asymptotics of S (0) m is more delicate. First, we extend summation in (76) in the index n from 1 till 2M + 1, where In (80) we have taken into account the reflection symmetry R m (q) = R m (2π − q) of the function (81), which leads to R m (k 2n−1 ) = R m (k 2(2M +1−n)+1 ).
Since k 2M +1 = π, and η(π) = 0, ω(π) = 4, we get from (81) The sum in the second line of (80) reads as Since R m (q) ∈ C ∞ (R/2πZ), this sum can be replaced with exponential accuracy by the integral at large M → ∞: where µ is an arbitrary positive number, see formula 25.4.3 in [13]. Furthermore, since Taking into account (81), the integral on the right-hand side of (84) can be written as .
The principal value integral in the first term on the right-hand side vanishes due to the equality which can be easily proved for ω(p) = 2 − 2 cos p, all ν ∈ N, and 0 < k < π. The integral in the second line of (85) reduces after the change λ = ω(q) of the integration variable to the form where Collecting (80)-(88), we get  (77), (79), respectively. In the second term, we can replace with sufficient accuracy the summation in m by integration in the momentum k: The first integral on the right-hand side vanishes due to equality (87) with 0 < k < π, and ν = 1. The second line in (91) can be transformed as follows The first term on the right-hand side vanishes due to equality (87) with ν = 2. Then, substituting for λ 1 = ω(k), λ 2 = ω(q), and integrating by parts, we obtain from the second term on the right-hand side of (92) Let us turn now to calculation of the first term on the left-hand side of equality (90). At large M , one obtains The first term on the right-hand side equals to I(4)/(2π). The large-M asymptotics of the sum on the right-hand side can be found as follows where µ is an arbitrary positive number. In deriving (95) we have taken into account, that I[(ω(k)] is the 2π-periodical function of p in R, which is continuous with all its derivatives [i.e., I[(ω(k)] ∈ C ∞ (R/2πZ)], and formula 25.4.3 in [13]. The integral on the right-hand side vanishes due to equality (87) with ν = 1: Derivation of (56) from (55) is straightforward being based on (47), and exploits properties (i), (ii), which were supposed to hold for the Jost functionF (z) in Section 3.
The integrand of the integral on the left-hand side of equation (56) is meromorphic in the region |z| > 1 having there only simple poles at z = z n , n = 1, . . . , 2J − 1 with residues lnF (1/z n ). After straightforward application of the Cauchy's residue theorem, one finds from (56), Substitution of the right-hand side of (52) into this relation yields, In order prove equality (57), it remains to show that the product in the square brackets on the left-hand side is positive. The latter statement follows immediately from the two properties of the zeroes z n of the Jost functionF (z).
2. The numbers z n are either real, or else occur as complex conjugate pares in the set {z n } 2J−1 n=1 of roots of the real polynomial (52). E x a m p l e 0.1. Let us check directly that equality (57) holds for J = 2. In this case, the Jost function (52) is the polynomial of the third degree, which is parametrized by two parameters v 1 , v 2 . Let us rewrite equation (57) in the form in terms of the symmetric polynomials of the Jost function zeroes, Exploiting the Vieta's theorem, one obtains from (99), Upon substitution of the right-hand sides into (100), we convince ourselves that equality (100) holds for arbritrary complex v 1 and v 2 .

Conclusions and open problems
We have studied some spectral properties of two classes of Jacobi operators. The first class is represented by the real 2M -dimensional Jacobi matrices with the refection symmetry (3). We have proved in Theorem 1 a new polynomial identity (8), which relates the eigenvalues of such Jacobi matrices with their matrix elements.
The second class of Jacobi operators considered in the present paper is represented by the discrete Schrödinger operators acting in l 2 (N) with purely continuous spectrum, which potentials have a compact support. Note, that the infinite-dimensional Jacobi matrix corresponding to the discrete Schrödinger operator has the super-and sub-diagonal matrix elements a j = −1 for all j ∈ N. The scattering data of such operators must satisfy three identities (55), (56), and (57) which are proved in Theorem 2.
Obtained result could be extended and generalized in several directions. We did not try to prove identities (55), (56) for the most general case of the discrete semiinfinite scattering problem. It is natural to expect, however, that they should hold for some more general class of potentials, which vanish fast enough at infinity, though do not have a compact support. On the other hand, in the case of the potentials with non-empty discrete spectrum and/or semi-bound states (the latter appear if the Jost function has zeroes at the end points of the continuous spectrum, see [12]), some modified forms of (55), (56) should P r o o f. Denote g(α) = 2M +1 m=1 4 sin 2 (2m − 1)π 2(2M + 1) − α − 4 sin 2 2nπ 2(2M + 1) .
It follows from (106) that the function is analytic and has no zeroes in the complex α-plane. Furthermore, this function is rational in the variable z = e iα and approaches to 1 at z → ∞ and at z → 0. Therefore, R(α) = 1.
Putting α = 0 in (104) we find 2M +1 (109) To fix the sign of the right-hand side of (109), we have taken into account that just the first n factors in the product on the left-hand side are negative at n = 1, . . . , M . Thus, . (110) To determine the product on the right-hand side we use formula 1.392.1 in Ref. [14]: sin nx = 2 n−1 n−1 k=0 sin x + πk n .