Advances in Mathematical Physics
Volume 2018 (2018), Article ID 9784091, 11 pages
https://doi.org/10.1155/2018/9784091

Research Article

A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry

S. B. Rutkevich

Fakultät für Physik, Universität Duisburg-Essen, Duisburg 47048, Germany

Correspondence should be addressed to S. B. Rutkevich; sergei.rutkevich@uni-due.de

Received 11 July 2018; Accepted 1 October 2018; Published 16 October 2018

Academic Editor: Pavel Kurasov

Copyright © 2018 S. B. Rutkevich. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the -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 this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, of which super- and subdiagonal matrix elements are equal to . We obtain such requirements in the simplest case of the discrete Schrödinger operator acting in , which does not have bound and semibound states and whose potential has a compact support.

1. Introduction

The theory of tridiagonal (Jacobi) matrices has rich applications in different fields of physics and mathematics including the theory of Anderson localization [1–3], the thermodynamic Casimir effect [4, 5], orthogonal polynomials [6], and nonlinear integrable lattice models [7–9]. 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 . Throughout this paper, the dimension of this space is supposed to be even,

The (real) entries of the matrix will be subject to the following constrains:

Jacobi matrices satisfying this symmetry property were studied by Hochstadt [10].

Since the Hamiltonian commutes with the reflection operator

where , the eigenvectors of the matrix (1) can be classified by their parities ,

There are exactly eigenvectors of the matrix , which are even under the action of the reflection (4), and eigenvectors , which are odd. We shall use different notation and for the corresponding eigenvalues,

with , .

As the main result, we prove the following.

Theorem 1. Let be the real Jacobi matrix (1) satisfying (2) and (3). Then, the eigenvalues and of the matrix specified by relations (7) obey the equality

Proceeding then to the limit , we obtain from (8) identities (60), (61) for the scattering data corresponding to the discrete Schrödinger operator (i.e., the Jacobi operator acting in , which is described by the infinite-dimensional matrix (1) with and , ) under certain assumptions on the potential . These results are described in Section 4 and Theorem 5 therein.

A part of the results described here were given in the preprint [11].

2. Proof of Theorem 1

First, let us prove the theorem for the particular choice of the Jacobi matrix (1), which is specified by the relations

In this case, the solution of the eigenvalue problem (5) is well known,

with , and

This yields

and equality (8) reduces to the form

Proof of this equality is given in Appendix.

Turning to the eigenvalue problem for a general Jacobi matrix specified in the Theorem, we first rewrite it as a difference equation

subjected to constrains (3) and supplemented with the Dirichlet boundary conditions

Let us consider two associated eigenvalue problems for the even and odd states, which are restricted to the -dimensional quotient vector space . We shall use the lower case to denote vectors in this quotient space, . Of course, for .

The vectors , , are the eigenstates of the tridiagonal matrix :

corresponding to eigenvalues , . Similarly, the vectors , , are the eigenstates of the tridiagonal matrix :

corresponding to eigenvalues , . Note that the matrices and are simply related

with the projecting matrix , , and rank .

It is convenient now to allow parameters in the matrices (17), (18) to take complex values.

Lemma 2. The matrices and defined by (17), (18) have no common eigenvalues for arbitrary complex numbers , if for all .

Proof. For a contradiction, we shall assume that the matrices and have a common eigenvalue .
So, let us suppose that

with nonzero vectors , . We get

Here we have taken into account the fact that the matrix is symmetric. Therefore,

Since , this means that at least one of the numbers and is zero. However, if , we conclude immediately (really, if , then , since for all . And since the wave-function takes zero values at two neighbor sites , and for all , one can check, recursively, from (14) that , and, therefore, for all ) that for all . Therefore, is not an eigenvalue of . Similarly, if , we conclude that for all , and, hence, is not an eigenvalue of . This contradiction proves the lemma.

The eigenvalues and are the zeroes of the characteristic polynomials

and

respectively. It follows from Vieta’s theorem that the symmetric polynomials of eigenvalues , as well as the symmetric polynomials of eigenvalues , can be written as polynomial functions of the variables and . This is true, in particular for the function

since the product on the left-hand side is a symmetric polynomial of , and it is also a symmetric polynomial of .

However, the polynomial in the right-hand side of (25) must not depend on , i.e.,

To prove this, let us fix all at some arbitrary nonzero complex values , for all . Then (25) takes the form

where we have explicitly indicated the algebraic dependence of the eigenvalues and on the parameters . One can easily see from (27) that the polynomial has no zeroes. Really, if at some , then at least one eigenvalue of the matrix must coincide with some eigenvalue of the matrix , as it is implied by equality (27). However, this is a contradiction with Lemma 2, which guaranties that matrices and have no common eigenvalues. Since the polynomial has no zeroes, it is just a constant, and we arrive at (26).

Thus, we have proven that

with some polynomial . In order to find this polynomial explicitly, let us address some of its properties.

Lemma 3. (a) The polynomial defined by (29) is homogeneous, and its degree equals to :

(b) The polynomial can be represented in the form

where is also some polynomial of .

Proof. The statement (a) follows immediately from (29). To prove (b), let us first note that vanishes, if one of the variables takes the zero value. Really, if for some , the matrices and defined by (17), (18) become block-diagonal containing two blocks of dimensions and . Furthermore, the -dimensional blocks in their top-left corners are identical. Therefore, eigenvalues of the matrix merge with eigenvalues of the matrix in this case,

First, let us choose and vary the parameter near the zero point and keeping the rest , at some fixed nonzero values. Then, a simple perturbative analysis of equations

yields for small :

Combining this result with (29), we conclude that the polynomial has at least the second order zero at .
Second, let us choose and tune the parameter near the origin keeping all others nonzero. Then, for , one can easily obtain from (32), (33)

Therefore, two brackets and in the product on the left-hand side of (29) vanish at . Since the both brackets are of the order , the polynomial must have at least the fourth order zero at .
Finally, if vanishes for some , while all others remain nonzero, one can derive from (32), (33) for ,

Repeating for the analysis described above leads to representation (31).

Combining (30) and (31), one concludes that the polynomial on the right-hand side of (31) is uniform, and its degree is zero. Therefore, does not depend on parameters being just a constant, To fix this constant, we choose parameters according to (9). For such a choice of the Jacobi matrix, the product specified on the left-hand side of (29) is given by (13). Comparing the latter with (31), we find

and arrive at equality (8).

Remark 4. If all and the matrix (1) satisfying (3) is real, its eigenvalues are also real and nondegenerate. 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  

with the potential , supplemented with the Dirichlet boundary condition

In order to simplify further analysis, only the potentials with a compact support will be considered, i.e.,

with some . The latter requirement means that (40) looks as the free equation

for .

The Jacobi operator considered in the previous Section reduces to (41) at and .

The spectrum of the discrete Schrödinger operator defined by (40)-(44) consists of the continuous part and a finite number of discrete eigenvalues.

At a given , equations (40), (41) with omitted boundary condition (43) have two linearly independent solutions, and the general solution of (40), (41) can be written as their linear combination. Following [8], we introduce the Wronskian of two sequences , and ,

It does not depend on , if and are solutions of (40), (41).

Let us turn now to the scattering problem associated with (40)-(44), which corresponds to the case . Instead of the parameter , it is also convenient to use the momentum and the related complex parameter :

Three solutions of (40) are important for the scattering problem.

(i)The fundamental solution , which is fixed by the boundary conditions

(ii)Two Jost solutions, and , which are determined by their behavior at large and describe the out- and in-waves, respectively,

 Note that this formula is exact only in the considered case of the potential with a compact support. In the general case, it must be replaced by the asymptotical formula , as . This comment also relates to (53) and (66).

The fundamental solution can be represented as a linear combination of two Jost solutions,

The complex coefficient in the above equation is known as the Jost function. It is determined by (50) for real momenta in the interval , where it satisfies the relation

and can be written as

with real and . At large , the fundamental solution behaves as

where is the scattering amplitude and is the scattering phase. The latter can be defined in such a way that for .

Note that the Jost function can be also represented as the Wronskian of the Jost and fundamental solutions

with arbitrary .

The following exact representation holds for the Jost function in terms of the fundamental solution :

cf. (1.4.4) in [12] in the continuous case.

For , denote by the Jost function expressed in terms of the complex parameter : . The function can be analytically continued into the circle , where it has finite number of zeroes . These zeroes determine the discrete spectrum of problem (40)-(43):

Of course, these eigenvalues are real in the boundary problem with a real potential.

Besides (44), two further restrictions will be imposed in the sequel on the potential .

(i)The Jost function associated with such a potential should not have zeroes inside the circle , i.e., . In other words, the spectrum should be purely continuous.(ii)The Jost function should take nonzero values at : , and .

Conditions (i) and (ii) imply that the operator does not have bound and semibound states [12], respectively.

Since is a polynomial of the spectral parameter of the degree , the Jost function (55) expressed in the parameter is a polynomial of the degree :

where the real coefficients polynomially depend on the potential , . This indicates that the set of Jost polynomial (57) has the codimension in the -dimensional linear space of polynomials of the degree . Due to the constrains (i), (ii), we get

Let us periodically continue the scattering phase from the interval to the whole real axis . It follows from (58) that, for a potential satisfying (44), (i), (ii), the scattering phase is a smooth odd -periodical function in the whole real axis: .

The notation will be used for the scattering phase expressed in terms of the spectral parameter : , for and . Conditions (i), (ii) guarantee that

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 , and prove the following

Theorem 5. The scattering phase for the discrete Schrödinger operator which is defined by (41) with the Dirichlet boundary condition (43) and arbitrary potential obeying (44), (i), and (ii), must satisfy the equality

where indicates the principal value integral.
Equality (60) can be rewritten as well in terms of the Jost function :

where the integration path is counter-clockwise oriented.
Equality (61) can be also written in terms the zeroes of the Jost function :

Proof. To prove (60), let us consider the discrete Schrödinger eigenvalue problem (14)-(16) in the finite interval , with , , and with the even potential , of which restriction to the interval coincides with that of the potential :

As it was explained in the Introduction, the eigenfunctions of such a problem are either even, or odd with respect to the reflection (4); see (7). In both cases, the equality

must hold. On the other hand, the eigenfunction corresponding to the eigenvalue obeys the Dirichlet boundary condition at and therefore is proportional to the fundamental solution ,

where and are related according to (47). Since the potential is zero at the sites near the middle of the interval , the fundamental solution has the form (53) at . Substitution of (53) and (65) into (64) leads to the equation

which determines the spectrum in terms of the scattering phase . The equivalent but more convenient form of (66) reads

where were defined in (10).
For an arbitrary , let us put for all in equality (8) and take the logarithm of the ratio of two versions of this equality written for the potential (63) and for the zero potential, respectively. As a result, we get

where

In order to proceed to the large- limit on the right-hand side, let us introduce three one-parametric sets of the -functions , , and , where is a small parameter lying in the interval and is some fixed positive number such that

The function from the first set is defined in the real axis by the equation

At a fixed , the function monotonically increases in being odd and quasiperiodic,

For , the inverse function is a well-defined odd quasiperiodic -function of , which also smoothly depends on the parameter . The Taylor expansion of in reads as

where are -periodic odd bounded functions of .
The function representing the third set

is -periodic and even in both arguments and . Its Taylor expansion in

converges at uniformly in . The two initial partial derivatives in this expansion read as

These and all higher derivatives , , in the Taylor expansion (75) are the -periodic -functions of their arguments , which are therefore bounded in together with all their partial derivatives in and .
Let us now rewrite the sum (69) as

and expand the right-hand side in to the second order,

It follows from the above analysis that the correction term in the square brackets is uniformly bounded in and ,

with some positive independent on . Taking this into account, one obtains from (76)-(79)

where

where

and the -dependent correction term is again uniformly bounded in . Replacing in (84) the Riemann sum of the regular function of the momenta by the corresponding integral in the limit of large , one obtains

The -dependent correction term is uniformly bounded in , since .
Calculation of the large- asymptotics of is more delicate. First, we extend summation in (82) in the index from 1 till ,

where

In (86) we have taken into account the reflection symmetry of the function (87), which leads to .
Since , , and , we get from (87)

The sum in the second line of (86) reads as

Since , this sum can be replaced with exponential accuracy by the integral at large :

where is an arbitrary positive number; see formula 25.4.3 in [13]. Furthermore, since , the -dependent correction term is uniformly bounded in .
Taking into account (87), the integral on the right-hand side of (90) 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 , all , and . The integral in the second line of (91) reduces after the change of the integration variable to the form

where

Collecting (86)-(95), we get

where the -dependent error term is uniformly bounded in . Thus, we obtain from (68) and (81) the equality

where and are given by (96), (83), and (85), respectively.
In the second term, we can replace with sufficient accuracy the summation in by integration in the momentum :

The first integral on the right-hand side vanishes due to equality (93) with , and . The second line in (98) can be transformed as follows:

The first term on the right-hand side vanishes due to equality (93) with . Then, substituting for and and integrating by parts, we obtain from the second term on the right-hand side of (99)

Let us turn now to calculation of the first term on the left-hand side of equality (97). At large , one obtains

The first term on the right-hand side equals . The large- asymptotics of the sum on the right-hand side can be found as follows:

where is an arbitrary positive number. In deriving (102) we have taken into account the fact that is the -periodical function of in , which is continuous with all its derivatives [i.e., ], and used formula 25.4.3 of the monograph [13] by Abramowitz and Stegun. The integral on the right-hand side vanishes due to equality (93) with :

Collecting (101)-(103), we come to the simple formula

From (95), (97), (100), and (104), we arrive at the final result (60).
Derivation of (61) from (60) is straightforward being based on (52) and exploits properties (i), (ii), which were supposed to hold for the Jost function in Section 3.
The integrand of the integral on the left-hand side of (61) is meromorphic in the region having there only simple poles at , with residues . After straightforward application of the Cauchy’s residue theorem, one finds from (61)

Substitution of the right-hand side of (57) into this relation yields

In order prove equality (62), 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 of the Jost function .

(1) for all ; see (58).(2)The numbers either are real or else occur as complex conjugate pares in the set of roots of the real polynomial (57).

Example 6. Let us check directly that equality (62) holds for . In this case, the Jost function (57) is the polynomial of the third degree,

which is parametrized by two parameters . Let us rewrite (62) in the form

in terms of the symmetric polynomials of the Jost function zeroes,

Exploiting Vieta’s theorem, one obtains from (107)

Upon substitution of the right-hand sides into (108), we convince ourselves that equality (108) holds for arbitrary complexes and .

5. Conclusions and Open Problems

We have studied some spectral properties of two classes of Jacobi operators.

The first class is represented by the real -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 with purely continuous spectrum, of which potentials have a compact support. Note that the infinite-dimensional Jacobi matrix corresponding to the discrete Schrödinger operator has the super- and subdiagonal matrix elements for all . The scattering data of such operators must satisfy three identities (60), (61), and (62) which are proved in Theorem 5.

Obtained result could be extended and generalized in several directions.

We did not try to prove identities (60), (61) 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 they do not have a compact support. On the other hand, in the case of the potentials with nonempty discrete spectrum and/or semibound states (the latter appear if the Jost function has zeroes at the end points of the continuous spectrum; see [12]), some modified forms of (60), (61) should also exist.

It should be noted also that no counterparts of Theorems 1 and 5 are known in the continuous Sturm-Liouville theory. Simple heuristic arguments exploiting the formal continuous limit of the discrete boundary problem (14)-(16) and identity (8) lead to the following tentative form of the extension of Theorem 1 to the continuous case.

Hypothesis. Let be two Schrödinger operators

acting in “the appropriate space of function” defined in the interval , which satisfy the Dirichlet boundary conditions . Let the potential in (111) be a real even function,

which belongs to some space of “good enough” potentials. The space should contain the piecewise continuous potentials defined on .

Denote by the naturally ordered eigenvalues of the operators , which correspond to the eigenfunctions with different parities:

Then the following identity should hold:

We have performed numerical check of formula (114) for the probe singular potential with several values of the parameter in the interval by cutting the infinite products in and in (114) at different values of , with . The results of our numerical calculations agree well with (114).

It would be interesting to find the precise form and the direct proof of the above statement.

Appendix

A. Proof of Equality (13)

Let us start from the following auxiliary.

Lemma A.1. The following equality holds for all natural and integer :

Proof. Denote

Function is analytic in the complex -plane and has the second order zeroes at the points

It follows from (A.3) that the function

is analytic and has no zeroes in the complex -plane. Furthermore, this function is rational in the variable and approaches to 1 at and at . Therefore, .

Putting in (A.1) we find

Since

we get

or

To fix the sign of the right-hand side of (A.8), we have taken into account that just the first factors in the product on the left-hand side are negative at .

Thus,

To determine the product on the right-hand side we use formula 1.392.1 in [14]:

For , , we get from (A.10)

yielding

Substitution of (A.12) into (A.9) leads finally to (13).

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

The author is thankful to H. W. Diehl for many fruitful discussions. This work was funded by Deutsche Forschungsgemeinschaft (DFG), via Grant Ru 1506/1.

References

I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1988.

V. A. Chulaevsky and Y. G. Sinai, “Anderson localization for the 1-D discrete Schrödinger operator with two-frequency potential,” Communications in Mathematical Physics, vol. 125, no. 1, pp. 91–112, 1989.

M. Aizenman and S. Molchanov, “Localization at large disorder and at extreme energies: an elementary derivation,” Communications in Mathematical Physics, vol. 157, no. 2, pp. 245–278, 1993.

H. W. Diehl, D. Grüneberg, M. Hasenbusch, A. Hucht, S. B. Rutkevich, and F. M. Schmidt, “Large- N approach to thermodynamic Casimir effects in slabs with free surfaces,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 89, no. 6, 2014.

S. B. Rutkevich and H. W. Diehl, “Inverse-scattering-theory approach to the exact n→∞ solutions of O(n)ϕ4 models on films and semi-infinite systems bounded by free surfaces,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 91, no. 6, Article ID 062114, 48 pages, 2015.

P. A. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, vol. 3 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, New York, NY, USA, 1999.

M. Toda, Theory of Nonlinear Lattices, vol. 20, Springer-Verlag, Berlin, Germany, 2nd edition, 1989.

G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, vol. 72, The American Mathematical Society, Providence, RI, USA, 2000.

S. P. Novikov, “Riemann surfaces, operator fields, strings---analogues of the Fourier-Laurent bases,” Progress of Theoretical and Experimental Physics, no. 102, pp. 293–300, 1990.

H. Hochstadt, “On the construction of a Jacobi matrix from spectral data,” Linear Algebra and Its Applications, vol. 8, no. 5, pp. 435–446, 1974.

S. B. Rutkevich, “On the spectrum of the discrete 1d Schrödinger operator with an arbitrary even potential,” arXiv1404.4325v1, 2014.

K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, Texts and Monographs in Physics, Springer, New York, NY, USA, 2nd edition, 1989.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, 1972.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, London, 4th edition, 1980.