# A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry

Authors: S. B. Rutkevich
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the2M-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→∞ 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 -1. We obtain such requirements in the simplest case of the discrete Schrödinger operator acting in l2(N), which does not have bound and semibound states and whose potential has a compact support.


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## Body

## 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(1)H=b1a1000a1b2⋱000⋱⋱⋱000aN-2bN-1aN-1000aN-1bN,which acts in the Hilbert space CN. Throughout this paper, the dimension of this space is supposed to be even,(2)N=2M.The (real) entries of the matrix H will be subject to the following constrains:(3)bN+1-j=bj,j=1,…N,aN-j=aj,j=1,…N-1.Jacobi matrices satisfying this symmetry property were studied by Hochstadt [10].Since the HamiltonianH commutes with the reflection operator(4)R:Ψj→ΨN+1-j,where {Ψj}j=1N∈CN, the eigenvectors {Ψn}n=1N of the matrix (1) can be classified by their parities κn=±1,(5)HΨn=λnΨn,(6)RΨn=κnΨn.There are exactly M eigenvectors Ψm(ev) of the matrix H, which are even under the action of the reflection (4), and M eigenvectors Ψm(od), which are odd. We shall use different notation μm and νn for the corresponding eigenvalues,(7)HΨmev=μmΨmev,ΨmevN+1-j=Ψmevj,HΨmod=νmΨmod,ΨmodN+1-j=-Ψmodj,with j=1,…,N, m=1,…,M.As the main result, we prove the following.Theorem 1.
LetH be the real Jacobi matrix (1) satisfying (2) and (3). Then, the eigenvalues {μm}m=1M and {νm}m=1M of the matrix H specified by relations (7) obey the equality(8)∏m=1M∏n=1Mμm-νn=-1MM-1/22aMM∏j=1M-1aj2j.Proceeding then to the limitM→∞, we obtain from (8) identities (60), (61) for the scattering data corresponding to the discrete Schrödinger operator (i.e., the Jacobi operator acting in l2(N), which is described by the infinite-dimensional matrix (1) with aj=-1 and bj=2+vj, j∈N) under certain assumptions on the potential {vj}j∈N. 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 Theorem1

First, let us prove the theorem for the particular choice of the Jacobi matrix (1), which is specified by the relations(9)aj=-1,bj′=2for allj=1,…,N-1,j′=1,…,N.In this case, the solution of the eigenvalue problem (5) is well known,(10)Ψnj=sin⁡knj,λn=ωkn,kn=πnN+1,with n=1,…,N, and(11)ωp=4sin2⁡p2.This yields(12)μm=4sin2⁡2m-1π22M+1,νm=4sin2⁡2mπ22M+1,and equality (8) reduces to the form(13)∏m=1M∏n=1M4sin2⁡2m-1π22M+1-4sin2⁡2nπ22M+1=2M-1MM+1/2.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(14)ajΨnj+1+bjΨnj+aj-1Ψnj-1=λnΨnj,(15)j=1,…,N,n=1,…,N,subjected to constrains (3) and supplemented with the Dirichlet boundary conditions(16)Ψn0=ΨnN+1=0.Let us consider two associated eigenvalue problems for the even and odd states, which are restricted to theM-dimensional quotient vector space CM. We shall use the lower case ψ to denote vectors in this quotient space, {ψ(j)}j=1M∈CM. Of course, Ψ(j)=ψ(j) for j=1,…,M.The vectorsψm(ev)(j), j=1,…,M, are the eigenstates of the tridiagonal M×M matrix H(ev):(17)Hev=b1a1000…0a1b2a200…00a2b3a30…0…………………00…0aM-2bM-1aM-100…00aM-1bM+aM,corresponding to eigenvalues μm, m=1,…,M. Similarly, the vectors ψm(od)(j), j=1,…,M, are the eigenstates of the tridiagonal M×M matrix H(od):(18)Hod=b1a1000…0a1b2a200…00a2b3a30…0…………………00…0aM-2bM-1aM-100…00aM-1bM-aM,corresponding to eigenvalues νm, m=1,…,M. Note that the matrices H(ev) and H(od) are simply related(19)Hod-Hev=-2aMP,with the projecting matrix Pm,m′=δm,Mδm′,M, m,m′=1,…,M, and rank P=1.It is convenient now to allow parameters{aj,bj}j=1M in the matrices (17), (18) to take complex values.Lemma 2.
The matricesH(od) and H(ev) defined by (17), (18) have no common eigenvalues for arbitrary complex numbers {bj}j=1M, if aj≠0 for all j=1,…,M.Proof.
For a contradiction, we shall assume that the matricesH(ev) and H(od) have a common eigenvalue Λ.
So, let us suppose that(20)∑j′=1MHj,j′evxj′=Λxj,∑j′=1MHj,j′odyj′=Λyj,with nonzero vectors {xj}j=1M, {yj}j=1M. We get (21)Λ∑j=1Myjxj=∑j=1M∑j′=1MyjHj,j′evxj′=∑j=1M∑j′=1MyjHj,j′od+2aMPj,j′xj′=∑j=1M∑j′=1MyjHj,j′odxj′+2aM∑j=1M∑j′=1MyjPj,j′xj′=2aMyMxM+Λ∑j=1Myjxj.Here we have taken into account the fact that the matrix H(od) is symmetric. Therefore,(22)aMyMxM=0.Since aM≠0, this means that at least one of the numbers yM and xM is zero. However, if yM=0, we conclude immediately (really, if 0=yM≡ψ(j=M)≡Ψ(j=M), then Ψ(j=M+1)=-Ψ(j=M)=0, since Ψ(j)=-Ψ(2M+1-j) for all j=1,…,2M. And since the wave-function Ψ(j) takes zero values at two neighbor sites Ψ(M)=Ψ(M+1)=0, and aj≠0 for all j=1,…,M, one can check, recursively, from (14) that Ψ(M-1)=0,Ψ(M-2)=0,…,Ψ(1)=0, and, therefore, ψ(j)=0 for all j=1,…,M) that ym=0 for all m=1,…,M. Therefore, Λis not an eigenvalue of H(od). Similarly, if xM=0, we conclude that xm=0 for all m=1,…,M, and, hence, Λis not an eigenvalue of H(ev). This contradiction proves the lemma.The eigenvalues{μm}m=1M and {νm}m=1M are the zeroes of the characteristic polynomials(23)Pevμ=det⁡μ-Hev,and(24)Podν=det⁡ν-Hod,respectively. It follows from Vieta’s theorem that the symmetric polynomials of eigenvalues {μm}m=1M, as well as the symmetric polynomials of eigenvalues {νm}m=1M, can be written as polynomial functions of the variables {aj}j=1M and {bj}j=1M. This is true, in particular for the function(25)∏m=1M∏n=1Mμm-νn=Q~a,b,since the product on the left-hand side is a symmetric polynomial of {μm}m=1M, and it is also a symmetric polynomial of {νm}m=1M.However, the polynomialQ~(a,b) in the right-hand side of (25)must not depend on {bj}j=1M, i.e.,(26)Q~a,b≡Qa.To prove this, let us fix allaj at some arbitrary nonzero complex values aj(0)∈C, aj(0)≠0 for all j=1,…,M. Then (25) takes the form(27)∏m=1M∏n=1Mμmb-νnb=Pb,(28)Pb≡Q~a0,b,where we have explicitly indicated the algebraic dependence of the eigenvalues μm and νm on the parameters bj. One can easily see from (27) that the polynomial P(b) has no zeroes. Really, if P(b)=0 at some b=b(0), then at least one eigenvalue μm0(b(0)) of the matrix H(ev)(b(0)) must coincide with some eigenvalue νn0(b(0)) of the matrix H(od)(b(0)), as it is implied by equality (27). However, this is a contradiction with Lemma 2, 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 (26).Thus, we have proven that(29)∏m=1M∏n=1Mμm-νn=Qa,with some polynomial Q(a). In order to find this polynomial explicitly, let us address some of its properties.Lemma 3.
(a) The polynomialQ(a) defined by (29) is homogeneous, and its degree equals to M2:(30)Qta=tM2Qa.(b) The polynomial Q(a) can be represented in the form(31)Qa=a12a24…aM-12M-1aMMQ0a,where Q0(a) is also some polynomial of {aj}j=1M.Proof.
The statement (a) follows immediately from (29). To prove (b), let us first note that Q(a) vanishes, if one of the variables aj takes the zero value. Really, if aj=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,(32)μmaj=0=νmaj=0≡Λm,form=1,…,j.
First, let us choosej=1 and vary the parameter a1 near the zero point a1→0 and keeping the rest aj′, j′=2,…,M at some fixed nonzero values. Then, a simple perturbative analysis of equations(33)det⁡μ-Hev=0,det⁡ν-Hod=0,yields for small a1:(34)μ1a1=Λ1+Oa12,ν1a1=Λ1+Oa12.Combining this result with (29), we conclude that the polynomial Q(a) has at least the second order zero at a1.
Second, let us choosej=2 and tune the parameter a2 near the origin a2→0 keeping all others aj nonzero. Then, for j=1,2, one can easily obtain from (32), (33)(35)μja2=Λj+Oa22,νja2=Λj+Oa22.Therefore, two brackets (μ1-ν1) and (μ2-ν2) in the product on the left-hand side of (29) vanish at a2→0. Since the both brackets are of the order O(a22), the polynomial Q(a) must have at least the fourth order zero at a2.
Finally, ifaj vanishes for some j∈[1,M], while all others aj′ remain nonzero, one can derive from (32), (33) for m=1,…,j,(36)μmaj=Λm+Oaj2,ifj=1,…,M-1,Λm+Oaj,ifj=M,νmaj=Λm+Oaj2,ifj=1,…,M-1,Λm+Oaj,ifj=M..Repeating for j=3,…,M the analysis described above leads to representation (31).Combining (30) and (31), one concludes that the polynomial Q0(a) on the right-hand side of (31) is uniform, and its degree is zero. Therefore, Q0(a) does not depend on parameters aj being just a constant, Q0(a)≡CM. To fix this constant, we choose parameters {aj,bj}j=1M 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(37)CM=-1MM-1/22M,and arrive at equality (8).Remark 4.
If allaj≠0 and the matrix (1) satisfying (3) is real, its eigenvalues λn are also real and nondegenerate. In this case, one can order them so that(38)λ1<λ2<…<λN.It is possible to show that(39)κn=-1nsign⁡aM,forn=1,…,2M.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 l2(N) (40)Hψj=λψj,(41)Hψj=vjψj+2ψj-ψj+1-ψj-1,(42)j=1,2,…,∞,with the potential vj∈R, supplemented with the Dirichlet boundary condition(43)ψ0=0.In order to simplify further analysis, only the potentials V={vj}j∈N with a compact support will be considered, i.e.,(44)vj=0,ifj>J,with some J∈N. The latter requirement means that (40) looks as the free equation(45)2ψj-ψj-1-ψj+1=λψj,for j≥J+1.The Jacobi operator considered in the previous Section reduces to (41) at aj=-1 and bj=2+vj.The spectrumσ[H] of the discrete Schrödinger operator defined by (40)-(44) consists of the continuous part σcont[H]=[0,4] and a finite number of discrete eigenvalues.At a givenλ, 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 {ψ1(j)}j∈N, and {ψ2(j)}j∈N,(46)Wjψ1,ψ2=-ψ1jψ2j+1-ψ1j+1ψ2j,j=0,1,2,…It does not depend on j, if ψ1(j) and ψ2(j) are solutions of (40), (41).Let us turn now to the scattering problem associated with (40)-(44), which corresponds to the case 0<λ<4. Instead of the parameter λ∈σcont[H], it is also convenient to use the momentum p and the related complex parameter z=eip:(47)λ=2-2cos⁡p=2-z-z-1.Three solutions of (40) are important for the scattering problem. (i)
Thefundamental solutionφ(j,p), which is fixed by the boundary conditions(48)φ0,p=0,φ1,p=1.(ii)
TwoJost solutionsf(j,p), and f(j,-p), which are determined by their behavior at large j and describe the out- and in-waves, respectively,(49)fj,±p=exp⁡±ipj=z±j,atj≥J+1.Note that this formula is exact only in the considered case of the potentialV with a compact support. In the general case, it must be replaced by the asymptotical formula f(j,±p)~exp⁡(±ipj)=z±j, as j→∞. This comment also relates to (53) and (66).The fundamental solutionφ(j,p) can be represented as a linear combination of two Jost solutions,(50)φj,p=i2sin⁡pFpfj,-p-F-pfj,p,0<p<π.The complex coefficient F(p) in the above equation is known as the Jost function. It is determined by (50) for real momenta p in the interval p∈(-π,π), where it satisfies the relation(51)F-p=Fp∗,and can be written as(52)Fp=exp⁡σp-iηp,with real σ(p) and η(p). At large j≥J+1, the fundamental solution behaves as(53)φj,p=Apsin⁡psin⁡pj+ηp,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 functionF(p) can be also represented as the Wronskian of the Jost and fundamental solutions(54)Fp=Wjφp,fp,with arbitrary j∈N.The following exact representation holds for the Jost functionF(p) in terms of the fundamental solution φ(j,p):(55)Fp=1+∑j=1Jeipjvjφj,p,cf. (1.4.4) in [12] in the continuous case.Forz=1, denote by F^z the Jost function F(p) expressed in terms of the complex parameter z: F(p)=F^(z=eip). The function F^(z) can be analytically continued into the circle |z|<1, where it has finite number of zeroes {αn}n=1N. These zeroes determine the discrete spectrum {λn}n=1N of problem (40)-(43):(56)λn=2-αn-αn-1,forn=1,…,N.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 V.(i)
The Jost functionF^(z) associated with such a potential should not have zeroes inside the circle z<1, i.e., N=0. In other words, the spectrum σ[H] should be purely continuous.(ii)
The Jost functionF^(z) should take nonzero values at z=±1: F^(1)≠0, and F^(-1)≠0.Conditions (i) and (ii) imply that the operatorH does not have bound and semibound states [12], respectively.Sinceφ(j,p) is a polynomial of the spectral parameter λ=2-z-z-1 of the degree j-1, the Jost function (55) expressed in the parameter z is a polynomial of the degree 2J-1:(57)F^z=1+∑j=12J-1cjVzj=∏n=12J-11-zznV,where the real coefficients cj(V) polynomially depend on the potential vj, j=1,…,J. This indicates that the set of Jost polynomial (57) has the codimension J-1 in the 2J-dimensional linear space of polynomials of the degree 2J-1. Due to the constrains (i), (ii), we get(58)znV>1,for alln=1,…,2J-1.Let us periodically continue the scattering phaseη(p) from the interval (-π,π) to the whole real axis p∈R. It follows from (58) that, for a potential V satisfying (44), (i), (ii), the scattering phase is a smooth odd 2π-periodical function in the whole real axis: η(p)∈C∞(R/2πZ).The notationδ(λ) will be used for the scattering phase η(p) expressed in terms of the spectral parameter λ: η(p)=δ(λ=2-2cos⁡p), for 0≤λ≤4 and 0≤p≤π. Conditions (i), (ii) guarantee that(59)δ0=δ4=0.


## 4. Identities on the Scattering Data for the Discrete Schrödinger Operator

In this Section we apply Theorem1 to study some spectral properties of the discrete Schrödinger operator acting in l2(N), and prove the followingTheorem 5.
The scattering phaseδ(λ) for the discrete Schrödinger operator H∈End[l2(N)] which is defined by (41) with the Dirichlet boundary condition (43) and arbitrary potential V={vj}j=1∞ obeying (44), (i), and (ii), must satisfy the equality(60)∫04dλδλλ-2λλ-4+1π∫04dλ1δλ1P∫04dλ2δ′λ2λ2-λ1=0,where P∫‍ indicates the principal value integral.
Equality (60) can be rewritten as well in terms of the Jost function F^(z):(61)log⁡F^z=1+log⁡F^z=-1+∮z=1dz2πilog⁡F^1zdlog⁡F^zdz=0,where the integration path is counter-clockwise oriented.
Equality (61) can be also written in terms the zeroes {zn(V)}n=12J-1 of the Jost function F^(z):(62)∏1≤m<n≤2J-11-1znVzmV=1.Proof.
To prove (60), let us consider the discrete Schrödinger eigenvalue problem (14)-(16) in the finite interval 1≤j≤N=2M, with aj=-1, M>J, and with the even potential V(M)={vj(M)=bj-2}j=12M, of which restriction to the interval [1,M] coincides with that of the potential V={vj}j=1∞:(63)vjM=vj,ifj≤J,0,ifJ<j≤2M-J,v2M+1-j,if2M-J<j≤2M.As it was explained in the Introduction, the eigenfunctions Ψl(j) of such a problem are either even, or odd with respect to the reflection (4); see (7). In both cases, the equality(64)ΨlM2=ΨlM+12must hold. On the other hand, the eigenfunction Ψl(j) corresponding to the eigenvalue λl obeys the Dirichlet boundary condition at j=0 and therefore is proportional to the fundamental solution φ(j,pl),(65)Ψlj=Clφj,pl,where pl and λl are related according to (47). Since the potential vj is zero at the sites j=M-2,M-1,M,M+1,M+2 near the middle of the interval [1,2M], the fundamental solution φ(j,pl) has the form (53) at j=M,M+1. Substitution of (53) and (65) into (64) leads to the equation(66)sin2⁡plM+ηpl=sin2⁡plM+1+ηpl,which determines the spectrum λl≡ω(pl) in terms of the scattering phase η(p). The equivalent but more convenient form of (66) reads(67)2M+1pl+2ηpl=2M+1kl,where kl were defined in (10).
For an arbitraryM>J, let us put aj=-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 (63) and for the zero potential, respectively. As a result, we get(68)∑m=1MSm=0,where(69)Sm=∑n=1Mln⁡ωp2n-1-ωp2m-ln⁡ωk2n-1-ωk2m.
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;ϵ), p(k;ϵ), and F(k,k′;ϵ), where ϵ is a small parameter lying in the interval 0≤ϵ≤δ and δ is some fixed positive number such that(70)δ<12maxp∈R⁡η′p-1.The function k(p;ϵ) from the first set is defined in the real axis p∈R by the equation(71)kp;ϵ=p+2ϵηp.At a fixed ϵ∈[0,δ], the function k(p;ϵ) monotonically increases in p being odd and quasiperiodic,(72)∂pkp;ϵ>0,k-p;ϵ=-kp,ϵ,kp+2π;ϵ=kp;ϵ+2π.
Forϵ∈[0,δ], the inverse function p(k;ϵ) is a well-defined odd quasiperiodic C∞-function of k∈R, which also smoothly depends on the parameter ϵ. The Taylor expansion of p(k,ϵ) in ϵ reads as(73)pk;ϵ=k-ϵηk+4ϵ2ηkη′k+∑j=3∞ϵjAjk,where Aj(k) are 2π-periodic odd bounded functions of k.
The function representing the third set(74)Fk,k′;ϵ=ln⁡ωpk;ϵ-ωpk′;ϵωk-ωk′is 2π-periodic and even in both arguments k,k′∈R and F(k,k′;ϵ)∈C∞R×R×[0,δ]. Its Taylor expansion in ϵ(75)Fk,k′;ϵ=∑n=1∞ϵnn!∂ϵnFk,k′;ϵϵ=0converges at 0≤ϵ≤δ uniformly in k,k′∈R2. The two initial partial derivatives in this expansion read as(76)∂ϵFk,k′;ϵϵ=0=2ω′k′ηk′-ω′kηkωk-ωk′,(77)∂ϵ2Fk,k′;ϵϵ=0=4ω′kη2k′-ω′k′η2k′′ωk-ωk′-ω′k′ηk′-ω′kηkωk-ωk′2.These and all higher derivatives ∂nFk,k′;ϵ/∂ϵnϵ=0, n=3,…, in the Taylor expansion (75) are the 2π-periodic C∞[R2]-functions of their arguments k,k′∈R2, which are therefore bounded in R2 together with all their partial derivatives in k and k′.
Let us now rewrite the sum (69) as(78)Sm=∑n=1MFk2n-1,k2m;ϵϵ=2M+1-1,and expand the right-hand side in ϵ to the second order,(79)Sm=∑n=1M∂ϵFk2n-1,k2m;ϵϵ=02M+1+∂ϵ2Fk2n-1,k2m;ϵϵ=022M+12+δ3Fk2n-1,k2m;ϵϵ=2M+1-1.It follows from the above analysis that the correction term in the square brackets is uniformly bounded in n and m,(80)δ3Fk2n-1,k2m;ϵ<ϵ3C,with some positive C independent on n,m. Taking this into account, one obtains from (76)-(79)(81)Sm=Sm0+Sm1+OM-2,where(82)Sm0=12M+1∑n=1M∂ϵFk2n-1,k2m;ϵϵ=0,(83)Sm1=Xmπ2M+1.where(84)Xm=π22M+1∑n=1M∂ϵ2Fk2n-1,k2m;ϵϵ=0,and the m-dependent correction term O(M-2) is again uniformly bounded in m. Replacing in (84) the Riemann sum of the regular function of the momenta k2n-1 by the corresponding integral in the limit of large M, one obtains(85)Xm=14∫0πdq∂ϵ2Fq,k2m;ϵϵ=0+OM-1.The m-dependent correction term O(M-1) is uniformly bounded in m, since ∂ϵ2Fq,k2m;ϵϵ=0∈C∞[R/(2πZ)×R/(2πZ)].
Calculation of the large-M asymptotics of Sm(0) is more delicate. First, we extend summation in (82) in the index n from 1 till 2M+1,(86)Sm0=22M+1∑n=1MRmk2n-1=22M+1-Rmk2M+12+12∑n=12M+1Rmk2n-1,where(87)Rmq=12∂ϵFq,k2m;ϵϵ=0=ω′k2mηk2m-ω′qηqωq-ωk2m.In (86) we have taken into account the reflection symmetry Rm(q)=Rm(2π-q) of the function (87), which leads to Rm(k2n-1)=Rm(k2(2M+1-n)+1).
Sincek2M+1=π, η(π)=0, and ω(π)=4, we get from (87)(88)Rmk2M+1=ω′k2mηk2m4-ωk2m.The sum in the second line of (86) reads as(89)∑n=12M+1Rmk2n-1=∑n=12M+1Rm2πn-1/22M+1.Since Rm(q)∈C∞(R/2πZ), this sum can be replaced with exponential accuracy by the integral at large M→∞:(90)∑n=12M+1Rm2πn-1/22M+1=2M+12π∫02πdqRmq+oM-μ,where μ is an arbitrary positive number; see formula 25.4.3 in [13]. Furthermore, since ∂ϵFq,k;ϵϵ=0∈C∞[R/(2πZ)×R/(2πZ)], the m-dependent correction term o(M-μ) is uniformly bounded in m.
Taking into account (87), the integral on the right-hand side of (90) can be written as(91)∫02πdqRmq=ω′k2mηk2mP∫02πdqωq-ωk2m-P∫02πdqω′qηqωq-ωk2m.The principal value integral in the first term on the right-hand side vanishes(92)P∫02πdqωq-ωk2m=2P∫0πdqωq-ωk2m=0due to the equality(93)P∫0πdqωq-ωkν≡12limϵ→+0⁡∫0πdqωq+iϵ-ωkν+∫0πdqωq-iϵ-ωkν=0,which can be easily proved for ω(p)=2-2cos⁡p, all ν∈N, and 0<k<π. The integral in the second line of (91) reduces after the change λ=ω(q) of the integration variable to the form(94)P∫02πdqω′qηqωq-ωk2m=2Iωk2m,where(95)IΛ=P∫04dλδλλ-Λ,with0<Λ<4.
Collecting (86)-(95), we get(96)Sm0=-Iωk2mπ-12M+1ω′k2mηk2m4-ωk2m+oM-μ,where the m-dependent error term o(M-μ) is uniformly bounded in m. Thus, we obtain from (68) and (81) the equality(97)limM→∞⁡∑m=1MSm0+limM→∞⁡∑m=1MSm1=0,where Sm(0) and Sm(1) are given by (96), (83), and (85), respectively.
In the second term, we can replace with sufficient accuracy the summation inm by integration in the momentum k:(98)∑m=1MSm1=12π2∫0πdk∫0πdqω′qη2q′-ω′kη2k′ωq-ωk-12π2∫0πdk∫0πdqω′kηk-ω′qηqωq-ωk2+OM-1.The first integral on the right-hand side vanishes due to equality (93) with 0<k<π, and ν=1. The second line in (98) can be transformed as follows:(99)-12π2∫0πdk∫0πdqω′kηk-ω′qηqωq-ωk2=-1π2∫0πdkω′kηk2P∫0πdqωq-ωk2+1π2∫0πdkP∫0πdqω′kηkω′qηqωq-ωk2.The first term on the right-hand side vanishes due to equality (93) with ν=2. Then, substituting for λ1=ω(k) and λ2=ω(q) and integrating by parts, we obtain from the second term on the right-hand side of (99)(100)limM→∞⁡∑m=1MSm1=1π2∫04dλ1δλ1P∫04dλ2δ′λ2λ2-λ1.
Let us turn now to calculation of the first term on the left-hand side of equality (97). At large M, one obtains(101)∑m=1MSm0=-12π∫0πdkω′kηk4-ωk-1π∑m=1MIωk2m+OM-1.The first term on the right-hand side equals I(4)/(2π). The large-M asymptotics of the sum on the right-hand side can be found as follows:(102)∑m=1MIωk2m=-I02+12∑m=12M+1Iωk2m=-I02+2M+14π∫02πdkIωk+OM-μ,where μ is an arbitrary positive number. In deriving (102) we have taken into account the fact 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 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 ν=1:(103)∫02πdkIωk=∫02πdkP∫04dλδλλ-ωk=∫04dλδλP∫02πdkλ-ωk=0.Collecting (101)-(103), we come to the simple formula(104)limM→∞⁡∑m=1MSm0=I0+I42π.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 F^(z) in Section 3.
The integrand of the integral on the left-hand side of (61) is meromorphic in the region z>1 having there only simple poles at z=zn, n=1,…,2J-1 with residues ln⁡F^(1/zn). After straightforward application of the Cauchy’s residue theorem, one finds from (61)(105)ln⁡F^1+ln⁡F^-1-∑n=12J-1ln⁡F^1zn=0.Substitution of the right-hand side of (57) into this relation yields (106)∏1≤m<n≤2J-11-1znzm2=1.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 zn of the Jost function F^(z). (1)
zn>1 for all n=1,…,2J-1; see (58).(2)
The numberszn either are real or else occur as complex conjugate pares in the set {zn}n=12J-1 of roots of the real polynomial (57).Example 6.
Let us check directly that equality (62) holds for J=2. In this case, the Jost function (57) is the polynomial of the third degree,(107)F^z=1+zv1+v2+z2v1v2+z3v2,which is parametrized by two parameters v1,v2. Let us rewrite (62) in the form(108)σ32-σ3σ1+σ2-1σ32=1in terms of the symmetric polynomials of the Jost function zeroes, (109)σ1=z1+z2+z3,σ2=z1z2+z1z3+z2z3,σ3=z1z2z3.Exploiting Vieta’s theorem, one obtains from (107) (110)σ1=-v1,σ2=v1+v2v2,σ3=-1v2.Upon substitution of the right-hand sides into (108), we convince ourselves that equality (108) holds for arbitrary complexes v1 and v2.


## 5. Conclusions and Open Problems

We have studied some spectral properties of two classes of Jacobi operators.The first class is represented by the real2M-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 inl2(N) 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 aj=-1 for all j∈N. 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 Theorems1 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 H± be two Schrödinger operators(111)H±=-∂z2±vzacting in “the appropriate space of function” defined in the interval 0≤z≤π, which satisfy the Dirichlet boundary conditions ψ(0)=ψ(π)=0. Let the potential v(z) in (111) be a real even function,(112)vπ-z=vz,which belongs to some space L of “good enough” potentials. The space L should contain the piecewise continuous potentials defined on [0,π].Denote by{μm(±),νm(±)}m∈N the naturally ordered eigenvalues of the operators H(±), which correspond to the eigenfunctions with different parities: (113)H±ψm±,ev=μm±ψm±,ev,ψm±,evπ-z=ψm±,evz,H±ψm±,od=νm±ψm±,od,ψm±,odπ-z=-ψm±,odz,μm±<μm+1±,νm±<νm+1±,m=1,…,∞.Then the following identity should hold:(114)∏m=1∞∏n=1∞μm+-νn+2m2-2n-12=∏m=1∞∏n=1∞2m2-2n-12μm--νn-.We have performed numerical check of formula (114) for the probe singular potential v(z)=Aδ(z-π/2) with several values of the parameter A in the interval 1≤A≤5 by cutting the infinite products in n and m in (114) at different values of nmax=mmax=N, with 10≤N≤100. 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.



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*Source: 9784091-2018-10-16.xml*