Quadratic eigenparameter dependent discrete Sturm–Liouville equations with spectral singularities

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Abstract

Let us consider the boundary value problem (BVP) for the discrete Sturm–Liouville equation(0.1)an-1yn-1+bnyn+anyn+1=λyn,nN,(0.2)(γ0+γ1λ+γ2λ2)y1+(β0+β1λ+β2λ2)y0=0,where (an) and (bn),nN are complex sequences, γi,βiC,i=0,1,2, and λ is a eigenparameter. Discussing the point spectrum, we prove that the BVP (0.1), (0.2) has a finite number of eigenvalues and spectral singularities with a finite multiplicities, ifsupnNexp(εnδ)1-an+bn<for some ε>0 and 12δ1.

Introduction

Spectral analysis of difference equations with spectral singularities has been intensively investigated in the last decade. The modeling of certain problems from engineering, economics, control theory and other areas of study has led to the rapid development of the theory of difference equations. Also, some problems for spectral theory of difference equations in connection with the classical moment problem were studied by various authors (see the monographs of Agarwal [1], Akhiezer [2], Kelley–Peterson [3] and the references therein). The spectral theory of discrete equations has also been applied to the solution of classes of nonlinear discrete equations and Toda lattices [4].

Let us consider the boundary value problem (BVP)-y+q(x)y=λ2y,0x<,y(0)-hy(0)=0where hC,q is a complex valued function, and λ is a spectral parameter. The spectral analysis of the BVP (1.1) has been studied in [5]. In that paper, the spectrum of the BVP (1.1) was investigated and it is proved that it is composed of the eigenvalues, the continuous spectrum and spectral singularities. The spectral singularities are poles of the resolvent that are imbedded in the continuous spectrum and are not eigenvalues. In the spectral expansion of the BVP (1.1) the effect of the spectral singularities in terms of the principal functions has been analyzed in [6]. The spectral analysis of a non-selfadjoint second order difference equation has been investigated in [7]. In that study, it is shown that the Jost solution of this equation has an analytic continuation to the lower half-plane and the finiteness of the eigenvalues and the spectral singularities of the difference equation is obtained as a result of this analytic continuation. The discrete spectrum of general difference equations has been studied in [8]. Some problems for spectral theory of difference equations with spectral singularities have been discussed in [9], [10], [11], [12]. The spectral analysis of eigenparameter dependent nonselfadjoint difference equation and Sturm Liouville equation was studied in [13], [14], [15].

Let us consider the non-selfadjoint BVP for the difference equation of second orderan-1yn-1+bnyn+anyn+1=λyn,nN=1,2,,(γ0+γ1λ+γ2λ2)y1+(β0+β1λ+β2λ2)y0=0,where an,bn,nN are complex sequences, an0 for all nN0,γ0β1-γ1β00,γ2+β20 and γ2-β1a0 where γi,βiC,i=0,1,2. Note that we can write the difference Eq. (1.2) in the following Sturm–Liouville form:(anΔyn)+hnyn=λyn,nN,where hn=an-1+an+bn,Δ is the forward difference operator, and is the backward difference operator.

Differently other studies in the literature, the specific feature of this paper which is one of the articles have applicability in study areas such as control theory, economics, medicine and biology is the presence of the spectral parameter not only in the difference equation but also it’s in the boundary condition at quadratic form. In this paper, we investigate the eigenvalues and the spectral singularities of the BVP (1.2), (1.3) and show that this BVP has a finite number of eigenvalues and spectral singularities with finite multiplicities, ifsupnNexp(εnδ)1-an+bn<,for some ε>0 and 12δ1.

Section snippets

Jost solution and Jost function of (1.2) and (1.3)

Assume that for some ε>0 and 12δ1, the complex sequences (an) and (bn) satisfysupnNexp(εnδ)1-an+bn<.From [16], [17] the following result is obtained: Under the condition (2.1), Eq. (1.2) has the solutionen(z)=αneinz1+m=1Anmeimz,nN0,for λ=2cosz, where zC+z:zC,Imz0, and αn,Anm are given in terms of (an) and (bn). MoreoverAnmBk=n+m21-ak+bk,holds, where B>0 is constant and m2 is the integer part of m2 . Therefore en(z) is analytic with respect to z in C+z:zC,Imz>0 and continuous

Eigenvalues and spectral singularities of (1.2) and (1.3)

We will denote the set of all eigenvalues and spectral singularities of the BVP (1.2), (1.3) by σd and σss, respectively. Soσd=λ:λ=2cosz,zP0,f(z)=0,σss=λ:λ=2cosz,z[0,2π],f(z)=00.from (2.5), (2.6) and the definition of the eigenvalues and the spectral singularities [18]. From (2.2), (2.4), we getf(z)=γ0+γ1eiz+e-iz+γ2(2+e2iz+e-2iz)α1eiz1+m=1A1meimz+β0+β1eiz+e-iz+β2(2+e2iz+e-2iz)α01+m=1A0meimz=α0β2e-2iz+α1γ2+α0β1e-iz+α1γ1+α0β0+2β2+α1γ0+2γ2+α0β1eiz+(α1γ1+α0β2)e2iz+α1γ2e3iz+m=1α0β2A0mei(m-2)

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