Quadratic eigenparameter dependent discrete Sturm–Liouville equations with spectral singularities
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)where is a complex valued function, and 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 orderwhere are complex sequences, for all and where . Note that we can write the difference Eq. (1.2) in the following Sturm–Liouville form:where 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, iffor some and .
Section snippets
Jost solution and Jost function of (1.2) and (1.3)
Assume that for some and , the complex sequences and satisfyFrom [16], [17] the following result is obtained: Under the condition (2.1), Eq. (1.2) has the solutionfor , where , and are given in terms of and . Moreoverholds, where is constant and is the integer part of . Therefore is analytic with respect to z in 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 and , respectively. Sofrom (2.5), (2.6) and the definition of the eigenvalues and the spectral singularities [18]. From (2.2), (2.4), we get
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