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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 924628, 15 pages
Research Article

Principal Functions of Nonselfadjoint Discrete Dirac Equations with Spectral Parameter in Boundary Conditions

1Department of Mathematics, Ankara University, Tandogan, 06100 Ankara, Turkey
2Department of Mathematics, Kastamonu University, 37100 Kastamonu, Turkey

Received 18 July 2012; Accepted 12 September 2012

Academic Editor: Patricia J. Y. Wong

Copyright © 2012 Yelda Aygar et al. 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.


Let denote the operator generated in by , , , and the boundary condition , where , , , and , are complex sequences, , , and is an eigenparameter. In this paper we investigated the principal functions corresponding to the eigenvalues and the spectral singularities of .

1. Introduction

Consider the boundary value problem (BVP) in , where is a complex-valued function and is a spectral parameter. The spectral theory of the above BVP with continuous and point spectrum was investigated by Naĭmark [1]. He showed the existence of the spectral singularities in the continuous spectrum of (1.1). Note that the eigen and associated functions corresponding to the spectral singularities are not the elements of .

In [2, 3] the effect of the spectral singularities in the spectral expansion in terms of the principal vectors was considered. Some problems related to the spectral analysis of difference equations with spectral singularities were discussed in [47]. The spectral analysis of eigenparameter dependent nonselfadjoint difference equation was studied in [8, 9].

Let us consider the nonselfadjoint BVP for the discrete Dirac equations where are vector sequences, , for all ,  , and and, is a spectral parameter.

In [10] the authors proved that eigenvalues and spectral singularities of (1.2)-(1.3) have a finite number with finite multiplicities, if the condition, holds, for some and .

In this paper, we aim to investigate the principal functions corresponding to the eigenvalues and the spectral singularities of the BVP (1.2)-(1.3).

2. Discrete Spectrum of (1.2)-(1.3)

Let for some and , be satisfied. It has been shown that [10] under the condition (2.1), (1.2) has the solution for and , where Note that and () are uniquely expressed in terms of , , , and ,  as follows and for Moreover holds, where is the integer part of and is a constant. Therefore is vector-valued analytic function with respect to in and continuous in [10]. The solution is called Jost solution of (1.2).

Let us define It follows (2.2) and (2.3) that the function is analytic in , continuous up to the real axis, and We denote the set of eigenvalues and spectral singularities of by and , respectively. From the definition of the eigenvalues and spectral singularities we have [10] where . The finiteness of the multiplicities of eigenvalues and spectral singularities has been proven in [10]. Using (2.2), (2.3), and (2.8) we obtain

Definition 2.1. The multiplicity of a zero of in is called the multiplicity of the corresponding eigenvalue or spectral singularity of the BVP (1.2), (1.3).

3. Principal Functions

In this section we also assume that (2.1) holds.

Let and denote the zeros of in and with multiplicities and , respectively.

Let us define where

Definition 3.1. Let be an eigenvalue of . If the vectors , , satisfy the equations then the vector is called the eigenvector corresponding to the eigenvalue of . The vectors are called the associated vectors corresponding to . The eigenvector and the associated vectors corresponding to are called the principal vectors of the eigenvalue . The principal vectors of the spectral singularities of are defined similarly.

We define the vectors where and If is a solution of (1.2), then satisfies From (3.4) and (3.8) we get that The vectors , , and , , are the principal vectors of eigenvalues and spectral singularities of , respectively.

Theorem 3.2.

Proof. Using (3.5) we get that where , , and , are constant depending on . From (2.2) we obtain that For the principal vectors , , corresponding to the eigenvalues of we get Since for from (3.14) we obtain that or From (3.16), Holds, where Now we define the function So we get Using the boundness of and , we obtain that where If we take , we can write where Therefore we have From (3.17) and (3.25) , , .
On the other hand, since for ; using (3.12) we find that but the other terms in (3.12) belongs , so . Similarly from (3.13) we get , then we obtain that , ; .

Let us introduce Hilbert space with

Theorem 3.3. , , , .

Proof. Using (3.4), (3.14) we have for , . Since , using (3.29) we obtain Using (3.30), (2.1), and (2.7) we first obtain that where So we get from (3.31) Secondly, using (3.30) and (3.31) we obtain that and also the first part of the (3.31) obviously convergent so, we get from (3.33) and (3.34) and similarly Finally , , .


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