Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 608329, 10 pages

http://dx.doi.org/10.1155/2011/608329

## Principal Functions of Non-Selfadjoint Difference Operator with Spectral Parameter in Boundary Conditions

Department of Mathematics, Ankara University, 06100 Ankara, Turkey

Received 21 January 2011; Accepted 6 April 2011

Academic Editor: SvatoslavΒ Staněk

Copyright Β© 2011 Murat Olgun 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.

#### Abstract

We investigate the principal functions corresponding to the eigenvalues and the spectral singularities of the boundary value problem (BVP) , and , where and are complex sequences, is an eigenparameter, and , for , 1.

#### 1. Introduction

Let us consider the (BVP) in , where is a complex-valued function and is a spectral parameter and . The spectral theory of the above BVP with continuous and point spectrum was investigated by Naĭmark [1]. He showed that the existence of the spectral singularities in the continuous spectrum of the BVP. He noted that the spectral singularities that belong to the continuous spectrum are the poles of the resolvents kernel but they are not the eigenvalues of the BVP. Also he showed that eigenfunctions and the associated functions (principal functions) corresponding to the spectral singularities are not the element of . The spectral singularities in the spectral expansion of the BVP in terms of principal functions have been investigated in [2]. The spectral analysis of the quadratic pencil of Schrödinger, Dirac, and Klein-Gordon operators with spectral singularities was studied in [3–8]. The spectral analysis of a non-selfadjoint difference equation with spectral parameter has been studied in [9]. In this paper, it is proved that the BVP has a finite number of eigenvalues and spectral singularities with a finite multiplicities if for some and .

Let denote difference operator of second order generated in by and with boundary condition where , are complex sequences and for all and for .

In this paper, which is extension of [9], we aim to investigate the properties of the principal functions corresponding to the eigenvalues and spectral singularities of the BVP (1.2)-(1.3).

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

Let for some and . The following result is obtained in [10, 11]: under the condition (2.1), equation (1.2) has the solution for , where and , are expressed in terms of and as Moreover, satisfies where is the integer part of and is a constant. So is continuous in and analytic in with respect to .

Let us define using (2.2) and the boundary condition (1.3) as The function is analytic in , continuous in , 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 [12] From (2.2) and (2.5), we get Let then the function is analytic in , continuous in , and . It follows from (2.6) and (2.8) that

*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) and (1.3).

#### 3. Principal Functions

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

*Definition 3.1. *Let be an eigenvalue of . If the vectors ; satisfy the equations
then 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 Moreover, if is a solution of (1.2), then satisfies From (3.2) and (3.4), we get that Consequently, the vectors ; , and ; , are the principal vectors of eigenvalues and spectral singularities of , respectively.

Theorem 3.2.

*Proof. *Using , we obtain that
where ; ; is a constant depending on .

From (2.2), we find that
For the principal vectors , , , corresponding to the eigenvalues , , of , we get
then
for , .

Since , from (3.10) we obtain that
where is a constant. Now we define the function
From (2.4), we obtain that
where . Therefore, we have
It follows from (3.11) and (3.14) that , , .

If we consider (3.10) for the principal vectors corresponding to the spectral singularities , , of and consider that for the spectral singularities, then we have
for , .

Since , from (3.15) we find that
Now we define , and using (2.4) we get
where

If we use (3.17), we obtain that
So , , .

Let us introduce Hilbert spaces with , , respectively. It is obvious that and

Theorem 3.3. *, .*

*Proof. *From (3.15), we have
for , . Therefore, we obtain that , , .

Let us choose . By Theorem 3.2 and (3.21), we get the following.

Theorem 3.4. *. *

*Proof. *The proof of theorem is trivial.

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