Principal Functions of Non-Selfadjoint Difference Operator with Spectral Parameter in Boundary Conditions
Murat Olgun,1Turhan Koprubasi,1and Yelda Aygar1
Academic Editor: Svatoslav Staněk
Received21 Jan 2011
Accepted06 Apr 2011
Published16 Jun 2011
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).
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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