Abstract and Applied Analysis

Volume 2012 (2012), Article ID 924628, 15 pages

doi:10.1155/2012/924628

Research Article

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

Yelda Aygar,¹ Murat Olgun,¹ and Turhan Koprubasi²

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.

Abstract.

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 [4–7]. 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 , , .

References

M. A. Naĭmark, “Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis,” American Mathematical Society Translations, vol. 16, pp. 103–193, 1960.

V. E. Lyance, “A differential operator with spectral singularities, I, II,” American Mathematical Society Translations, vol. 2, no. 60, pp. 185–225, 1967.

M. Adıvar, “Quadratic pencil of difference equations: jost solutions, spectrum, and principal vectors,” Quaestiones Mathematicae, vol. 33, no. 3, pp. 305–323, 2010.

M. Adıvar and M. Bohner, “Spectral analysis of -difference equations with spectral singularities,” Mathematical and Computer Modelling, vol. 43, no. 7-8, pp. 695–703, 2006.

M. Adivar and M. Bohner, “Spectrum and principal vectors of second order -difference equations,” Indian Journal of Mathematics, vol. 48, no. 1, pp. 17–33, 2006.

E. Bairamov and A. O. Çelebi, “Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators,” The Quarterly Journal of Mathematics, vol. 50, no. 200, pp. 371–384, 1999.

E. Bairamov and C. Coskun, “Jost solutions and the spectrum of the system of difference equations,” Applied Mathematics Letters, vol. 17, no. 9, pp. 1039–1045, 2004.

E. Bairamov, Y. Aygar, and T. Koprubasi, “The spectrum of eigenparameter-dependent discrete Sturm-Liouville equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4519–4523, 2011.

M. Adıvar and E. Bairamov, “Spectral properties of non-selfadjoint difference operators,” Journal of Mathematical Analysis and Applications, vol. 261, no. 2, pp. 461–478, 2001.

E. Bairamov and T. Koprubasi, “Eigenparameter dependent discrete Dirac equations with spectral singularities,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4216–4220, 2010.