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# A study of impulsive discrete Dirac system with hyperbolic eigenparameter

## DOI

10.3906/mat-2010-29

## Abstract

Let $L$ denote the discrete Dirac operator generated in $\ell _{2}\left( %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion ,% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion ^{2}\right)$ by the difference operators of first order% \begin{equation*} \left\{ \begin{array}{cc} {\bigtriangleup y_{n}^{\left( 2\right) }+p_{n}y_{n}^{\left( 1\right) }=\lambda y_{n}^{(1)}} & \\ {\bigtriangleup y_{n-1}^{\left( 1\right) }+q_{n}y_{n}^{\left( 2\right) }=\lambda y_{n}^{(2)}}, \end{array} \text{ }n\in \mathbb{N} \setminus \left\{ k-1,k,k+1\right\} \right. \end{equation*} with boundary and impulsive conditions% \begin{equation*} \begin{array}{c} y_{0}^{(1)}=0\text{ }, \\ \\ \left( \begin{array}{c} y_{k+1}^{(1)} \\ y_{k+2}^{(2)}% \end{array}% \right) =\theta \left( \begin{array}{c} y_{k-1}^{(2)} \\ y_{k-2}^{(1)}% \end{array}% \right) ;\text{ }\theta =\left( \begin{array}{cc} \theta _{1} & \theta _{2} \\ \theta _{3} & \theta _{4}% \end{array}% \right) ,\text{ }\left\{ \theta _{i}\right\} _{i=1,2,3,4}\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion % \end{array}% \end{equation*}% where $\left\{ p_{n}\right\} _{n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion },$ $\left\{ q_{n}\right\} _{n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion }$ are real sequences, $\lambda =2\sinh \left( \frac{z}{2}\right)$ is a hyperbolic eigenparameter and $\bigtriangleup$ is forward operator. In this paper, the spectral properties of $L$ such as the spectrum, the eigenvalues, the scattering function and their properties are given with an example in the special cases under the condition% \begin{equation*} \sum\limits_{n=1}^{\infty }n\left( \left\vert p_{n}\right\vert +\left\vert q_{n}\right\vert \right)

## Keywords

Discrete Dirac equations, impulsive condition, hyperbolic eigenparameter, scattering function, eigenvalues

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