TAILIEUCHUNG - Spectrum and scattering function of the impulsive discrete Dirac systems

In this paper, we investigate analytical and asymptotic properties of the Jost solution and Jost function of the impulsive discrete Dirac equations. We also study eigenvalues and spectral singularities of these equations. Then we obtain characteristic properties of the scattering function of the impulsive discrete Dirac systems. | Turk J Math (2018) 42: 3182 – 3194 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Spectrum and scattering function of the impulsive discrete Dirac systems Elgiz BAIRAMOV,, Şeyda SOLMAZ∗, Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we investigate analytical and asymptotic properties of the Jost solution and Jost function of the impulsive discrete Dirac equations. We also study eigenvalues and spectral singularities of these equations. Then we obtain characteristic properties of the scattering function of the impulsive discrete Dirac systems. Therefore, we find the Jost function, point spectrum, and scattering function of the unperturbed impulsive equations. Key words: Dirac systems, Jost solution, scattering function, eigenvalues 1. Introduction Impulsive differential and discrete equations appear as natural descriptions of observed evolution phenomena of several real-world problems. Many physical phenomena involving thresholds, bursting rhythm models in medicine, and mathematical models in economics do exhibit impulsive differential and discrete equations [6,7,17]. Therefore the theory of impulsive equations is a new and important branch of applied mathematics, which has extensive physical and realistic mathematical models. For the general theory of impulsive differential equations, we refer to the monographs [1,2,8]. In the literature, impulsive equations are called different kinds of names. Some of these names are equations with jump condition, equations with interface condition, and equations with transmission condition. In particular, impulsive Sturm–Liouville problems have been investigate in detail in [3,4,10–15,18–22]. In the following, we will use these notations: N := {1, 2, 3, .} , N0 := {0, 1, 2, .} , N∗m0 := N\ {m0 } , Nm0

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