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Journal of Mathematical Chemistry

, Volume 55, Issue 5, pp 1202–1212 | Cite as

On the structure of discrete spectrum of a non-selfadjoint system of differential equations with integral boundary condition

  • Şeyhmus Yardimci
  • Esra Kir ArpatEmail author
  • Çağla Can
Original Paper
  • 152 Downloads

Abstract

In this paper, we investigated the spectrum of the operator \(L(\lambda )\) generated in Hilbert Space of vector-valued functions \(L_{2}(\mathbb {R}_{+},C_{2})\) by the system
$$\begin{aligned} iy_{1}^{'}(x,\lambda )+q_{1}(x)y_{2}(x,\lambda )&=\lambda y_{1}(x,\lambda )\\ -iy_{2}^{'}(x,\lambda )+q_{2}(x)y_{1}(x,\lambda )&=\lambda y_{2}(x,\lambda ),x\in \mathbb {R}_{+}:=(0,\infty ), \end{aligned}$$
and the integral boundary condition of the type
$$\begin{aligned} \int _{0}^{\infty }K(x,t)y(t,\lambda ){\mathrm {dt}}+\alpha y_{2}(0,\lambda )-\beta y_{1}(0,\lambda )=0 \end{aligned}$$
where \(\lambda \) is a complex parameter, \(q_{i},\,i=1,2\) are complex-valued functions and \(\alpha ,\beta \in \mathbb {C}\). K(xt) is vector fuction such that \(K(x,t)=(K_{1}(x,t),K_{2}(x,t)),\,K_{i}(x,t)\in L_{1}(0,\infty )\cap L_{2}(0,\lambda ),\,i=1,2\). Under the condition
$$\begin{aligned} \left| q_{i}(x)\right| \le ce^{-\varepsilon \sqrt{x}},c>0,\varepsilon >0,i=1,2 \end{aligned}$$
we proved that \(L(\lambda )\) has a finite number of eigenvalues and spectral singularities with finite multiplicities.

Keywords

Spectrum Spectral singularities Non-selfadjoint system of differential equations 

Mathematics Subject Classification

34L05 34B05 47E05 34L40 

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Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Şeyhmus Yardimci
    • 1
  • Esra Kir Arpat
    • 2
    Email author
  • Çağla Can
    • 3
  1. 1.Department of Mathematics, Faculty of SciencesAnkara UniversityTandoğanTurkey
  2. 2.Department of Mathematics, Faculty of SciencesGazi UniversityTeknikokullarTurkey
  3. 3.Department of MathematicsThe Graduate School of Natural and Applied Sciences, Ankara UniversityDışkapıTurkey

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