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\def\ms{\textsc{2010 Mathematics Subject Classifications} : }
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\begin{document}
	\begin{center}
		\LARGE
		On the Spectral Theory of Multidimensional Schr\"{o}dinger Operator
	\end{center}
	\vspace{1cm}
	\normalsize
	\centerline{Setenay AKDUMAN}
		\vspace{.5cm} \centerline{Department of Mathematics,
		Faculty of  Science, Dokuz Eyl\"{u}l University,}
	\centerline{T{\i}naztepe Camp., Buca, 35160, Izmir, Turkey}
	\centerline{setenayakduman@gmail.com}
	
	\vspace{.5cm} 
\begin{abstract}
The main object of my talk is the Schr\"{o}dinger operator $L(V)$ with a matrix potential on a $d-$ dimensional rectangle  with Neumann boundary conditions. The operator $L(V)$ is considered as a perturbation of the free Hamiltonian $L(0)$, when $V=0$.  Perturbative analysis of eigenvalues and eigenfunctions of $L(V)$ meets a serious difficulty due to presence of the very close eigenvalues of unperturbed operator. In general, this leads to the so-called `small denominators' problem. However, the set of unperturbed eigenvalues can be splitted into two parts called resonance and non-resonance domains, respectively. This opens a door to study the perturbations of each group for obtaining various asymptotic formulas for the eigenvalues and eigenfunctions of $L(V)$. I will talk about this problem which is a part of my PhD dissertation. 
\end{abstract}
	\vspace{.5cm} 
\ms {47F05, 35P15}
\key {Schr\"{o}dinger operator, Neumann condition, Perturbation theory.}
\begin{thebibliography}{9}
		\bibitem{veliev}Veliev, O. (2015). Multidimensional Periodic Schrödinger Operator. Springer. 
\bibitem{Kato} Kato, T. \textit{Perturbation Theory for Linear
	Operators}, Springer, Berlin (1980)

	\bibitem{Karakilicsetenay} Karak{\i}l{\i}\c{c}, S. 
	\& Akduman S. (2015). Eigenvalue Asymptotics for the Schr\"odinger Operator with a Matrix Potential in a Single Resonance Domain. \textit{Filomat, 29:1, 21-38}.
	
\end{thebibliography}



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