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Journal of Mathematical Chemistry

, Volume 51, Issue 8, pp 2196–2213 | Cite as

An eigenfunction expansion of the non-selfadjoint Sturm–Liouville operator with a singular potential

  • Esra Kir Arpat
Original Paper

Abstract

In this paper, we consider the operator \(L\) generated in \(L^{2}\left( \mathbb{R }_{+}\right) \) by the differential expression
$$\begin{aligned} l\left( y\right) =-y^{\prime \prime }+\left[ \frac{\nu ^{2}-\frac{1}{4}}{x^{2}}+q\left( x\right) \right] y,\,\,x\in \mathbb{R }_{+}:=\left( 0,\infty \right) \end{aligned}$$
and the boundary condition
$$\begin{aligned} \underset{x\rightarrow 0}{\lim }x^{-\nu -\frac{1}{2}}y\left( x\right) =1, \end{aligned}$$
where \(q\) is a complex valued function and \(\nu \) is a complex number with \(Re\nu >0\). We have proved a spectral expansion of L in terms of the principal functions under the condition
$$\begin{aligned} \underset{x\in \mathbb{R }_{+}}{Sup}\left\{ e^{\epsilon \sqrt{x}}\left| q(x)\right| \right\} <\infty , \epsilon >0 \end{aligned}$$
taking into account the spectral singularities. We have also investigated the convergence of the spectral expansion.

Keywords

Eigenvalues Spectral singularities Resolvent  Spectral expansion 

Mathematics Subject Classification

47E05 34B05 34L05 47A10 

Notes

Acknowledgments

The author would like to thank Professor E. Bairamov for his helpful suggestions during the preparation of this work.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science Gazi UniversityTeknikokullar, AnkaraTurkey

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