On the Spectrum of the Non-Selfadjoint Differential Operator with an Integral Boundary Condition and Negative Weight Function
Abstract
In this paper, we shall study the spectral properties of the non-selfadjoint operator in the space $L_{\varrho }^{2}\left(\mathbb{R}_{+}\right) $ generated by the Sturm-Liouville differential equation \begin{equation*} -y^{^{\prime \prime }}+q\left( x\right) y=\omega ^{2}\varrho \left( x\right) y, \quad x \in \mathbb{R}_{+} \end{equation*} with the integral type boundary condition \begin{equation*} \int \limits_{0}^{\infty }G\left( x \right) y\left( x\right) dx+ \gamma y^{\prime }\left( 0\right) -\theta y\left( 0\right) =0 \end{equation*} and the non-standard weight function \begin{equation*} \varrho \left( x\right) =-1 \end{equation*} where $\left \vert \gamma \right \vert +\left \vert \theta \right \vert \neq 0$. There are an enormous number of papers considering the positive values of $ \varrho \left( x\right) $ for both continuous and discontinuous cases. The structure of the weight function affects the analytical properties and representations of the solutions of the equation. Differently from the classical literature, we used the hyperbolic type representations of the fundamental solutions of the equation to obtain the spectrum of the operator. Moreover, the conditions for the finiteness of the eigenvalues and spectral singularities were presented. Hence, besides generalizing the recent results, Naimark's and Pavlov's conditions were adopted for the negative weight function case.
Keywords
resolvent operator, spectral analysis, spectral singularities, Sturm-Liouville equations
Supporting Institution
None
Project Number
There is no funding for this work.
Thanks
The authors would like to express their sincere thanks to the editor and the anonymous reviewers
for their helpful comments and suggestions.
References
- [1] M. Kudu, G. M. Amiraliyev, Finite Difference Method for a Singularly Perturbed Differential Equations with Integral Boundary Condition, International Journal of Mathematics and Computation, 26(3) (2015).
- [2] A. N. Tikhonov, A. A. Samarskii, Equations of Mathematical Physics, Dover Books on Physics, Courier Corporation, 800 p., (2013).
- [3] B. S. Pavlov, On the spectral theory of non-selfadjoint differential operators, Doklady Akademii Nauk, Russian Academy of Sciences, 146(6) (1962).
- [4] V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser Verlag, (Basel), (1986).
- [5] M. A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of second order on a semi-axis, AMS Transl. 2 (1960), 103-193.
- [6] M. A. Naimark, Linear Differential Operators I, II. Ungar, (New York), (1968).
- [7] W. O. Amrein, A. M. Hinz, D. B. Pearson, (Eds.) Sturm-Liouville theory: past and present, Springer Science & Business Media, (2005).
- [8] E. Bairamov, O¨ . C¸ akar, A. M. Krall, Spectral properties, including spectral singularities, of a quadratic pencil of Schro¨dinger operators on the whole real axis, Quaestiones Mathematicae 26(1) (2003), 15-30.
- [9] A. M. Krall, E. Bairamov, O¨ . C¸ akar, Spectrum and spectral singularities of a quadratic pencil of a Schro¨dinger operator with a general boundary condition, Journal of Differential Equations 151(2) (1999), 252-267.
- [10] E. Bairamov, O¨ . Karaman, Spectral singularities of Klein-Gordon s-wave equations with an integral boundary condition, Acta Mathematica Hungarica 97(1-2) (2002), 121-131.
