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.
In this article, we focus on the scattering analysis of Sturm-Liouville type singular operator including an impulsive condition and turning point. In the classical literature, there are plenty of papers considering the positive values of the weight function in both continuous and discontinuous cases. However, this article differs from the others in terms of the impulsive condition appearing at the turning point. We generate the scattering function, resolvent operator, and discrete spectrum of the operator using the hyperbolic representations of the fundamental solutions. Finally, we create an example to show the article’s primary conclusions.
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