On the Spectrum of the Non-Selfadjoint Differential Operator with an Integral Boundary Condition and Negative Weight Function

Authors : Nimet COSKUN, Merve GÖRGÜLÜ

Pages : 23-29

Doi:10.32323/ujma.1216691

View : 20 | Download : 11

Publication Date : 2023-03-28

Article Type : Research Paper

Abstract :In this paper, we shall study the spectral properties of the non-selfadjoint operator in the space $L_{\\varrho }^{2}\\leftinsert ignore into journalissuearticles values(\\mathbb{R}_{+}\\right); $ generated by the Sturm-Liouville differential equation \\begin{equation*} -y^{^{\\prime \\prime }}+q\\leftinsert ignore into journalissuearticles values( x\\right); y=\\omega ^{2}\\varrho \\leftinsert ignore into journalissuearticles values( x\\right); y, \\quad x \\in \\mathbb{R}_{+} \\end{equation*} with the integral type boundary condition \\begin{equation*} \\int \\limits_{0}^{\\infty }G\\leftinsert ignore into journalissuearticles values( x \\right); y\\leftinsert ignore into journalissuearticles values( x\\right); dx+ \\gamma y^{\\prime }\\leftinsert ignore into journalissuearticles values( 0\\right); -\\theta y\\leftinsert ignore into journalissuearticles values( 0\\right); =0 \\end{equation*} and the non-standard weight function \\begin{equation*} \\varrho \\leftinsert ignore into journalissuearticles values( 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 \\leftinsert ignore into journalissuearticles values( 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