Spectral Singularities of Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions

Abstract

Let $L$ denote the operator generated in $L_2\left(R_{+}\right)$ by Sturm-Liouville equation $-y^{\mathrm{\prime \prime }}+q\left(x\right)y={\lambda }^{2}y$, $x\in R_{+}=\left[0,\infty \right)$, $y^{\prime }\left(0\right)/y\left(0\right)={\alpha }_0+{\alpha }_1\lambda +{\alpha }_2{\lambda }^2$, where $q$ is a complex-valued function and ${\alpha }_i\in ℂ$, $i=0,1,2$ with ${\alpha }_2\ne 0$. In this article, we investigate the eigenvalues and the spectral singularities of $L$ and obtain analogs ofNaimark and Pavlov conditions for $L$.