Non-Self-Adjoint Singular Sturm-Liouville Problems with Boundary Conditions Dependent on the Eigenparameter

Abstract

Let $A$ denote the operator generated in $L_2\left({\mathcal{R}}_{+}\right)$ by the Sturm-Liouville problem: $-y^{\mathrm{\prime \prime }}+q\left(x\right)y={\lambda }^{2}y$, $x\in {\mathcal{R}}_{+}=[0,\infty )$, $(y^{\prime }/y)\left(0\right)=({\beta }_1\lambda +{\beta }_0)/({\alpha }_1\lambda +{\alpha }_0)$, where $q$ is a complex valued function and ${\alpha }_0,{\alpha }_1,{\beta }_0,{\beta }_1\in \mathcal{C}$, with ${\alpha }_0{\beta }_1-{\alpha }_1{\beta }_0\ne 0.$ In this paper, using the uniqueness theorems of analytic functions, we investigate the eigenvalues and the spectral singularities of $A$. In particular, we obtain the conditions on $q$ under which the operator $A$ has a finite number of the eigenvalues and the spectral singularities.