Spectral properties of non-selfadjoint Sturm-Liouville operator equation on the real axis

Abstract

In this paper, we analyze the non-selfadjoint Sturm-Liouville operator $L$ defined in the Hilbert space $L_{2}(\mathbb{R},H)$ of vector-valued functions which are strongly-measurable and square-integrable in $ \mathbb{R} $. $L$ is defined \[L(y)=-y''+Q(x)y,\, x\in\mathbb{R} \] for every $ y \in L_{2}(\mathbb{R},H) $ where the potential $Q(x)$ is a non-selfadjoint, completely continuous operator in a separable Hilbert space $H$ for each $x\in \mathbb{R}.$ We obtain the Jost solutions of this operator and examine the analytic and asymptotic properties. Moreover, we find the point spectrum and the spectral singularities of $ L $ and also obtain the sufficient condition which assures the finiteness of the eigenvalues and spectral singularities of $ L $.