Phillips symmetric operators and their extensions

Abstract

This article is devoted to the investigation of self-adjoint (and, more generally, proper) extensions of Phillips symmetric operators (PSO). A closed densely defined symmetric operator with equal defect numbers is considered a Phillips symmetric operator if its characteristic function is a constant on ${\mathbb{C}}_{+}$. We present equivalent definitions of PSO and prove that proper extensions with real spectra of a given PSO are similar to each other. Our results imply that one-point interaction of the momentum operator $i\frac{d}{dx}+\alpha \delta (x-y)$ leads to unitarily equivalent self-adjoint operators with Lebesgue spectra. Self-adjoint operators with nontrivial spectral properties can be obtained as a result of more complicated perturbations of the momentum operator. In this way, we study special classes of perturbations which can be characterized as one-point interactions defined by the nonlocal potential $\gamma \in L_2\left(\mathbb{R}\right)$.