Complete Non-Selfadjointness of Extensions of Symmetric Operators with Bounded Dissipative Perturbations

Abstract

Using boundary triples, we develop an abstract framework to investigate the complete non-selfadjointness of the maximally dissipative extensions of dissipative operators of the form $S+iV$, where $S$ is symmetric with equal finite defect indices and $V$ is a bounded non-negative operator. Our key example is the dissipative Schr\"odinger operator $-\tfrac{d^2}{dx^2}+\mathrm{i} V$ on the interval.