Complete Non-Selfadjointness for Schrödinger Operators on the Semi-Axis

In this note we investigate complete non-selfadjointness for all maximally dissipative extensions of a Schr¨odinger operator on a half-line with dissipative bounded potential and dissipative boundary condition. We show that all maximally dissipative extensions that preserve the differential expression are completely non-selfadjoint. However, it is possible for maximally dissipative extensions to have a one-dimensional reducing subspace on which the operator is selfadjoint. We give a characterisation of these extensions and the corresponding subspaces and present a specific example.