  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.
