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

Complete Non-Selfadjointness for Schr\"odinger Operators on the Semi-Axis