Half-line Schrödinger operators with no bound states

We consider Schödinger operators on the half-line, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if Δ+V has no spectrum outside of the interval [−2,2], then it has purely absolutely continuous spectrum. In the continuum case we show that if both −Δ+V and −Δ−V have no spectrum outside [0,∞), then both operators are purely absolutely continuous. These results extend to operators with finitely many bound states.