We consider Sch\"odinger 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.Comment: 34 page