  We show that whole-line Schr\"odinger operators with finitely many bound
states have no embedded singular spectrum. In contradistinction, we show that
embedded singular spectrum is possible even when the bound states approach the
essential spectrum exponentially fast.
  We also prove the following result for one- and two-dimensional Schr\"odinger
operators, $H$, with bounded positive ground states: Given a potential $V$, if
both $H\pm V$ are bounded from below by the ground-state energy of $H$, then
$V\equiv 0$.
