  After reviewing the general properties of zero-energy quantum states, we give
the explicit solutions of the \seq with $E=0$ for the class of potentials
$V=-|\gamma|/r^{\nu}$, where $-\infty < \nu < \infty$. For $\nu > 2$, these
solutions are normalizable and correspond to bound states, if the angular
momentum quantum number $l>0$. [These states are normalizable, even for $l=0$,
if we increase the space dimension, $D$, beyond 4; i.e. for $D>4$.] For $\nu
<-2$ the above solutions, although unbound, are normalizable. This is true even
though the corresponding potentials are repulsive for all $r$. We discuss the
physics of these unusual effects.
