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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 $C=|\gamme|/r^{\nu}$, where $-\infty<\nu<\infty$. For $\nu>2$, these solutions are normalizable and correspond to bound staes, if the angular momentum quantum number $|>0$. [These states are normalizable, even for $1=0$, 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.
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