Quantum Bound States with Zero Binding Energy

Abstract

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 $V=-|γ|/r^ν$, where $-\infty < ν< \infty$. For $ν> 2$, 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 $ν<-2$ 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.