Quantum bound states with zero binding energy

After reviewing the general properties of zero-energy quantum states, we give the explicit solutions of the Schrödinger equation with E = 0 for the class of potentials V = -|γ|/r^ν, where -∞ < ν < ∞. 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.