Unusual properties of some E=0E=0 localized states and the quantum-classical correspondence

For zero energy, E=0E=0, we propose some states which are perfectly well localized on the classical solutions for all the central power-law potentials V(r)=−γν/rνV(r)=−γν/rν, with γν>0γν>0, and ν<−2ν<−2, and ν>2ν>2. The correspondence principle is thus shown to hold even for very low values of the angular momentum quantum number. One of the unusual effects described here is that the exact quantum-classical connection is observed even for ν<−2ν<−2 where closed classical trajectories do not exist.