Repulsive 1 / r 3 interaction

Solutions of the Schrödinger equation for 1/r-type longrange potentials play a key role in quantum physics especially in the understanding of states, both bound and continuum, that are close to a threshold @1–4#, and in the understanding of small-angle forward scattering which is dominated by contributions from large impact parameters. In the context of atomic collisions, it can be stated that cold atom collisions and highly excited molecular vibration spectra can be described by the solution of the Schrödinger equation for the proper long-range potential plus a few parameters that characterize the interactions of a shorter range @5–8#. For a potential that is asymptotically a repulsive 1/r, the importance of the pure long-range solution becomes even more profound. The repulsive nature of the potential keeps the particles away from each other so that unless the energy is sufficiently large, the particles do not ‘‘see’’ the interactions at the short range and scattering is described by the pure 1/r solution over a wide range of energies. The 1/r interaction, of which the repulsive case is studied in detail here, represents the radial dependence of resonant electric dipole-dipole, magnetic dipole-dipole, and quadrupole-monopole interactions. It plays an important role in many physical processes including collisions of similar atoms in a radiation field @9–12#, molecular spectra converging to thresholds where two fragments can interact via resonant dipole-dipole interactions @13–23#, and atom-surface interactions @24,25#. It is also important in the understanding of atom-electron and atom-ion interactions when the atom involved is in a state that possesses a permanent quadrupole moment. Despite its significance, the 1/r interaction is one of the most poorly understood in the sense that it is the only longrange potential of the form of 1/r (n being a positive integer! for which even the threshold behavior has not been rigorously derived @26–28#. Mathematically, this difficulty originates from the fact that the radial Schrödinger equation for a potential of the form of 1/r with n.2 has two irregular singularities, one at r50 and the other at r5` . A second-order differential equation of this type cannot be solved by a straightforward application of the power-series expansion that gave the Coulomb and harmonic-oscillator solutions. Using methods that have been developed through the so-