Near-threshold quantization and level densities for potential wells with weak inverse-square tails

For potential tails consisting of an inverse-square term and an additional attractive [Formula Presented] term, [Formula Presented] we derive the near-threshold quantization rule [Formula Presented] which is related to the level density via [Formula Presented] For a weak inverse-square term, [Formula Presented] (and [Formula Presented] the leading contributions to [Formula Presented] are [Formula Presented] so ρ has a singular contribution proportional to [Formula Presented] near threshold. The constant B in the near-threshold quantization rule also determines the strength of the leading contribution to the transmission probability through the potential tail at small positive energies. For [Formula Presented] we recover results derived previously for potential tails falling off faster than [Formula Presented] The weak inverse-square tails bridge the gap between the more strongly repulsive tails, [Formula Presented] where [Formula Presented] and ρ remains finite at threshold, and the strongly attractive tails, [Formula Presented] where [Formula Presented] which corresponds to an infinite dipole series of bound states and connects to the behavior [Formula Presented] describing infinite Rydberg-like series in potentials with longer-ranged attractive tails falling off as [Formula Presented] [Formula Presented] For [Formula Presented] (and [Formula Presented] we obtain [Formula Presented] which remains finite at threshold.