Near-threshold quantization and level densities for potential wells with weak inverse-square tails - art. no. 022101

For potential tails consisting of an inverse-square term and an additional attractive 1/r(m) term, V(r) similar to[(h) over bar (2)/(2M)][(gamma /r(2) ) - (beta (m-2)/r(m))], we derive the near-threshold quantization rule n = n (E) which is related to the level density via rho = dn/dE. For a weak inv erse-square term, -1/4 <<gamma><3/4 (and m>2), the leading contributions to n(E) are [GRAPHICS] so rho has a singular contribution proportional to -E)root gamma +1/4-1 nea r threshold. The constant B in the near-threshold quantization rule also de termines the strength of the leading contribution to the transmission proba bility through the potential tail at small positive energies. For gamma =0 we recover results derived previously for potential tails falling off faste r than 1/r(2). The weak inverse-square tails bridge the gap between the mor e strongly repulsive tails, gamma greater than or equal to3/4, where [GRAPHICS] and p remains finite at threshold, and the strongly attractive tails, y<-1/ 4, where [GRAPHICS] which corresponds to an infinite dipole series of bound states and connects to the behavior [GRAPHICS] describing infinite Rydberg-like series in potentials with longer-ranged at tractive tails falling off as 1/r(m), 0<m<2. For <gamma>= -1/4 (and m>2) we obtain [GRAPHICS] which remains finite at threshold.