Threshold properties of attractive and repulsive 1/r² potentials

We study the near-threshold (E→0) behavior of quantum systems described by an attractive or repulsive 1/r² potential in conjunction with a shorter-ranged 1/r^m (m>2) term in the potential tail. For an attractive 1/r² potential supporting an infinite dipole series of bound states, we derive an explicit expression for the threshold value of the pre-exponential factor determining the absolute positions of the bound-state energies. For potentials consisting entirely of the attractive 1/r² term and a repulsive 1/r^m term, the exact expression for this prefactor is given analytically. For a potential barrier formed by a repulsive 1/r² term (e.g., the centrifugal potential) and an attractive 1/r^m term, we derive the leading near-threshold behavior of the transmission probability through the barrier analytically. The conventional treatment based on the WKB formula for the tunneling probability and the Langer modification of the potential yields the right energy dependence, but the absolute values of the near-threshold transmission probabilities are overestimated by a factor which depends on the strength of the 1/r² term (i.e., on the angular momentum quantum number l) and on the power m of the shorter ranged 1/r^m term. We derive a lower bound for this factor. It approaches unity for large l, but it can become arbitrarily large for fixed l and large values of m. For the realistic example l=1 and m=6, the conventional WKB treatment overestimates the exact near-threshold transmission probabilities by at least 38%.