Threshold properties of attractive and repulsive 1/r(2) potentials - art. no. 042102

We study the near-threshold (E-->0) behavior of quantum systems described b y an attractive or repulsive 1/r(2) potential in conjunction with a shorter -ranged 1/r(m) (m>2) term in the potential tail. For an attractive 1/r(2) p otential 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 potenti als consisting entirely of the attractive 1/r(2) 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/r(2) term (e.g., the centrifuga l potential) and an attractive 1/r(m) term, we derive the leading near-thre shold behavior of the transmission probability through the barrier analytic ally. 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 transmissi on probabilities are overestimated by a factor which depends on the strengt h of the 1/r(2) term (i.e., on the angular momentum quantum number l) and o n the power m of the shorter ranged 1/r(m) term. We derive a lower bound fo r this factor. It approaches unity for large l, but it can become arbitrari ly 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-thr eshold transmission probabilities by at least 38%.