Near-threshold properties of bound and continuum states in a deep potential
with an attractive tail depend essentially on a few 'tail parameters', whi
ch are determined by the properties of the potential tail beyond the region
of r-values where WKB wavefunctions are accurate solutions of the Schrodin
ger equation. One of these tail parameters is a length parameter which defi
nes the singular contribution to the level density just below threshold and
the reflectivity of the tail of the potential just above threshold; anothe
r is a phase difference which, together with the length parameter, determin
es the mean scattering length. The near-threshold quantization rule and the
actual scattering length are determined by the tail parameters together wi
th a dimensionless constant depending on the zero-energy value of the WKB a
ction integral. We study potentials with tails consisting of two inverse-po
wer terms, V(r) similar to -C-alpha/r(alpha) - C-alpha1/r(alpha1), alpha (1
) > alpha > 2 and we derive exact analytical expressions for the tail param
eters in the special case alpha (1) = 2(alpha - 1). This enables us to demo
nstrate the effect of a significant non-homogeneity of the potential tail o
n the results derived previously for homogeneous tails.