Near-threshold quantization and scattering for deep potentials with attractive tails

Near-threshold properties of bound and continuum states in a deep potential with an attractive tail depend essentially on a few `tail parameters', which are determined by the properties of the potential tail beyond the region of r-values where WKB wavefunctions are accurate solutions of the Schroedinger equation. One of these tail parameters is a length parameter which defines the singular contribution to the level density just below threshold and the reflectivity of the tail of the potential just above threshold; another is a phase difference which, together with the length parameter, determines the mean scattering length. The near-threshold quantization rule and the actual scattering length are determined by the tail parameters together with a dimensionless constant depending on the zero-energy value of the WKB action integral. We study potentials with tails consisting of two inverse-power terms, V(r) approximately -C_α/r^α-C_α(1)/r^α(1), α₁>α>2 and we derive exact analytical expressions for the tail parameters in the special case α₁ = 2(α-1). This enables us to demonstrate the effect of a significant non-homogeneity of the potential tail on the results derived previously for homogeneous tails.