Abstract
Near-threshold properties of bound and continuum states in a deeppotential with an attractive tail depend essentially on a few `tailparameters', which are determined by the properties of the potential tailbeyond the region of r-values where WKB wavefunctions are accuratesolutions of the Schrödinger equation. One of these tail parameters is alength parameter which defines the singular contribution to the leveldensity just below threshold and the reflectivity of the tail of thepotential just above threshold; another is a phase difference which,together with the length parameter, determines the mean scatteringlength. The near-threshold quantization rule and the actual scatteringlength are determined by the tail parameters together with adimensionless constant depending on the zero-energy value of the WKBaction integral. We study potentials with tails consisting of twoinverse-power terms,V(r)~-Cα/rα-Cα1/rα1,α1>α>2 and we derive exact analytical expressions for thetail parameters in the special case α1 = 2(α-1). This enablesus to demonstrate the effect of a significant non-homogeneity of thepotential tail on the results derived previously for homogeneous tails.
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