Jost functions and singular attractive potentials

We use Jost functions to determine the leading and next-to-leading terms of the phase shifts delta(l)(k) in the case of homogeneous attractive singular potentials -1/r(alpha), alpha>2, for arbitrary angular momentum l with incoming boundary conditions at small distances. The Jost solutions are obtained by solving a Volterra equation and a more general ansatz is used to fit the Jost solutions to the WKB waves in the inner region, where the WKB approximation is accurate. A connection between the phase shifts of attractive and repulsive homogeneous singular potentials is presented. Finally, an alternative formulation of the effective range theory for arbitrary l is proposed, where both, the scattering length and the effective range have the dimension of a length. The ratio of effective range and scattering length is then the same for attractive and repulsive potentials.