Semiclassical approach for calculating Regge-pole trajectories for singular potentials

A simple semiclassical approach, based on the investigation of the anti-Stokes line topology is presented for calculating Regge-poles trajectories for singular potentials, viz. potentials more singular than $r^{-2}$ at the origin. It uses the explicit solution of the Bohr-Sommerfeld quantization condition with the proviso that the positions of two turning points of the effective potential responsible for the Regge poles be relatively close together. We also demonstrate that due to this closeness the Regge trajectories asymptotically approach parallel equidistant straight lines with a slope of $\cot (\phi /m),$ m being the power and $\phi$ the argument of the coefficient of the potential. Illustrative results are presented for the polarization and Lennard-Jones potentials.