Analytical accurate Regge-trajectory calculation for singular potentials

Abstract

An entirely analytical method for computing Regge-pole trajectories for singular potentials (potentials diverging faster than ${r}^{\ensuremath{-}2}$ at the origin) is presented. Explicit results are presented for polarization ${(1/r}^{4})$ and Lennard-Jones (6,12) potentials. Numerical precision of the calculations is fully controlled and often better than the state of the art numerical methods. The present method combines two different analytical approaches. The first one obtains a perturbation expansion of the Regge trajectories in terms of an appropriate parameter. Then the convergence of the resulting asymptotic series is improved through Pad\'e approximations. Highly accurate results are then possible within a wide range of energies. For typical parameters used in the literature, the two-term expansion, for which explicit formulas are presented, gives at least five digits of accuracy, while higher approximations give up to 14 digits of accuracy.