A new expansion in ℏ: accurate calculations of Regge trajectories for singular potentials

An analytic method for computing Regge pole trajectories for potentials more singular than r−2 at the origin, is presented and a new expansion in ℏ is derived. To obtain results of higher accuracy, we use an appropriate accelerator of convergence applied to the expansion in ℏ, by building its continued fraction expansion. For the general (12,6) Lennard–Jones potential, we give an explicit analytic Regge trajectory formula that provides highly accurate results over a wide range of energies. For typical parameters used in the literature, the first continued fraction truncation gives at least five digits of accuracy, while higher approximations give more than 14 digits of accuracy.