On Regge pole trajectories for a rational function approximation of Thomas–Fermi potentials

Abstract

The positions of two Regge poles caused by the rational function approximate Thomas–Fermi potential (a screened attractive Coulomb potential) are studied in detail as a function of the scattering energy. The leading pole is traced from the right-hand (physical) complex angular momentum plane ℜ(ℓ) > −1/2 to the left-hand (unphysical) complex angular momentum plane ℜ(ℓ) < −1/2 as the scattering energy is increased indefinitely. The second Regge pole is located in the unphysical half-plane at all energies. In this study an exact numerical and an approximate semiclassical WKB methods are discussed in some detail, where the boundary conditions of the regular (physical) Schrödinger solution become important. Transitions in behavior of Regge poles are explained in terms of Stokes lines and turning points evolution. The major change in the direction of Regge trajectories is demonstrated as a transition in the topology of Stokes lines. This phenomenon is caused by switching from one pair of turning points to another pair as the third turning point approaches the Stokes line connecting to the original turning points.