An Extension of the Pais Variational Phase Shift Approximation
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The Pais variational phase shift approximation has been modified so that it can be applied to systems in which either long range interactions are present or potential resonances exist, or both. Numerical calculations arising from the modified Pais formalism are relatively simple and lead to accurate results even for the lowest partial waves. Short-range and Coulomb potentials as well as their combinations have been considered in the applications of the extended Pais method. The formalism also enables the calculation of partial derivatives of the phase shift with respect to physical quantities such as linear and orbital angular momentum with only minor additional effort. It has possible application to both electron scattering from ions, heavy-ion scattering in nuclear physics and resonance scattering. The generalized Coulomb potential (GCP) has supersymmetry and shape invariance in the r and θ dimensions, and the energy eigenvalues, the Pais variational phase shifts and corresponding wavefunctions can also be obtained by the method of supersymmetric (SUSYQM) quantum mechanics, and shape invariance. The partial derivatives of the Pais phase shifts are also relevant for orbit and glory scattering, and rainbows in collision theory and for the case of scalar waves scattering in a gravitational field by rotating black holes.