A parabolic quasi-Sturmian approach to quantum scattering by a Coulomb-like potential

Abstract

A computational method in parabolic coordinates is proposed to treat the scattering of a charged particle from both spherically and axially symmetric Coulomb-like potentials. Specifically, the long-range part of the Hamiltonian is represented in parabolic quasi-Sturmian basis functions, while the short-range part is approximated by a Sturmian $$L^2$$ -basis-set truncated expansion. We establish an integral representation of the Coulomb Green’s function in parabolic coordinates from which we derive a convenient closed form for its matrix elements in the chosen $$L^2$$ basis set. From the Green’s function, we build quasi-Sturmian functions that are also given in closed form. Taking advantage of their adequate built-in Coulomb asymptotic behavior, scattering amplitudes are extracted as simple analytical sums that can be easily computed. The scheme, based on the proposed quasi-Sturmian approach, proves to be numerically efficient and robust as illustrated with converged results for three different scattering potentials, one of spherical and two of axial symmetry.