Parabolic Sturmians approach to antiproton-impact ionization of atomic hydrogen

Abstract

A square integrable parabolic Sturmian functions approach is suggested to calculate fully differential cross sections for the hydrogen ionization by impact of fast antiprotons. The problem is recast as an inhomogeneous Schrödinger equation for the Coulomb three-body system $(e^-, p^+, p^-)$. The transition amplitude is expressed as an expansion in terms of ’basis amplitudes’ that contain the angular dependence. They are provided here in closed form thereby avoiding numerical integration of very rapidly oscillating functions. The angular dependence is weighted by expansion coefficients whose evaluation is the computationally demanding part. These coefficients are expressed in terms of overlaps between the basis Sturmian functions and the result of the action of the incident channel potential action onto the initial state. Convergence with respect to the basis size is achieved numerically. Our approach makes the high enough incident energy assumption that the Born series for the Coulomb three-body Green’s function converges pretty fast, leaving the antiproton-electron interaction to be treated as a perturbation. The performed numerical calculations validate this assumption at 500 keV incident energy, and provide cross sections for a momentum transfer of 0.25 a.u. in agreement with three other theoretical results.