A potential-splitting approach applied to the Temkin-Poet model for electron scattering off the hydrogen atom and the helium ion

TL;DR

This paper introduces a potential-splitting method for three-body scattering problems involving long-range interactions, demonstrated on electron scattering off hydrogen and helium ions within the Temkin-Poet model.

Contribution

A novel potential-splitting approach that simplifies three-body scattering calculations with long-range potentials, enabling analytical solutions for the tail part and improved boundary condition handling.

Findings

Successfully applied to electron-hydrogen and electron-helium scattering

Reduces the problem to a boundary value problem with zero boundary conditions

Enables analytical treatment of the long-range tail Hamiltonian

Abstract

The study of scattering processes in few body systems is a difficult problem especially if long range interactions are involved. In order to solve such problems, we develop here a potential-splitting approach for three body systems. This approach is based on splitting the reaction potential into a finite range core part and a long range tail part. The solution to the Schr\"odinger equation for the long range tail Hamiltonian is found analytically, and used as an incoming wave in the three body scattering problem. This reformulation of the scattering problem makes it suitable for treatment by the exterior complex scaling technique in the sense that the problem after the complex dilation is reduced to a boundary value problem with zero boundary conditions. We illustrate the method with calculations on the electron scattering off the hydrogen atom and the positive helium ion in the frame of the Temkin-Poet model.