Parabolic Sturmians approach to the three-body continuum Coulomb problem

TL;DR

This paper introduces a novel approach using parabolic Sturmians and generalized coordinates to approximate solutions for the three-body Coulomb continuum problem, demonstrating numerical convergence in a helium-electron system.

Contribution

It develops a new method employing parabolic Sturmians and a Lippmann-Schwinger type equation for three-body Coulomb problems, with numerical validation.

Findings

Numerical solutions converge as basis set size increases.

The approach effectively models the three-body Coulomb continuum.

Potential expansion improves with larger basis sets.

Abstract

The three-body continuum Coulomb problem is treated in terms of the generalized parabolic coordinates. Approximate solutions are expressed in the form of a Lippmann-Schwinger type equation, where the Green's function includes the leading term of the kinetic energy and the total potential energy, whereas the potential contains the non-orthogonal part of the kinetic energy operator. As a test of this approach, the integral equation for the $(e^-,\, e^-,\, {{He}^{++}})$ system is solved numerically by using the parabolic Sturmian basis representation of the (approximate) potential. Convergence of the expansion coefficients of the solution is obtained as the basis set used to describe the potential is enlarged.