The impact of sharp screening on the Coulomb scattering problem in three dimensions

TL;DR

This paper investigates how sharp cut-offs in Coulomb potentials affect three-dimensional scattering, deriving explicit operators and proposing a driven Schrödinger equation approach for long-range interactions.

Contribution

It introduces a new operator-based framework for Coulomb scattering with sharp cut-offs and a driven Schrödinger equation formulation for long-range potentials.

Findings

Explicit operator representation in Legendre basis.

Wave function proportionality operator derived.

Proposed driven Schrödinger equation with outgoing asymptotics.

Abstract

The scattering problem for two particles interacting via the Coulomb potential is examined for the case where the potential has a sharp cut-off at some distance. The problem is solved for two complimentary situations, firstly when the interior part of the Coulomb potential is left in the Hamiltonian and, secondly, when the long range tail is considered as the potential. The partial wave results are summed up to obtain the wave function in three dimensions. It is shown that in the domains where the wave function is expected to be proportional to the known solutions, the proportionality is given by an operator acting on the angular part of the wave function. The explicit representation for this operator is obtained in the basis of Legendre polynomials. We proposed a driven Schr\"odinger equation including an inhomogeneous term of the finite range with purely outgoing asymptotics for its solution in the case of the three dimensional scattering problem with long range potentials.