On scattering problem off the potential, decreasing as inverse square of distance

TL;DR

This paper presents an exact solution to the scattering problem involving a potential that decreases as the inverse square of the distance, relevant for charged particle collisions with complex systems, and analyzes its low-energy behavior.

Contribution

It provides an exact integral representation for the $K$-matrix in the Schrödinger equation with inverse square potential, including analysis of threshold behavior for various dipole moments.

Findings

Derived the $K$-matrix representation for inverse square potentials

Analyzed low-energy threshold behavior of scattering cross sections

Applied results to electron, positron, and antiproton systems

Abstract

A solution of the scattering problem is obtained for the Schr\"odinger equation with the potential of induced dipole interaction, which decreases as the inverse square of the distance. Such a potential arises in the collision of an incident charged particle with a complex of charged particles (for example, in the collision of electrons with atoms). For the wave function, an integral equation is constructed for an arbitrary value of the orbital momentum of relative motion. By solving this equation, an exact integral representation for the $K$-matrix of the problem is obtained in terms of the wave function. This representation is used to analyze the behavior of the $K$-matrix at low energies and to obtain comprehensive information on its threshold behavior for various values of the dipole momentum. The resulting solution is applied to study the behavior of the scattering cross sections in the electron, positron and antiproton system.