# Coulomb transition matrix at negative energy and integer values of interaction parameter

## TL;DR

This paper derives analytical expressions for the Coulomb transition matrix at negative energy for both repulsive and attractive interactions, focusing on integer Coulomb parameters using stereographic projection.

## Contribution

It introduces a novel method employing stereographic projection to analytically solve the Coulomb transition matrix at specific energy and parameter values.

## Key findings

- Analytical formulas for repulsive Coulomb transition matrix at positive integer parameters.
- Extension of the method to attractive Coulomb interaction at negative integer parameters.
- Simplification of the Coulomb transition matrix calculation at negative energy.

## Abstract

With the use of the stereographic projection of momentum space into the four-dimensional sphere of unit radius. the possibility of the analytical solution of the three-dimensional two-body Lippmann-Schwinger equation with the Coulomb interaction at negative energy has been studied. Simple analytical expressions for the three-dimensional Coulomb transition matrix in the case of the repulsive Coulomb interaction and positive integer values of the Coulomb parameter have been obtained. The worked out method has been also applied for the generalized three-dimensional Coulomb transition matrix in the case of the attractive Coulomb interaction and negative integer values of the Coulomb parameter.