# The 3-body Coulomb problem

## TL;DR

This paper introduces an exact, simple, and fast integral equation method for solving the three-body Coulomb problem, demonstrated on the Helium atom with promising applications to semiconductor trions.

## Contribution

It presents a novel integral equation approach leveraging the two-body T-matrix for efficient three-body Coulomb problem solutions, validated on the Helium atom.

## Key findings

- Accurate ground state energy for Helium atom obtained
- Wave function directly computed from the solution
- Method shows potential for semiconductor trion problems

## Abstract

We present a general approach for the solution of the three-body problem for a general interaction, and apply it to the case of the Coulomb interaction. This approach is exact, simple and fast. It makes use of integral equations derived from the consideration of the scattering properties of the system. In particular this makes full use of the solution of the two-body problem, the interaction appearing only through the corresponding known T-matrix. In the case of the Coulomb potential we make use of a very convenient expression for the T-matrix obtained by Schwinger. As a check we apply this approach to the well-known problem of the Helium atom ground state and obtain a perfect numerical agreement with the known result for the ground state energy. The wave function is directly obtained from the corresponding solution. We expect our method to be in particular quite useful for the trion problem in semiconductors.