# Analytical Expressions for a Hyperspherical Adiabatic Basis Suitable for a particular Three Particle Problem in 2 Dimensions

## TL;DR

This paper derives analytical expressions for the hyperspherical adiabatic basis and potential in a specific three-particle scattering problem in two dimensions, providing insights into long-range interactions.

## Contribution

It introduces analytical formulas for the hyperspherical adiabatic basis and potential in a three-particle 2D scattering model with a step potential, enhancing understanding of long-range interactions.

## Key findings

- Analytical expressions for asymptotic potentials derived
- Insights into long-range 2-body interactions in hyperspherical coordinates
- Model captures key behaviors of the fully interacting 3-body problem

## Abstract

For a particular case of three-body scattering in two dimensions, and matching analytical expressions at a transition point, we obtain accurate solutions for the hyperspherical adiabatic basis and potential. We find analytical expressions for the respective, asymptotic, inverse logarithmic and inverse power potential behaviors, that arise as functions of the radial coordinate. The model that we consider is that of two particles interacting with a repulsive step potential, a third particle acting as a spectator. The model is simple but gives insight, as the 2-body interaction is long ranged in hyperspherical coordinates. The fully interacting 3-body problem is known, numerically, to yield similar behaviors that we can now begin to understand.That, clearly, is the ultimate aim.