# Integrable families of hard-core particles with unequal masses in a one-dimensional harmonic trap

## TL;DR

This paper identifies specific arrangements of unequal masses in a one-dimensional harmonic trap that lead to integrable and superintegrable dynamics, classified by Coxeter groups and solvable via Bethe ansatz-like methods.

## Contribution

It introduces new integrable mass families in a harmonic trap with hard-core particles, classified by Coxeter groups, and suggests superintegrability and maximal superintegrability.

## Key findings

- Existence of integrable mass families for any number of particles.
- Classification of integrable families by Coxeter reflection groups.
- Evidence and conjecture of superintegrability and maximal superintegrability.

## Abstract

We show that the dynamics of particles in a one-dimensional harmonic trap with hard-core interactions can be solvable for certain arrangements of unequal masses. For any number of particles, there exist two families of unequal mass particles that have integrable dynamics, and there are additional exceptional cases for three, four and five particles. The integrable mass families are classified by Coxeter reflection groups and the corresponding solutions are Bethe ansatz-like superpositions of hyperspherical harmonics in the relative hyperangular coordinates that are then restricted to sectors of fixed particle order. We also provide evidence for superintegrability of these Coxeter mass families and conjecture maximal superintegrability.