An exactly solvable quantum four-body problem associated with the symmetries of an octacube

TL;DR

This paper presents an exact solution for a four-body quantum system with specific masses in a hard-wall box, leveraging symmetries of a four-dimensional octacube and affine reflection groups, revealing its integrability structure.

Contribution

It introduces an exactly solvable four-body quantum problem linked to the symmetries of an octacube and identifies its integrability via affine and finite reflection groups.

Findings

Eigenenergies and eigenstates obtained exactly using Bethe Ansatz.

The system exhibits Liouville integrability with conserved quantities.

The problem is associated with the symmetries of a four-dimensional octacube.

Abstract

In this article, we show that eigenenergies and eigenstates of a system consisting of four one-dimensional hard-core particles with masses $6m$, $2m$, $m$, and $3m$ in a hard-wall box can be found exactly using Bethe Ansatz. The Ansatz is based on the exceptional affine reflection group $\tilde{F}_{4}$ associated with the symmetries and tiling properties of an octacube---a Platonic solid unique to four dimensions, with no three-dimensional analogues. We also uncover the Liouville integrability structure of our problem: the four integrals of motion in involution are identified as invariant polynomials of the finite reflection group $F_{4}$, taken as functions of the components of momenta.