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

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.