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 3min a hard-wall box can be found exactly using Bethe Ansatz. The Ansatz is based on the exceptional affine reflection group (F) over tilde (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.