XXXV Workshop on Geometric Methods in Physics | 26.06-2.07.2016 |

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## Steven Jackson## An exactly solvable quantum four-body problem associated with the symmetries of an octacubeIn this talk we discuss the application of the theory of discrete reflection groups to integrable systems via the Bethe ansatz. As an example, 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 a Bethe ansatz based on the exceptional affine reflection group F̃_4 associated with the symmetries of the 24-cell. We also relate conserved quantities to the invariant theory of the reflection group: 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. |

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