  The two-site Bose--Hubbard model is a simple model used to study Josephson
tunneling between two Bose--Einstein condensates. In this work we give an
overview of some mathematical aspects of this model. Using a classical
analysis, we study the equations of motion and the level curves of the
Hamiltonian. Then, the quantum dynamics of the model is investigated using
direct diagonalisation of the Hamiltonian. In both of these analyses, the
existence of a threshold coupling between a delocalised and a self-trapped
phase is evident, in qualitative agreement with experiments. We end with a
discussion of the exact solvability of the model via the algebraic Bethe
ansatz.
