  In this work we investigate the energy gap between the ground state and the
first excited state in a model of two single-mode Bose-Einstein condensates
coupled via Josephson tunneling. The energy gap is never zero when the
tunneling interaction is non-zero. The gap exhibits no local minimum below a
threshold coupling which separates a delocalised phase from a self-trapping
phase which occurs in the absence of the external potential. Above this
threshold point one minimum occurs close to the Josephson regime, and a set of
minima and maxima appear in the Fock regime. Analytic expressions for the
position of these minima and maxima are obtained. The connection between these
minima and maxima and the dynamics for the expectation value of the relative
number of particles is analysed in detail. We find that the dynamics of the
system changes as the coupling crosses these points.
