  Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an
external potential, such as that presented by an optical lattice. We present a
model of this phenomenon which we are able to analyze rigorously. The system is
a hard core lattice gas at half-filling and the optical lattice is modeled by a
periodic potential of strength $\lambda$. For small $\lambda$ and temperature,
BEC is proved to occur, while at large $\lambda$ or temperature there is no
BEC. At large $\lambda$ the low-temperature states are in a Mott insulator
phase with a characteristic gap that is absent in the BEC phase. The
interparticle interaction is essential for this transition, which occurs even
in the ground state. Surprisingly, the condensation is always into the $p=0$
mode in this model, although the density itself has the periodicity of the
imposed potential.
