  A strong-coupling expansion for the Green's functions, self-energies and
correlation functions of the Bose Hubbard model is developed. We illustrate the
general formalism, which includes all possible inhomogeneous effects in the
formalism, such as disorder, or a trap potential, as well as effects of thermal
excitations. The expansion is then employed to calculate the momentum
distribution of the bosons in the Mott phase for an infinite homogeneous
periodic system at zero temperature through third-order in the hopping. By
using scaling theory for the critical behavior at zero momentum and at the
critical value of the hopping for the Mott insulator to superfluid transition
along with a generalization of the RPA-like form for the momentum distribution,
we are able to extrapolate the series to infinite order and produce very
accurate quantitative results for the momentum distribution in a simple
functional form for one, two, and three dimensions; the accuracy is better in
higher dimensions and is on the order of a few percent relative error
everywhere except close to the critical value of the hopping divided by the
on-site repulsion. In addition, we find simple phenomenological expressions for
the Mott phase lobes in two and three dimensions which are much more accurate
than the truncated strong-coupling expansions and any other analytic
approximation we are aware of. The strong-coupling expansions and scaling
theory results are benchmarked against numerically exact QMC simulations in two
and three dimensions and against DMRG calculations in one dimension. These
analytic expressions will be useful for quick comparison of experimental
results to theory and in many cases can bypass the need for expensive numerical
simulations.
