  We derive the Bogoliubov+U formalism to study the thermodynamical properties
of the Bose-Hubbard model. The framework can be viewed as the zero-frequency
limit of bosonic dynamical mean-field theory (B-DMFT), but equally well as an
extension of the mean-field decoupling approximation in which pair creation and
annihilation of depleted particles is taken into account. The self-energy on
the impurity site is treated variationally, minimizing the grand potential. The
theory containing just three parameters that are determined self-consistently
reproduces the T=0 phase diagrams of the three-dimensional and two-dimensional
Bose-Hubbard model with an accuracy of 1% or better. The superfluid to normal
transition at finite temperature is also reproduced well and only slightly less
accurately than in B-DMFT.
