On mathematical models for Bose-Einstein condensates in optical lattices (expanded version)

Our aim is to analyze the various energy functionals appearing in the physics literature and describing the behavior of a Bose-Einstein condensate in an optical lattice. We want to justify the use of some reduced models. For that purpose, we will use the semi-classical analysis developed for linear problems related to the Schr\"odinger operator with periodic potential or multiple wells potentials. We justify, in some asymptotic regimes, the reduction to low dimensional problems and analyze the reduced problems.