Derivation of the Bose-Hubbard model in the multiscale limit
arXiv:1208.5867
Abstract
In this paper we consider a one-dimensional non-linear Schroedinger equation (NLSE) with a periodic potential. In the semiclassical limit we prove that the stationary solutions of the Bose-Hubbard equation approximate the stationary solutions of the (NLSE). In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential; as a result the phase transition from superfluid to Mott-insulator phase for Bose-Einstein condensates in a one-dimensional periodic lattice is rigorously proved.
29 pages; Keywords: Nonlinear Schroedinger and Discrete Nonlinear Schroedinger equations; Semiclassical approximation; Bose-Einstein condensates in periodic lattices