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

TL;DR

This paper analyzes energy functionals for Bose-Einstein condensates in optical lattices, using semi-classical analysis to justify reduced models and their asymptotic regimes.

Contribution

It provides a rigorous justification for reduced low-dimensional models of Bose-Einstein condensates in optical lattices using semi-classical analysis.

Findings

Validation of reduced models in certain asymptotic regimes

Analysis of energy functionals in periodic potentials

Justification of low-dimensional approximations

Abstract

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.