Effective mass theorems for nonlinear Schroedinger equations

TL;DR

This paper rigorously derives effective mass theorems for nonlinear Schrödinger equations with periodic potentials, justifying the effective mass approximation for Bose-Einstein condensates in optical lattices.

Contribution

It provides a rigorous homogenization analysis for nonlinear Schrödinger equations with lattice-periodic potentials, establishing effective dynamics with an effective mass tensor.

Findings

Effective mass tensor depends on initial Bloch eigenvalue

Homogenized nonlinear Schrödinger equation accurately describes dynamics

Justifies effective mass formalism for Bose-Einstein condensates

Abstract

We consider time-dependent nonlinear Schroedinger equations subject to smooth, lattice-periodic potentials plus additional confining potentials, slowly varying on the lattice scale. After an appropriate scaling we study the homogenization limit for vanishing lattice spacing. Assuming well prepared initial data, the resulting effective dynamics is governed by a homogenized nonlinear Schroedinger equation with an effective mass tensor depending on the initially chosen Bloch eigenvalue. The given results rigorously justify the use of the effective mass formalism for the description of Bose-Einstein condensates on optical lattices.