Bounds on the tight-binding approximation for the Gross-Pitaevskii equation with a periodic potential

TL;DR

This paper rigorously justifies the tight-binding approximation for the Gross-Pitaevskii equation with a periodic potential, focusing on the validity of the discrete nonlinear Schrödinger equation in this context.

Contribution

It provides a rigorous mathematical analysis confirming the tight-binding approximation for the Gross-Pitaevskii equation with periodic potentials, including energy estimates and time-dependent solutions.

Findings

Validation of the tight-binding approximation over finite time intervals

Construction of Wannier functions based on previous work

Energy estimates supporting the approximation's accuracy

Abstract

We justify the validity of the discrete nonlinear Schrodinger equation for the tight-binding approximation in the context of the Gross-Pitaevskii equation with a periodic potential. Our construction of the periodic potential and the associated Wannier functions is based on the previous work, while our analysis involving energy estimates and Gronwall's inequality addresses time-dependent localized solutions on large but finite time intervals.