The NLS limit for bosons in a quantum waveguide

TL;DR

This paper demonstrates that the dynamics of bosons confined in a thin, curved waveguide can be effectively described by a one-dimensional nonlinear Schrödinger equation as the number of particles becomes large and the confinement tightens.

Contribution

It derives a rigorous limit connecting 3D bosonic systems in waveguides to 1D nonlinear Schrödinger equations, incorporating geometric effects.

Findings

The effective 1D NLS captures the system's evolution.

The non-linearity strength depends on the waveguide's cross-section.

Geometric features induce potential terms in the limit equation.

Abstract

We consider a system of $N$ bosons confined to a thin waveguide, i.e.\ to a region of space within an $\varepsilon$-tube around a curve in $\mathbb{R}^3$. We show that when taking simultaneously the NLS limit $N\to \infty$ and the limit of strong confinement $\varepsilon\to 0$, the time-evolution of such a system starting in a state close to a Bose-Einstein condensate is approximately captured by a non-linear Schr\"odinger equation in one dimension. The strength of the non-linearity in this Gross-Pitaevskii type equation depends on the shape of the cross-section of the waveguide, while the "bending" and the "twisting" of the waveguide contribute potential terms. Our analysis is based on an approach to mean-field limits developed by Pickl.