Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space

TL;DR

This paper rigorously derives the 3D cubic defocusing NLS from quantum many-body dynamics, establishing near-optimal convergence rates in the energy space using a reformulated hierarchy approach.

Contribution

It introduces a reformulated hierarchy method with Klainerman-Machedon theory to prove convergence rates for the NLS in the energy space, improving previous results.

Findings

Established near-optimal $H^{1}$ convergence rates.

Proved a bi-scattering theorem for the NLS.

Enhanced convergence estimates with additional regularity.

Abstract

We consider the derivation of the defocusing cubic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{R}^{3}$ from quantum $N$-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering theorem for the NLS to obtain convergence rate estimates under $H^{1}$ regularity. The $H^{1}$ convergence rate estimate we obtain is almost optimal for $H^{1}$ datum, and immediately improves if we have any extra regularity on the limiting initial one-particle state.