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

Abstract

We consider the derivation of the defocusing cubic nonlinear Schrödinger 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.