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.

Revised according to the referee reports

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Cited by in corpus (4)

• The Unconditional Uniqueness for the Energy-critical Nonlinear Schrödinger Equation on ${\mathbb{T}}^4$
• Well/Ill-posedness of the Boltzmann Equation with Soft Potential
• Well/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel
• On the mean-field and semiclassical limit from quantum $N$-body dynamics