On the Rigorous Derivation of the 3D Cubic Nonlinear Schr\"odinger Equation with A Quadratic Trap

TL;DR

This paper rigorously derives the 3D cubic nonlinear Schrödinger equation with a quadratic trap from the N-body Schrödinger dynamics, extending previous results to a broader interaction regime.

Contribution

It provides a rigorous derivation of the 3D cubic NLS with a quadratic trap for a wider range of interaction parameters, confirming the space-time bound conjectured by Klainerman and Machedon.

Findings

Established the space-time bound for β in (0, 2/7]

Extended the derivation to include quadratic traps

Simplified the proof method from previous work

Abstract

We consider the dynamics of the 3D N-body Schr\"{o}dinger equation in the presence of a quadratic trap. We assume the pair interaction potential is N^{3{\beta}-1}V(N^{{\beta}}x). We justify the mean-field approximation and offer a rigorous derivation of the 3D cubic NLS with a quadratic trap. We establish the space-time bound conjectured by Klainerman and Machedon [30] for {\beta} in (0,2/7] by adapting and simplifying an argument in Chen and Pavlovi\'c [7] which solves the problem for {\beta} in (0,1/4) in the absence of a trap.