On the Rigorous Derivation of the 3D Cubic Nonlinear Schr\"odinger Equation with A Quadratic Trap
Abstract
We consider the dynamics of the 3D N-body Schr\"odinger equation in the presence of a quadratic trap. We assume the pair interaction potential is N3β-1V(Nβx). 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 β in (0,2/7] by adapting and simplifying an argument in Chen and Pavlovi\'c [7] which solves the problem for β in (0,1/4) in the absence of a trap.
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