On the Rigorous Derivation of the 3D Cubic Nonlinear Schrödinger Equation with A Quadratic Trap

Abstract

We consider the dynamics of the 3D N-body Schrödinger equation in the presence of a quadratic trap. We assume the pair interaction potential is N^{3β-1}V(N^{β}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 β in (0,2/7] by adapting and simplifying an argument in Chen and Pavlović [7] which solves the problem for β in (0,1/4) in the absence of a trap.

Revised according to the referee report. Accepted to appear in Archive for Rational Mechanics and Analysis

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• Pair excitations and the mean field approximation of interacting Bosons, II
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• Focusing Quantum Many-body Dynamics II: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D
• On the well-posedness and scattering for the Gross-Pitaevskii hierarchy via quantum de Finetti
• Local existence of solutions to Randomized Gross-Pitaevskii hierarchies
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• Dynamics of Large Boson Systems with Attractive Interaction and a Derivation of the Cubic Focusing NLS in ${\mathbb{R}}^3$
• Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space
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