Abstract
We consider the dynamics of the three-dimensional 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 three-dimensional cubic nonlinear Schrödinger equation (NLS) with a quadratic trap. We establish the space-time bound conjectured by Klainerman and Machedon (Commun Math Phys 279:169–185, 2008) for \({\beta \in (0, 2/7]}\) by adapting and simplifying an argument in Chen and Pavlović (Annales Henri Poincaré, 2013) which solves the problem for \({\beta \in (0, 1/4)}\) in the absence of a trap.
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Communicated by P.-L. Lions
Dedicated to Xuqing.
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Chen, X. On the Rigorous Derivation of the 3D Cubic Nonlinear Schrödinger Equation with a Quadratic Trap. Arch Rational Mech Anal 210, 365–408 (2013). https://doi.org/10.1007/s00205-013-0645-5
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DOI: https://doi.org/10.1007/s00205-013-0645-5