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

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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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Authors and Affiliations

  1. Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI, 02912, USA

    Xuwen Chen

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  1. Xuwen Chen
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Correspondence to Xuwen Chen.

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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

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