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Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schr\"odinger Equation

Authors
  • Chen, Xuwen
  • Holmer, Justin
Type
Published Article
Publication Date
Oct 01, 2013
Submission Date
Aug 18, 2013
Identifiers
DOI: 10.1007/s00205-016-0970-6
Source
arXiv
License
Yellow
External links

Abstract

We consider the dynamics of N bosons in one dimension. We assume that the pair interaction is attractive and given by N^{\beta -1}V(N^{\beta}\cdot) where \int V\leqslant 0. We develop new techniques in treating the N-body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity requirement in the uniqueness argument by half a derivative. We derive rigorously the one dimensional focusing cubic NLS with a quadratic trap as the N->\infty limit of the N-body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system.

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