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Annales Henri Poincaré

, Volume 17, Issue 12, pp 3321–3360 | Cite as

The NLS Limit for Bosons in a Quantum Waveguide

  • Johannes von Keler
  • Stefan Teufel
Article
  • 74 Downloads

Abstract

We consider a system of N bosons confined to a thin waveguide, i.e. to a region of space within an \({\epsilon}\)-tube around a curve in \({\mathbb{R}^3}\). We show that when taking simultaneously the NLS limit \({N \to \infty}\) and the limit of strong confinement \({\epsilon \to 0}\), the time-evolution of such a system starting in a state close to a Bose–Einstein condensate is approximately captured by a non-linear Schrödinger equation in one dimension. The strength of the non-linearity in this Gross–Pitaevskii type equation depends on the shape of the cross-section of the waveguide, while the “bending” and the “twisting” of the waveguide contribute potential terms. Our analysis is based on an approach to mean-field limits developed by Pickl (On the time-dependent Gross–Pitaevskii-and Hartree equation. arXiv:0808.1178, 2008).

Keywords

Einstein Condensate Pitaevskii Equation Quantum Waveguide German Science Foundation Einstein Conden 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany

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