The NLS limit for bosons in a quantum waveguide

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.

Final version to appear in Annales Henri Poincare

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• Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics
• Twisting versus bending in quantum waveguides
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• Generalised Quantum Waveguides
• On the Time Dependent Gross Pitaevskii- and Hartree Equation
• The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases
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Cited by in corpus (6)

• Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions
• Mean-field limits of particles in interaction with quantized radiation fields
• Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons
• Derivation of the 1d Gross-Pitaevskii equation from the 3d quantum many-body dynamics of strongly confined bosons
• Derivation of the 2d Gross-Pitaevskii equation for strongly confined 3d bosons
• Quantum strips in higher dimensions