The NLS Limit for Bosons in a Quantum Waveguide

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).