Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons

Abstract

We consider the dynamics of $N$ interacting bosons initially exhibiting Bose-Einstein condensation. Due to an external trapping potential, the bosons are strongly confined in two spatial directions, with the transverse extension of the trap being of order $\epsilon$. The non-negative interaction potential is scaled such that its scattering length is positive and of order $(N/{\epsilon }^2{)}^{-1}$, the range of the interaction scales as $(N/{\epsilon }^2{)}^{-\beta }$ for $\beta \in (0,1)$. We prove that in the simultaneous limit $N\to \infty$ and $\epsilon \to 0$, the condensation is preserved by the dynamics and the time evolution is asymptotically described by a cubic defocusing nonlinear Schrödinger equation in one dimension, where the strength of the nonlinearity depends on the interaction and on the confining potential. This is the first derivation of a lower-dimensional effective evolution equation for singular potentials scaling with $\beta \ge \frac{1}{2}$ and lays the foundations for the derivation of the physically relevant one-dimensional Gross-Pitaevskii equation ($\beta =1$). For our analysis, we adapt an approach by Pickl to the problem with strong confinement.