Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons
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 $\varepsilon$. The non-negative interaction potential is scaled such that its scattering length is positive and of order $(N/\varepsilon^2)^{-1}$, the range of the interaction scales as $(N/\varepsilon^2)^{-\beta}$ for $\beta\in(0,1)$. We prove that in the simultaneous limit $N\rightarrow\infty$ and $\varepsilon\rightarrow 0$, the condensation is preserved by the dynamics and the time evolution is asymptotically described by a cubic defocusing nonlinear Schr\"odinger 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 $\beta\geq\frac12$ 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.