Derivation of the 1d Gross-Pitaevskii Equation from the 3d Quantum Many-Body Dynamics of Strongly Confined Bosons

We consider the dynamics of N interacting bosons initially forming a Bose-Einstein condensate. Due to an external trapping potential, the bosons are strongly confined in two dimensions, where the transverse extension of the trap is of order epsilon. The non-negative interaction potential is scaled such that its range and its scattering length are both of order (N/epsilon 2)-1, corresponding to the Gross-Pitaevskii scaling of a dilute Bose gas. We show that in the simultaneous limit N and epsilon 0, the dynamics preserve condensation and the time evolution is asymptotically described by a Gross-Pitaevskii equation in one dimension. The strength of the nonlinearity is given by the scattering length of the unscaled interaction, multiplied with a factor depending on the shape of the confining potential. For our analysis, we adapt a method by Pickl (Rev Math Phys 27(01):1550003, 2015) to the problem with dimensional reduction and rely on the derivation of the one-dimensional NLS equation for interactions with softer scaling behaviour in Bo ss mann (Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons. arXiv preprint, 2018. arXiv:1803.11011).