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

Abstract

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 $N\to \infty$ and $\epsilon \to 0$, 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 to the problem with dimensional reduction and rely on the derivation of the one-dimensional NLS equation for interactions with softer scaling behaviour.

v3:This version is to appear in Annales Henri Poincare; v2:Gap in the argument closed; improved presentation of the results