Derivation of the 2d Gross-Pitaevskii equation for strongly confined 3d bosons

Abstract

We study the dynamics of a system of $N$ interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to a region of order $\varepsilon$. The interaction is non-negative and scaled in such a way that its scattering length is of order $(N/\varepsilon)^{-1}$, while its range is proportional to $(N/\varepsilon)^{-\beta}$ with scaling parameter $\beta\in(0,1]$. We consider the simultaneous limit $(N,\varepsilon)\to(\infty,0)$ and assume that the system initially exhibits Bose-Einstein condensation. We prove that condensation is preserved by the $N$-body dynamics, where the time-evolved condensate wave function is the solution of a two-dimensional non-linear equation. The strength of the non-linearity depends on the scaling parameter $\beta$. For $\beta\in(0,1)$, we obtain a cubic defocusing non-linear Schr\"odinger equation, while the choice $\beta=1$ yields a Gross-Pitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential.