Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions

Abstract

We present a microscopic derivation of the defocusing two-dimensional cubic nonlinear Schrödinger equation as a mean field equation starting from an interacting $N$-particle system of Bosons. We consider the interaction potential to be given either by $W_{\hat{I}}^2(x)=N^{-1+2{\hat{I}}^2}W(N^{2}x)$, for any ${\hat{I}}^2>0$, or to be given by $V_N(x)=e^{2N}V(e^{N}x)$, for some spherical symmetric, positive and compactly supported $W,V\in L^{\infty }({\mathbb{R}}^2,\mathbb{R})$. In both cases we prove the convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential $V_N$ we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.

63 pages, improved presentation, minor changes