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We present a microscopic derivation of the defocusing two-dimensional cubicnonlinear Schr\"odinger equation as a mean field equation starting from aninteracting $N$-particle system of Bosons. We consider the interactionpotential to be given either by $W_\beta(x)=N^{-1+2 \beta}W(N^\beta x)$, forany $\beta>0$, or to be given by $V_N(x)=e^{2N} V(e^N x)$, for some sphericalsymmetric, positive and compactly supported $W,V \inL^\infty(\mathbb{R}^2,\mathbb{R})$. In both cases we prove the convergence ofthe reduced density matrix corresponding to the exact time evolution to theprojector onto the solution of the corresponding nonlinear Schr\"odingerequation in trace norm. For the latter potential $V_N$ we show that it iscrucial to take the microscopic structure of the condensate into account inorder to obtain the correct dynamics.

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