Derivation of the Time Dependent Two Dimensional Focusing NLS Equation

We present a microscopic derivation of the two-dimensional focusing cubic nonlinear Schrödinger equation starting from an interacting N-particle system of Bosons. The interaction potential we consider is given by $W_{\beta }\left(x\right)=N^{-1+2\beta }W\left(N^{\beta }x\right)$ for some spherically symmetric and compactly supported potential $W\in L^{\infty }(R^2,R)$. The class of initial wave functions is chosen such that the variance in energy is small. Furthermore, we assume that the Hamiltonian $H_{W_{\beta },t}=-\sum j=1N{\Delta }_j+\sum _{1\le j<k\le N}{W}_{\beta }(x_j-x_k)+\sum j=1N{A}_t\left(x_j\right)$ fulfills stability of second kind, that is $H_{W_{\beta },t}\ge -CN$. We then 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 either Sobolev trace norm, if $\parallel A_t{\parallel }_p<\infty$ for some $p>2$, or in trace norm, for more general external potentials. For trapping potentials of the form $A\left(x\right)=C|x{|}^s,C>0$, the condition $H_{W_{\beta },t}\ge -CN$ can be fulfilled for a certain class of interactions $W_{\beta }$, for all $0<\beta <\frac{s+1}{s+2}$, see Lewin et al. (Proc Am Math Soc 145:2441–2454, 2017).