Jeblick, Maximilian; Pickl, Peter
(2018):
Derivation of the Time Dependent Two Dimensional Focusing NLS Equation.
In: Journal of Statistical Physics, Vol. 172, No. 5: pp. 13981426

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Abstract
We present a microscopic derivation of the twodimensional focusing cubic nonlinear Schrodinger equation starting from an interacting Nparticle system of Bosons. The interaction potential we consider is given by W(x) = N 1+ 2W(N x) for some spherically symmetric and compactly supported potential W. L 8 (R2, R). The class of initial wave functions is chosen such that the variance in energy is small. Furthermore, we assume that the Hamiltonian HW, t =  Nj = 1 j + 1= j< k= N W(x j xk) + Nj = 1 At (x j) fulfills stability of second kind, that is HW, t = 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 Schrodinger equation in either Sobolev trace norm, if At p < 8 for some p > 2, or in trace norm, for more general external potentials. For trapping potentials of the form A(x) = C x s, C > 0, the condition HW, t = CN can be fulfilled for a certain class of interactions W, for all 0 < < s+ 1 s+ 2, see Lewin et al. (Proc m Math Soc 145: 24412454, 2017).