Jeblick, Maximilian; Leopold, Nikolai; Pickl, Peter
(2019):
Derivation of the Time Dependent GrossPitaevskii Equation in Two Dimensions.
In: Communications in Mathematical Physics, Vol. 372, No. 1: pp. 169

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Abstract
We present microscopic derivations of the defocusing twodimensional cubic nonlinear Schrodinger equation and the GrossPitaevskii equation starting from an interacting Nparticle system of bosons. We consider the interaction potential to be given either by W beta(x) = N 1+2 beta W(N beta x), for any beta > 0, or to be given by VN (x) = e(2N) V(e(N) x), for some spherical symmetric, nonnegative and compactly supported W, V. Linfinity (R2, R). In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrodinger equation in trace norm. For the latter potential VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.