Browsing by Author "Jeblick, Maximilian"

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    Derivation of the time dependent Gross-Pitaevskii equation for a class of non purely positive potentials
    Jeblick, Maximilian; Pickl, Peter
    We present a microscopic derivation of the time-dependent Gross-Pitaevskii equation starting from an interacting N-particle system of Bosons. We prove convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the respective Gross-Pitaevskii equation. Our work extends a previous result by one of us (P.P.[44]) to interaction potentials which need not to be nonnegative, but may have a sufficiently small negative part. One key estimate in our proof is an operator inequality which was first proven by Jun Yin, see [49].
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    Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions
    Jeblick, Maximilian; Leopold, Nikolai; Pickl, Peter
    We present a microscopic derivation of the defocusing two-dimensional cubic nonlinear Schr\"odinger 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 $W_\beta(x)=N^{-1+2 \beta}W(N^\beta x)$, for any $\beta>0$, 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 Schr\"odinger 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.