We derive rigorously, for both R 2 and [−L, L] ×2 , the cubic nonlinear Schrödinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.

We consider a general class of discrete nonlinear Schrödinger equations (DNLS) on the lattice hZ with mesh size h > 0. In the continuum limit when h → 0, we prove that the limiting dynamics are given by a nonlinear Schrödinger equation (NLS) on R with the fractional Laplacian (−∆) α as dispersive symbol. In particular, we obtain that fractional powers 1 2 < α < 1 arise from long-range lattice interactions when passing to the continuum limit, whereas the NLS with the usual Laplacian −∆ describes the dispersion in the continuum limit for short-range or quick-decaying interactions (e. g., nearest-neighbor interactions).Our results rigorously justify certain NLS model equations with fractional Laplacians proposed in the physics literature. Moreover, the arguments given in our paper can be also applied to discuss the continuum limit for other lattice systems with long-range interactions. n =mHere 0 < s < ∞ is a fixed parameter controlling the decay behavior of the lattice interactions. In fact, we will formulate below a generalized version of problem (1.1), where we allow for more general interaction terms of the form β(h) −1 J(|n − m|), where J is defined below, in place of the kernel h(|x m − x n |) −(1+2s) .

We study the many body quantum evolution of bosonic systems in the mean field limit. The dynamics is known to be well approximated by the Hartree equation. So far, the available results have the form of a law of large numbers. In this paper we go one step further and we show that the fluctuations around the Hartree evolution satisfy a central limit theorem. Interestingly, the variance of the limiting Gaussian distribution is determined by a time-dependent Bogoliubov transformation describing the dynamics of initial coherent states in a Fock space representation of the system.

We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates that are consistent with central limit theorems that have been established in the last years.

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