Derivation of the Cubic NLS and Gross-Pitaevskii Hierarchy from Manybody Dynamics in d=3 Based on Spacetime Norms

We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension d = 3, from an N-body Schrodinger equation describing a gas of interacting bosons in the GP scaling, in the limit N -> a. The main result of this paper is the proof of convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies (Chen and PavloviA double dagger in Discr Contin Dyn Syst 27(2):715-739, 2010; http://arxiv.org/abs/0906.2984; Proc Am Math Soc 141:279-293, 2013), which are inspired by the solution spaces based on space-time norms introduced by Klainerman and Machedon (Comm Math Phys 279(1):169-185, 2008). We note that in d = 3, this has been a well-known open problem in the field. While our results do not assume factorization of the solutions, consideration of factorized solutions yields a new derivation of the cubic, defocusing nonlinear Schrodinger equation (NLS) in d = 3.