Thomas Chen and Natasa Pavlovic
Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d=2,3$ based on spacetime norms
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ABSTRACT. We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in
dimensions $d=2,3$, from an $N$-body Schr\"{o}dinger equation
describing a gas of interacting bosons in the GP scaling, in the limit $N
ightarrow\infty$.
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, which are inspired by
the solutions spaces based on space-time norms
introduced by Klainerman and Machedon.
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 Schr\"odinger equation (NLS) in $d=2,3$.