 11163 Thomas Chen and Natasa Pavlovic
 Derivation of the cubic NLS and GrossPitaevskii hierarchy from manybody dynamics in $d=2,3$ based on spacetime norms
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Oct 26, 11

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Abstract. We derive the defocusing cubic GrossPitaevskii (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 wellposedness
of the Cauchy problem for GP hierarchies, which are inspired by
the solutions spaces based on spacetime norms
introduced by Klainerman and Machedon.
We note that in $d=3$, this has been a wellknown 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 $d=2,3$.
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