Friday, February 3, 2012

1111.6222 (Thomas Chen et al.)

Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody
dynamics in $d=2,3$ based on spacetime norms

Thomas Chen, Nataša Pavlović
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 $N\rightarrow\infty$. 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,
\cite{chpa2,chpa3,chpa4}, which are inspired by the solutions spaces based on
space-time norms introduced by Klainerman and Machedon in \cite{klma}. 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 $d=2,3$.
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