11-163 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 (624K, pdf) Oct 26, 11

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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 $d=2,3$.

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