On the Klainerman-Machedon Conjecture of the Quantum BBGKY Hierarchy with Self-interaction

Abstract

We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$-particle Schrödinger equation. We assume the pair interaction is $N^{3\beta -1}V(N^{\beta }\bullet ).$ For interaction parameter $\beta \in (0,\frac{2}{3})$, we prove that, as $N\to \infty ,$ the limit points of the solutions to the BBGKY hierarchy satisfy the space-time bound conjectured by Klainerman-Machedon in 2008. This allows for the application of the Klainerman-Machedon uniqueness theorem, and hence implies that the limit is uniquely determined as a tensor product of solutions to the Gross-Pitaevski equation when the $N$-body initial data is factorized. The first result in this direction in 3D was obtained by T. Chen and N. Pavlović (2011) for $\beta \in (0,\frac{1}{4})$ and subsequently by X. Chen (2012) for $\beta \in (0,\frac{2}{7}]$. We build upon the approach of X. Chen but apply frequency localized Klainerman-Machedon collapsing estimates and the endpoint Strichartz estimate in the estimate of the potential part to extend the range to $\beta \in (0,\frac{2}{3})$. Overall, this provides an alternative approach to the mean-field program by Erdös-Schlein-Yau (2007), whose uniqueness proof is based upon Feynman diagram combinatorics.

v2, final version for Journal of the European Mathematical Society. v1 is a less technical version

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