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

We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$-particle Schr\"{o}dinger equation. We assume the pair interaction is $N^{3\beta -1}V(N^{\beta}\bullet).$ For interaction parameter $\beta \in(0,\frac23)$, we prove that, as $N\rightarrow \infty ,$ 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. Pavlovi\'{c} (2011) for $\beta \in (0,\frac14)$ and subsequently by X. Chen (2012) for $\beta\in (0,\frac27]$. 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 $\beta\in (0,\frac23)$. Overall, this provides an alternative approach to the mean-field program by Erd\"os-Schlein-Yau (2007), whose uniqueness proof is based upon Feynman diagram combinatorics.