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 $N^{3β-1}V(N^β\bullet).$ For interaction parameter $β\in(0,\frac23)$, 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. Pavlović (2011) for $β\in (0,\frac14)$ and subsequently by X. Chen (2012) for $β\in (0,\frac27]$. 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 $β\in (0,\frac23)$. 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.