On the Klainerman–Machedon conjecture for 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 $N^{3\beta –1}V(B^{\beta })$. For the interaction parameter $\beta \in (0,2/3)$, we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the $N\to \infty$ limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for $\beta \in (0,3/5)$ in [28]. This allows, in the case $\beta \in (0,3/5)$, for the application of the Klainerman–Machedon uniqueness theorem and hence implies that the $N\to \infty$ limit of BBGKY is uniquely determined as a tensor product of solutions to the Gross–Pitaevskii 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 [11] for $\beta \in (0,1/4)$ and subsequently by X. Chen [15] for $\beta \in (0,2/7)$. 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 $\beta \in (0,2/3)$. Overall, this provides an alternative approach to the mean-field program by L. Erdős, B. Schlein, and H.-T. Yau [28], whose uniqueness proof is based upon Feynman diagram combinatorics.