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 Schrödinger equation. We assume the pair interaction is N3β−1V (N•). For interaction parameter β ∈ (0, 23 ), we prove that, as N →∞, the limit points of the solutions to the BBGKY hierarchy satisfy the space-time bound conjectured by Klainerman-Machedon [37] 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 [11] for β ∈ (0, 14 ) and subsequently by X. Chen [15] for β ∈ (0, 27 ]. We build upon the approach of X. Chen but apply frequency localized Klainerman-Machedon collapsing estimates and the endpoint Strichartz estimate to extend the range to β ∈ (0, 23 ). Overall, this provides an alternative approach to the mean-field program by Erdös-Schlein-Yau [23], whose uniqueness proof is based upon Feynman diagram combinatorics.