Pair excitations and the mean field approximation of interacting Bosons, II

We consider a large number of Bosons with interaction potential $v_N(x)=N^{3 \beta}v(N^{\beta}x)$. In earlier papers we considered a set of equations for the condensate $\phi$ and pair excitation function $k$ and proved that they provide a Fock space approximation to the exact evolution of the condensate for $\beta <\frac{1}{3}$. This result was extended to the case $\beta<\frac{1}{2}$ by E. Kuz, where it was also argued informally that the equations of our earlier work do not provide an approximation for $\beta>\frac{1}{2}$. In 2013, we introduced a coupled refinement of our original equations and conjectured that they provide a Fock space approximation in the range $0 \le \beta < 1$. In the current paper we prove that this is indeed the case for $\beta < \frac{2}{3}$, at least locally in time. In order to do that, we re-formulate the equations of \cite{GMM} in a way reminiscent of BBGKY and apply harmonic analysis techniques in the spirit of X. Chen and J. Holmer to prove the necessary estimates. In turn, these estimates provide bounds for the pair excitation function $k$.