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Correlation structures, Many-body Scattering Processes and the Derivation of the Gross-Pitaevskii Hierarchy

We consider the dynamics of $N$ bosons in three dimensions. We assume the pair interaction is given by $N^{3β-1}V(N^{β}\cdot )$ . By studying an associated many-body wave operator, we introduce a BBGKY hierarchy which takes into account all of the interparticle singular correlation structures developed by the many-body evolution from the beginning. Assuming energy conditions on the $N$-body wave function, for $β\in \left( 0,1\right] $, we derive the Gross-Pitaevskii hierarchy with $2$-body interaction. In particular, we establish that, in the $N\rightarrow \infty $ limit, all $k$-body scattering processes vanishes if $k\geqslant 3$ and thus provide a direct answer to a question raised by Erdös, Schlein, and Yau in [31]. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for $β\in \left( 0,1\right) $ and weakly for $β=1$. Since this new BBGKY hierarchy converts the problem from a two-body estimate to a weaker three body estimate for which we have the estimates to achieve $β<1$, it then allows us to prove that all limit points of the ordinary BBGKY hierarchy satisfy the space-time bound conjectured by Klainerman and Machedon in [47] for $β\in \left( 0,1\right) $.