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 N3β−1V (N ·) . 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 ∈ (0, 1], we derive the Gross-Pitaevskii hierarchy with 2-body interaction. In particular, we establish that, in the N →∞ limit, all k-body scattering processes vanishes if k > 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 ∈ (0, 1) 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 ∈ (0, 1).