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 that the pair interaction is given by |N3β -1V(Nβ· )|⁠. By studying an associated many-body wave operator, we introduce a Bogoliubov–Born–Green–Kirkwood–Yvon (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 vanish if |k 3| and thus provide a direct answer to a question raised by Erdös et al. [30]. 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 [48] for |β ∈ (0,1)|⁠.