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

TL;DR

This paper derives the Gross-Pitaevskii hierarchy for a system of N bosons with specific interactions, showing that higher-order scattering processes vanish in the large N limit and connecting to longstanding conjectures.

Contribution

It introduces a new BBGKY hierarchy accounting for all correlation structures, enabling the derivation of the Gross-Pitaevskii hierarchy and proving the vanishing of higher-order scattering processes.

Findings

Higher-order scattering processes vanish as N approaches infinity for k ≥ 3.

The new BBGKY hierarchy shares limit points with the traditional hierarchy, facilitating analysis.

The work confirms the space-time bound conjecture for certain interaction regimes.

Abstract

We consider the dynamics of $N$ bosons in three dimensions. We assume the pair interaction is given by $N^{3\beta -1}V(N^{\beta }\cdot )$ . 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 $\beta \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 Erd\"{o}s, Schlein, and Yau in [31]. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for $\beta \in \left( 0,1\right) $ and weakly for $\beta =1$. 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 $\beta <1$, 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 $\beta \in \left( 0,1\right) $.