Derivation in strong topology and global well-posedness of solutions to the Gross-Pitaevskii hierarchy

Abstract

We derive the cubic defocusing GP hierarchy in ${\mathbb R}^3$ from a bosonic $N$-particle Schr\"odinger equation as $N\rightarrow\infty$, in the strong topology corresponding to the space ${\mathcal H}_\xi^1$ introduced in \cite{chpa}. In particular, we thereby eliminate the requirement of regularity ${\mathcal H}_\xi^{1+}$ for the initial data used in \cite{CPBBGKY}. Moreover, the marginal density matrices obtained in this strong limit are allowed to be of infinite rank. This contrasts previous results where weak-* limits were derived, and subsequently enhanced to strong limits based on the condition that the limiting density matrices have finite rank. Furthermore, we prove that positive semidefiniteness of marginal density matrices is preserved over time, which we combine with results in \cite{CPHE}, to obtain the global well-posedness of solutions.