Uniqueness of solutions to the 3D quintic Gross-Pitaevskii Hierarchy

TL;DR

This paper proves the unconditional uniqueness of solutions to the 3D quintic Gross-Pitaevskii hierarchy in the critical space, extending previous results to small and non-small solutions using the quantum de Finetti theorem.

Contribution

It establishes unconditional uniqueness for solutions in the critical space, including non-small solutions to the Hartree hierarchy, using an extension of existing proof techniques.

Findings

Unconditional uniqueness for small solutions in the critical space.

Unconditional uniqueness for solutions to the Hartree hierarchy without smallness.

Extension of previous methods using the quantum de Finetti theorem.

Abstract

In this paper, we study solutions to the three-dimensional quintic Gross-Pitaevskii hierarchy. We prove unconditional uniqueness among all small solutions in the critical space $\mathfrak{H}^1$ (which corresponds to $H^1$ on the NLS level). With slight modifications to the proof, we also prove unconditional uniqueness of solutions to the Hartree hierarchy without smallness condition. Our proof uses the quantum de Finetti theorem, and is an extension of the work by Chen-Hainzl-Pavlovi\'c-Seiringer \cite{CHPS}, and our previous work \cite{UniqueLowReg}.