The derivation of the $T^3$ energy-critical NLS from quantum many-body dynamics

We derive the 3D energy critical quintic NLS from quantum many-body dynamics with 3-body interaction in the $T^3$ (periodic) setting. Due to the known complexity of the energy critical setting, previous progress was limited in comparison to the 2-body interaction case yielding energy subcritical cubic NLS. Previously, the only result for the 3D energy critical case was Hong et al. (J Funct Anal 270(1):34–67, 2016), which proved the uniqueness part of the argument in the case of small solutions. In the main part of this paper, we develop methods to prove the convergence of the BBGKY hierarchy to the infinite Gross–Pitaevskii (GP) hierarchy, and separately, the uniqueness of large GP solutions. Since the trace estimate used in the previous proofs of convergence is the false endpoint trace estimate in our setting, we instead introduce a new frequency interaction analysis and apply the finite dimensional quantum de Finetti theorem. For the large solution uniqueness argument, we discover the new hierarchical uniform frequency localization (HUFL) property for the GP hierarchy and use it to prove a new type of uniqueness theorem. The HUFL property reduces to a new statement even for NLS. With the help of Colliander et al. (Ann Math (2) 167(3):767–865, 2008) and Ionescu and Pausader (Duke Math J 161:1581–1612, 2012) which proved the global well-posedness for the quintic NLS, this new uniqueness theorem establishes global uniqueness.