A rigorous derivation of the defocusing cubic nonlinear Schr\"{o}dinger equation on $\mathbb{T}^3$ from the dynamics of many-body quantum systems

Abstract

In this paper, we will obtain a rigorous derivation of the defocusing cubic nonlinear Schr\"{o}dinger equation on the three-dimensional torus $\mathbb{T}^3$ from the many-body limit of interacting bosonic systems. This type of result was previously obtained on $\mathbb{R}^3$ in the work of Erd\H{o}s, Schlein, and Yau \cite{ESY2,ESY3,ESY4,ESY5}, and on $\mathbb{T}^2$ and $\mathbb{R}^2$ in the work of Kirkpatrick, Schlein, and Staffilani \cite{KSS}. Our proof relies on an unconditional uniqueness result for the Gross-Pitaevskii hierarchy at the level of regularity $\alpha=1$, which is proved by using a modification of the techniques from the work of T. Chen, Hainzl, Pavlovi\'{c} and Seiringer \cite{ChHaPavSei} to the periodic setting. These techniques are based on the Quantum de Finetti theorem in the formulation of Ammari and Nier \cite{AmmariNier1,AmmariNier2} and Lewin, Nam, and Rougerie \cite{LewinNamRougerie}. In order to apply this approach in the periodic setting, we need to recall multilinear estimates obtained by Herr, Tataru, and Tzvetkov \cite{HTT}. Having proved the unconditional uniqueness result at the level of regularity $\alpha=1$, we will apply it in order to finish the derivation of the defocusing cubic nonlinear Schr\"{o}dinger equation on $\mathbb{T}^3$, which was started in the work of Elgart, Erd\H{o}s, Schlein, and Yau \cite{EESY}. In the latter work, the authors obtain all the steps of Spohn's strategy for the derivation of the NLS \cite{Spohn}, except for the final step of uniqueness. Additional arguments are necessary to show that the objects constructed in \cite{EESY} satisfy the assumptions of the unconditional uniqueness theorem. Once we achieve this, we are able to prove the derivation result. In particular, we show \emph{Propagation of Chaos} for the defocusing Gross-Pitaevskii hierarchy on $\mathbb{T}^3$ for suitably chosen initial data.