# A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $T_{3}$ from the dynamics of many-body quantum systems

### Vedran Sohinger

University of Pennsylvania, Department of Mathematics, David Rittenhouse Lab, Office 3N4B, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA

## Abstract

In this paper, we will obtain a rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on the three-dimensional torus $T_{3}$ from the many-body limit of interacting bosonic systems. This type of result was previously obtained on $R_{3}$ in the work of Erdős, Schlein, and Yau [54–57], and on $T_{2}$ and $R_{2}$ in the work of Kirkpatrick, Schlein, and Staffilani [78]. Our proof relies on an unconditional uniqueness result for the Gross–Pitaevskii hierarchy at the level of regularity $α=1$, which is proved by using a modification of the techniques from the work of T. Chen, Hainzl, Pavlović and Seiringer [20] to the periodic setting. These techniques are based on the Quantum de Finetti theorem in the formulation of Ammari and Nier [6,7] and Lewin, Nam, and Rougerie [83]. In order to apply this approach in the periodic setting, we need to recall multilinear estimates obtained by Herr, Tataru, and Tzvetkov [74].

Having proved the unconditional uniqueness result at the level of regularity $α=1$, we will apply it in order to finish the derivation of the defocusing cubic nonlinear Schrödinger equation on $T_{3}$, which was started in the work of Elgart, Erdős, Schlein, and Yau [50]. In the latter work, the authors obtain all the steps of Spohn's strategy for the derivation of the NLS [108], except for the final step of uniqueness. Additional arguments are necessary to show that the objects constructed in [50] satisfy the assumptions of the unconditional uniqueness theorem. Once we achieve this, we are able to prove the derivation result. In particular, we show *Propagation of Chaos* for the defocusing Gross–Pitaevskii hierarchy on $T_{3}$ for suitably chosen initial data.

## Cite this article

Vedran Sohinger, A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $T_{3}$ from the dynamics of many-body quantum systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 6, pp. 1337–1365

DOI 10.1016/J.ANIHPC.2014.09.005