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

Sohinger, Vedran

A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on

𝕋^{3}

from the dynamics of many-body quantum systems

Sohinger, Vedran

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 1337-1365 / Harvested from Numdam

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Résumé

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��

Publié le : 2015-01-01

MR 3425265 | Zbl 1328.35220

DOI : https://doi.org/10.1016/j.anihpc.2014.09.005
Classification: 35Q55, 70E55

@article{AIHPC_2015__32_6_1337_0,
     author = {Sohinger, Vedran},
     title = {A rigorous derivation of the defocusing cubic nonlinear Schr\"odinger equation on $ {\mathbb{T}}^{3}$ from the dynamics of many-body quantum systems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {32},
     year = {2015},
     pages = {1337-1365},
     doi = {10.1016/j.anihpc.2014.09.005},
     mrnumber = {3425265},
     zbl = {1328.35220},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2015__32_6_1337_0}
}

Sohinger, Vedran. A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $ {\mathbb{T}}^{3}$ from the dynamics of many-body quantum systems. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) pp. 1337-1365. doi : 10.1016/j.anihpc.2014.09.005. http://gdmltest.u-ga.fr/item/AIHPC_2015__32_6_1337_0/

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