A rigorous derivation of the defocusing cubic nonlinear Schrödinger 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 Schrödinger 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 Erdős, 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, Pavlović 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 Schrödinger equation on ${\mathbb{T}}^3$, which was started in the work of Elgart, Erdős, 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.

38 pages. 2 figures

References in corpus (7)

• Random data Cauchy theory for supercritical wave equations I: Local theory
• Rigorous Derivation of the Gross-Pitaevskii Equation
• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
• Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation
• Quantum Dynamics with Mean Field Interactions: a New Approach
• On the well-posedness and scattering for the Gross-Pitaevskii hierarchy via quantum de Finetti

Cited by in corpus (18)

• The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations
• Correlation structures, Many-body Scattering Processes and the Derivation of the Gross-Pitaevskii Hierarchy
• The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution
• The Derivation of the ${\mathbb{T}}^3$ Energy-critical NLS from Quantum Many-body Dynamics
• On the unconditional uniqueness of solutions to the infinite radial Chern-Simons-Schrödinger hierarchy
• Higher order corrections to the mean-field description of the dynamics of interacting bosons
• Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions $d\le 3$
• Focusing Quantum Many-body Dynamics II: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D
• The Gross-Pitaevskii hierarchy on general rectangular tori
• Unconditional Uniqueness Results for the Nonlinear Schrödinger Equation
• The rigorous derivation of the ${\mathbb{T}}^2$ focusing cubic NLS from 3D
• Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space
• The unconditional uniqueness for the energy-supercritical NLS
• Uniqueness of Solutions to the Spectral Hierarchy in Kinetic Wave Turbulence Theory
• The Unconditional Uniqueness for the Energy-critical Nonlinear Schrödinger Equation on ${\mathbb{T}}^4$
• Well/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel
• Well/Ill-posedness of the Boltzmann Equation with Soft Potential
• On the mean-field and semiclassical limit from quantum $N$-body dynamics