Focusing Quantum Many-body Dynamics II: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D

Abstract

We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is attractive and given by $a^{3\beta -1}V(a^{\beta }\cdot )$ where $\int V⩽0$ and $a$ matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D $N$-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to $\infty$. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.

arXiv admin note: text overlap with arXiv:1212.0787

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• The Derivation of the ${\mathbb{T}}^3$ Energy-critical NLS from Quantum Many-body Dynamics
• Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions $d\le 3$
• 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
• Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons
• Derivation of the 1d Gross-Pitaevskii equation from the 3d quantum many-body dynamics of strongly confined bosons
• Derivation of the 2d Gross-Pitaevskii equation for strongly confined 3d bosons
• 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
• On the mean-field and semiclassical limit from quantum $N$-body dynamics
• On the nonlinear Schrödinger equation with a toroidal trap in the strong confinement regime