Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1212.0787 · Archive for Rational Mechanics and Analysis

On the Rigorous Derivation of the 2D Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-Body Dynamics

Xuwen Chen, Justin Holmer

Published Sep 23, 2013• 42 citations
arXivHTMLPDF
DOI
10.1007/s00205-013-0667-z

Related papers

Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems

László Erdős, Benjamin Schlein, Horng‐Tzer Yau

arXiv:math-ph/0508010•Inventiones mathematicae•Dec 19, 2006•318 citations

A microscopic derivation of time-dependent correlation functions of the $1D$ cubic nonlinear Schrödinger equation

Jürg Martin Fröhlich, Antti Knowles, Benjamin Schlein +1 more

arXiv:1703.04465•Mar 13, 2017•2 citations

A microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation

Jürg Martin Fröhlich, Antti Knowles, Benjamin Schlein +1 more

Advances in Mathematics•Jul 1, 2019•15 citations

Derivation of 3D Energy-Critical Nonlinear Schrödinger Equation and Bogoliubov Excitations for Bose Gases

Phan Thành Nam, Robert Salzmann

arXiv:1810.09374•Communications in Mathematical Physics•Jun 20, 2019•21 citations

Focusing quantum many-body dynamics, II : The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D

Xuwen Chen, Justin Holmer

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 $a^{3β-1}V(a^β\cdot)$ where $\int V\leqslant 0$ 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:1407.8457•Analysis & PDE•Apr 17, 2017•26 citations

Derivation of the two dimensional nonlinear Schrodinger equation from many body quantum dynamics

Kay Kirkpatrick, Benjamin Schlein, Gigliola Staffilani

We derive rigorously, for both R^2 and [-L, L]^2, the cubic nonlinear Schrodinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.

arXiv:0808.0505•Aug 4, 2008•0 citations
A microscopic derivation of time-dependent correlation functions of the 1D1D1D cubic nonlinear Schrödinger equation
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 a3β−1V(aβ⋅)a^{3β-1}V(a^β\cdot)a3β−1V(aβ⋅) where ∫V⩽0\int V\leqslant 0∫V⩽0 and aaa 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 NNN-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.
We derive rigorously, for both R^2 and [-L, L]^2, the cubic nonlinear Schrodinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.