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

被引:0
作者
Xuwen Chen
Justin Holmer
机构
[1] University of Rochester,Department of Mathematics
[2] Brown University,Department of Mathematics
来源
Archive for Rational Mechanics and Analysis | 2016年 / 221卷
关键词
Marginal Density; Trace Class Operator; Rigorous Derivation; Pitaevskii Equation; Wigner Measure;
D O I
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中图分类号
学科分类号
摘要
We consider the dynamics of N bosons in 1D. We assume that the pair interaction is attractive and given by Nβ-1V(Nβ).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N^{\beta-1}V(N^{\beta}.) where }$$\end{document} where ∫V⩽0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int V \leqslant 0}$$\end{document}. We develop new techniques in treating the N-body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity requirement in the uniqueness argument by half a derivative. We derive rigorously the 1D focusing cubic NLS with a quadratic trap as the N→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N \rightarrow \infty}$$\end{document} limit of the N-body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system.
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页码:631 / 676
页数:45
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