Dynamics of Large Boson Systems with Attractive Interaction and a Derivation of the Cubic Focusing NLS in $\mathbb{R}^3$.

摘要: We consider a system of $N$ bosons where the particles experience short range two-body interaction given by $N^{-1}v_N(x)=N^{3\beta-1}v(N^\beta x)$ $v \in C^\infty_c(\mathbb{R}^3)$, without definite sign on $v$. extend results M. Grillakis and Machedon, Comm. Math. Phys., $\textbf{324}$, 601(2013) E. Kuz, Differ. Integral Equ., $\textbf{137}$, 1613(2015) regarding second-order correction to mean-field evolution systems with repulsive attractive for $0<\beta<\frac{1}{2}$. Our extension allows more general set initial data which includes coherent states. Inspired works X. Chen J. Holmer, Arch. Ration. Mech. Anal., $\textbf{221}$, 631(2016) Int. Res. Not., $\textbf{2017}$, 4173(2017), P. T. Nam Napi\'orkowski, Adv. Theor. $\textbf{21}$, 683(2017), we also provide both derivation focusing nonlinear Schr\"odinger equation (NLS) in $3$D from many-body its rate convergence toward $0<\beta<\frac{1}{3}$. In particular, give two derivations NLS, one based $N$-norm approximation work Napi\'orkowski other via method introduced Pickl, Stat. $\textbf{140}$, 76(2010). The techniques used this article are standard literature dispersive PDEs. Nevertheless, NLS had only previously been studied 1D \& 2D cases conditionally answered 3D case $0<\beta<\frac{1}{6}$.