A note on 2D focusing many-boson systems

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

We consider a system of $N$ bosons where the particles experience a short range two-body interaction given by $N^{-1}v_N(x)=N^{3β-1}v(N^βx)$ where $v \in C^\infty_c(\mathbb{R}^3)$, without a definite sign on $v$. We extend the results of M. Grillakis and M. Machedon, Comm. Math. Phys., $\textbf{324}$, 601(2013) and E. Kuz, Differ. Integral Equ., $\textbf{137}$, 1613(2015) regarding the second-order correction to the mean-field evolution of systems with repulsive interaction to systems with attractive interaction for $0<β<\frac{1}{2}$. Our extension allows for a more general set of initial data…

关于平均场极限下玻色子结合能的一个注记

A Note on the Binding Energy for Bosons in the Mean-field Limit

本文研究基态弱相互作用玻色气体的结合能，证明其渐近展开并显式计算均匀气体前几阶，回应 Nam 的猜想且给出玻色原子电离能展开。

The Grillakis-Machedon-Margetis Second order corrections to mean field evolution for weakly interacting Bosons in the case of 3-body interactions

This note shows that, with a little modification, the results in arXiv:0904.0158 hold in the 3-body interactions case.

Norm approximation for many-body quantum dynamics: focusing case in low dimensions

We study the norm approximation to the Schrödinger dynamics of $N$ bosons in $\mathbb{R}^d$ ($d=1,2$) with an interaction potential of the form $N^{dβ-1}w(N^β(x-y))$. Here we are interested in the focusing case $w\le 0$. Assuming that there is complete Bose-Einstein condensation in the initial state, we show that in the large $N$ limit, the evolution of the condensate is effectively described by a nonlinear Schrödinger equation and the evolution of the fluctuations around the condensate is governed by a quadratic Hamiltonian, resulting from Bogoliubov approximation. Our result holds true for a…

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

We consider the dynamics of N bosons in one dimension. We assume that the pair interaction is attractive and given by N^{β-1}V(N^β\cdot) where \int V\leqslant 0. 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 one dimensional focusing cubic NLS with a quadratic trap as the N->\infty limit of the N-body dynami…

Quasi-Classical Spin-Boson Models

In this short note we study Spin-Boson Models from the Quasi-Classical standpoint. In the Quasi-Classical limit, the field becomes macroscopic while the particles it interacts with, they remain quantum. As a result, the field becomes a classical environment that drives the particle system with an explicit effective dynamics.