Xuwen Chen

Second Order Corrections to Mean Field Evolution for Weakly Interacting Bosons in The Case of Three-body Interactions

In this paper, we consider the Hamiltonian evolution of N weakly interacting bosons. Assuming triple collisions, its mean field approximation is given by a quintic Hartree equation. We construct a second order correction to the mean field approximation using a kernel k(t, x, y) and derive an evolution equation for k. We show global existence for the resulting evolution equation for the correction and establish an a priori estimate comparing the approximation to the exact Hamiltonian evolution. Our error estimate is global and uniform in time. Comparing with the work of Rodnianski and Schlein (Commun Math Phys 291:31–61, 2009), and Grillakis, Machedon and Margetis (Commun Math Phys 294:273–301, 2010; Adv Math 288:1788–1815, 2011), where the error estimate grows in time, our approximation tracks the exact dynamics for all time with an error of the order

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

{O(1/\sqrt{N}).}

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

We consider the dynamics of the three-dimensional N-body Schrödinger equation in the presence of a quadratic trap. We assume the pair interaction potential is N3β-1V(Nβx). We justify the mean-field approximation and offer a rigorous derivation of the three-dimensional cubic nonlinear Schrödinger equation (NLS) with a quadratic trap. We establish the space-time bound conjectured by Klainerman and Machedon (Commun Math Phys 279:169–185, 2008) for

Velocity Reversal Criterion of a Body Immersed in a Sea of Particles

{\beta \in (0, 2/7]}

On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction

by adapting and simplifying an argument in Chen and Pavlović (Annales Henri Poincaré, 2013) which solves the problem for