Communications on Pure &amp Applied Analysis

Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

Jianjun Yuan

Published Jan 1, 2015• 7 citations

Abstract

In this paper, we investigate the dynamics of a boson gas with three-body interactionsin $T^2$. We prove that when the particlenumber $N$ tends to infinity, the BBGKY hierarchy of $k$-particle marginals converges to a infiniteGross-Pitaevskii(GP) hierarchy for which we prove uniqueness of solutions,and for the asymptotically factorized $N$-body initial datum, we show that this $N\rightarrow\infty$ limitcorresponds to the quintic nonlinear Schrödinger equation.Thus, the Bose-Einstein condensation is preserved in time.

DOI: 10.3934/cpaa.2015.14.1941

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Communications on Pure &amp Applied Analysis

Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$ Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

Jianjun Yuan

Published Jan 1, 2015• 7 citations

Publisher

Abstract

In this paper, we investigate the dynamics of a boson gas with three-body interactionsin

T^2

. We prove that when the particlenumber

N

tends to infinity, the BBGKY hierarchy of

k

-particle marginals converges to a infiniteGross-Pitaevskii(GP) hierarchy for which we prove uniqueness of solutions,and for the asymptotically factorized

N

-body initial datum, we show that this

N\rightarrow\infty

limitcorresponds to the quintic nonlinear Schrödinger equation.Thus, the Bose-Einstein condensation is preserved in time.

DOI: 10.3934/cpaa.2015.14.1941

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Xuwen Chen, Justin Holmer

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Derivation of the Quintic NLS from many-body quantum dynamics in

T^2

In this paper, we investigate the dynamics of a boson gas with three-body interactionsin

T^2

. We prove that when the particlenumber

N

tends to infinity, the BBGKY hierarchy of

k

-particle marginals converges to a infiniteGross-Pitaevskii(GP) hierarchy for which we prove uniqueness of solutions,and for the asymptotically factorized

N

-body initial datum, we show that this

N\rightarrow\infty

limitcorresponds to the quintic nonlinear Schrödinger equation.Thus, the Bose-Einstein condensation is preserved in time.

We derive rigorously, for both

{\Bbb R}^2

and

[{-}L,L]^{\times 2}

, the cubic nonlinear Schr\"odinger 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.

Consider a system of

N

bosons in three dimensions interacting via a repulsive short range pair potential

N^2V(N(x_i-x_j))

, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let

H_N

denote the Hamiltonian of the system and let

ψ_{N,t}

be the solution to the Schrödinger equation. Suppose that the initial data

ψ_{N,0}

satisfies the energy condition

< ψ_{N,0}, H_N^k ψ_{N,0} > \leq C^k N^k

for $k=1,2,... $. We also assume that the

k

-particle density matrices of the initial state are asymptotically factorized as

N\to\infty

. We prove that the

k

-particle density matrices of

ψ_{N,t}

are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential

V

. We also prove the same conclusion if the energy condition holds only for

k=1

but the factorization of

ψ_{N,0}

is assumed in a stronger sense.

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.