arXiv:1710.09743 · Annales de l Institut Henri Poincaré C Analyse Non Linéaire

Fluctuations of $N$-particle quantum dynamics around the nonlinear Schrödinger equation

Christian Brennecke, Phan Thành Nam, Marcin Napiórkowski, Benjamin Schlein

Published Nov 23, 2018• 33 citations

Abstract

We consider a system of N bosons interacting through a singular two-body potential scaling with N and having the form N^{3\beta −1}V(N^{\beta }x) , for an arbitrary parameter \beta \in (0,1) . We provide a norm-approximation for the many-body evolution of initial data exhibiting Bose–Einstein condensation in terms of a cubic nonlinear Schrödinger equation for the condensate wave function and of a unitary Fock space evolution with a generator quadratic in creation and annihilation operators for the fluctuations.

DOI: 10.1016/j.anihpc.2018.10.007

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arXiv:1710.09743 · Annales de l Institut Henri Poincaré C Analyse Non Linéaire

Fluctuations of $N$ -particle quantum dynamics around the nonlinear Schrödinger equationFluctuations of $N$-particle quantum dynamics around the nonlinear Schrödinger equation

Christian Brennecke, Phan Thành Nam, Marcin Napiórkowski, Benjamin Schlein

Published Nov 23, 2018• 33 citations

arXiv HTML PDF

Abstract

DOI: 10.1016/j.anihpc.2018.10.007

Related papers

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Fluctuations of

N

-particle quantum dynamics around the nonlinear Schrödinger equation

Abstract We consider a system ofNbosons in the limit

N \rightarrow \infty

N→∞ , interacting through singular potentials. For initial data exhibiting Bose–Einstein condensation, the many-body time evolution is well approximated through a quadratic fluctuation dynamics around a cubic nonlinear Schrödinger equation of the condensate wave function. We show that these fluctuations satisfy a (multi-variate) central limit theorem.

\mathit{N}

. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one-particle reduced density, the form of the initial data is preserved by the many-body evolution, up to a small error that vanishes as

\mathit{N}^{-1/2}

in the limit of large

\mathit{N}

d

d -dimensional bosons for large N. The bosons initially form a Bose–Einstein condensate and interact with each other via a pair potential of the form

(N-1)^{-1}N^{d\beta }v(N^\beta \cdot )

(N-1)-1Ndβv(Nβ·) for

\beta \in [0,\frac{1}{4d})

β∈[0,14d) . We derive a sequence of N-body functions which approximate the true many-body dynamics in

L^2({\mathbb {R}}^{dN})

L2(RdN) -norm to arbitrary precision in powers of

N^{-1}

N-1 . The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.