arXiv:2006.09825 · Forum of Mathematics Sigma

Asymptotic expansion of low-energy excitations for weakly interacting bosons

Lea Boßmann, Sören Petrat, Robert Seiringer

Published Jan 1, 2021• 34 citations

Abstract

Abstract We consider a system ofNbosons in the mean-field scaling regime for a class of interactions including the repulsive Coulomb potential. We derive an asymptotic expansion of the low-energy eigenstates and the corresponding energies, which provides corrections to Bogoliubov theory to any order in $1/N$ .

DOI: 10.1017/fms.2021.22

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arXiv:2006.09825 · Forum of Mathematics Sigma

Asymptotic expansion of low-energy excitations for weakly interacting bosons

Lea Boßmann, Sören Petrat, Robert Seiringer

Published Jan 1, 2021• 34 citations

arXiv HTML PDF

Abstract

1/N

DOI: 10.1017/fms.2021.22

Related papers

Low-energy spectrum and dynamics of the weakly interacting Bose gas

Lea Boßmann

Jan 1, 20220 citations

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Lea Boßmann

We consider a gas of N bosons with interactions in the mean-field scaling regime. We review the proof of an asymptotic expansion of its low-energy spectrum, eigenstates, and dynamics, which provides corrections to Bogoliubov theory to all orders in 1/N. This is based on joint works with Petrat, Pickl, Seiringer, and Soffer. In addition, we derive a full asymptotic expansion of the ground state one-body reduced density matrix.

arXiv:2203.00730Journal of Mathematical PhysicsJun 1, 20225 citations

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Lea Boßmann, Nataša Pavlović, Peter Pickl +1 more

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1/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.

1/\sqrt{N}

1 / N . For suitable excited states, we show that the limiting distribution is a polynomial times a normal distribution, and that higher-order corrections are given by an Edgeworth-type expansion.

We consider systems of

N

bosons in

\mathbb{R}^3

, trapped by an external potential. The interaction is repulsive and has a scattering length of the order

N^{-1}

(Gross-Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum up to errors that vanish in the limit

N\to \infty