# Fluctuations around Hartree states in the mean-field regime

@article{Lewin2013FluctuationsAH,
title={Fluctuations around Hartree states in the mean-field regime},
author={Mathieu Lewin and Phan Th{\a}nh Nam and Benjamin Schlein},
journal={American Journal of Mathematics},
year={2013},
volume={137},
pages={1613 - 1650}
}`
• Published 2 July 2013
• Physics
• American Journal of Mathematics
We consider the dynamics of a large system of $N$ interacting bosons in the mean-field regime where the interaction is of order $1/N$. We prove that the fluctuations around the nonlinear Hartree state are generated by an effective quadratic Hamiltonian in Fock space, which is derived from Bogoliubov's approximation. We use a direct method in the $N$-particle space, which is different from the one based on coherent states in Fock space.
• Physics
Annales Henri Poincaré
• 2020
We consider the quantum mechanical many-body problem of a single impurity particle immersed in a weakly interacting Bose gas. The impurity interacts with the bosons via a two-body potential. We study
• Physics, Mathematics
• 2017
We consider the dynamics of a large quantum system of $N$ identical bosons in 3D interacting via a two-body potential of the form $N^{3\beta-1} w(N^\beta(x-y))$. For fixed $0\leq \beta <1/3$ and
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
We review recent results about the derivation of the Gross-Pitaevskii equation and of the Bogoliubov excitation spectrum, starting from many-body quantum mechanics. We focus on the mean-field regime,
We review recent progress towards a rigorous understanding of the excitation spectrum of bosonic quantum many-body systems. In particular, we explain how one can rigorously establish the predictions
• R. Seiringer
• Physics
Jahresbericht der Deutschen Mathematiker-Vereinigung
• 2014
We review recent progress towards a rigorous understanding of the excitation spectrum of bosonic quantum many-body systems. In particular, we explain how one can rigorously establish the predictions
• Physics, Mathematics
Annales de l'Institut Henri Poincaré C, Analyse non linéaire
• 2019
• Mathematics, Physics
Calculus of Variations and Partial Differential Equations
• 2018
We study a system of N fermions in the regime where the intensity of the interaction scales as 1 / N and with an effective semi-classical parameter $$\hbar =N^{-1/d}$$ħ=N-1/d where d is the space
• Physics
• 2022
We study the time-evolution of an initially trapped weakly interacting Bose gas at positive temperature, after the trapping potential has been switched off. It has been recently shown in [24] that
• Physics, Mathematics
• 2017
We review some recent results on the norm approximation to the Schrodinger dynamics. We consider N bosons in $$\mathbb{R}^{3}$$ with an interaction potential of the form N 3β−1 w(N β (x − y)) with 0

## References

SHOWING 1-10 OF 22 REFERENCES

• Physics, Mathematics
• 2011
We study the many body quantum evolution of bosonic systems in the mean field limit. The dynamics is known to be well approximated by the Hartree equation. So far, the available results have the form
• Mathematics, Physics
• 2007
The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this paper we provide estimates on the rate of convergence of the
• Mathematics, Physics
• 2011
We consider a system of N bosons interacting through a two-body potential with, possibly, Coulomb-type singularities. We show that the difference between the many-body Schrödinger evolution in the
• Physics, Mathematics
• 2010
We consider the time evolution of a system of N identical bosons whose interaction potential is rescaled by N−1. We choose the initial wave function to describe a condensate in which all particles
• Mathematics, Physics
• 2001
We consider the time evolution of N bosonic particles interacting via a mean field Coulomb potential. Suppose the initial state is a product wavefunction. We show that at any finite time the
• Physics
• 2012
We study the large‐N limit of a system of N bosons interacting with a potential of intensity 1/N. When the ground state energy is to the first order given by Hartree's theory, we study the next
• Physics, Mathematics
• 1979
We study the classical field limit of non relativistic many-boson theories in space dimensionn≧3, extending the results of a previous paper to more singular interactions. We prove the expected
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