Higher order corrections to the mean-field description of the dynamics of interacting bosons

Abstract

In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of $N$ $d$-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 $(N-1{)}^{-1}{N}^{d\beta }v(N^{\beta }\cdot )$ for $\beta \in [0,\frac{1}{4d})$. We derive a sequence of $N$-body functions which approximate the true many-body dynamics in $L^2({\mathbb{R}}^{dN})$-norm to arbitrary precision in powers of $N^{-1}$. The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.

References in corpus (34)

• Derivation of the Cubic Non-linear Schrödinger Equation from Quantum Dynamics of Many-Body Systems
• Mean-Field Dynamics: Singular Potentials and Rate of Convergence
• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
• Second-order corrections to mean field evolution for weakly interacting Bosons. I
• The excitation spectrum for weakly interacting bosons in a trap
• The Excitation Spectrum for Weakly Interacting Bosons
• Pair excitations and the mean field approximation of interacting Bosons, I
• Rate of Convergence Towards Hartree Dynamics
• Second-order corrections to mean-field evolution of weakly interacting Bosons, II
• Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti
• Optimal Rate for Bose-Einstein Condensation in the Gross-Pitaevskii Regime
• Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation
• Mean-Field- and Classical Limit of Many-Body Schrödinger Dynamics for Bosons
• Complete Bose-Einstein condensation in the Gross-Pitaevskii regime
• On the Rigorous Derivation of the 3D Cubic Nonlinear Schrödinger Equation with A Quadratic Trap
• Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d=3$ based on spacetime norms
• Quantum many-body fluctuations around nonlinear Schrödinger dynamics
• Pair excitations and the mean field approximation of interacting Bosons, II
• Bogoliubov corrections and trace norm convergence for the Hartree dynamics
• Bogoliubov correction to the mean-field dynamics of interacting bosons
• A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on ${\mathbb{T}}^3$ from the dynamics of many-body quantum systems
• Norm approximation for many-body quantum dynamics: focusing case in low dimensions
• Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions
• The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution
• Mean-field quantum dynamics for a mixture of Bose-Einstein condensates
• A simple proof of convergence to the Hartree dynamics in Sobolev trace norms
• Well-posedness of non-autonomous linear evolution equations in uniformly convex spaces
• Derivation of the Time Dependent Two Dimensional Focusing NLS Equation
• On the size of chaos in the mean field dynamics
• Dynamics of Large Boson Systems with Attractive Interaction and a Derivation of the Cubic Focusing NLS in ${\mathbb{R}}^3$
• Derivation of the 1d Gross-Pitaevskii equation from the 3d quantum many-body dynamics of strongly confined bosons
• Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons
• Derivation of the Bogoliubov Time Evolution for a Large Volume Mean-field Limit

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• Asymptotic expansion of low-energy excitations for weakly interacting bosons
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• On Bose-Einstein condensates in the Thomas-Fermi regime
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• Convergence rate towards the fractional Hartree-equation with singular potentials in higher Sobolev norms
• On the characterisation of fragmented Bose-Einstein condensation and its emergent effective evolution
• Global estimates for the Hartree-Fock-Bogoliubov equations
• Two-term expansion of the ground state one-body density matrix of a mean-field Bose gas
• Asymptotic analysis of the weakly interacting Bose gas: A collection of recent results and applications
• On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate
• Many-body excitations in trapped Bose gas: A non-Hermitian view
• Field moment expansion method for interacting Bosonic systems
• Vortex Lattices and the Bogoliubov-De Gennes Equations
• Low-energy spectrum and dynamics of the weakly interacting Bose gas
• Derivation of the Gross-Pitaevskii Dynamics through Renormalized Excitation Number Operators