Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1708.07278 · Journal of Mathematical Physics

Rate of convergence toward Hartree dynamics with singular interaction potential

Li Chen, Ji Oon Lee, Jinyeop Lee

Published Mar 1, 2018• 12 citations
arXivHTMLPDF

Abstract

We consider a system of N-bosons with a two-body interaction potential V∈L2(R3)+L∞(R3), possibly more singular than the Coulomb interaction. We show that, with H1(R3) initial data, the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding Hartree dynamics is of order 1/N, for any fixed time. The N-dependence of the bound is optimal.

DOI
10.1063/1.5003665

Related papers

RATE OF CONVERGENCE TOWARDS SEMI-RELATIVISTIC HARTREE DYNAMICS

Ji Oon Lee

We consider the semi-relativistic system of N gravitating Bosons with gravitation constant G. The time evolution of the system is described by the relativistic dispersion law, and we assume the mean-field scaling of the interaction where N → ∞ and G → 0 while GN = λ fixed. In the super-critical regime of large λ, we introduce the regularized interaction where the cutoff vanishes as N → ∞. We show that the difference between the many-body semi-relativistic Schrödinger dynamics and the corresponding semi-relativistic Hartree dynamics is at most of order N −1 for all λ, i.e., the result covers the sub-critical regime and the super-critical regime. The N dependence of the bound is optimal.

•Aug 14, 2016•16 citations

Rate of convergence towards mean-field evolution for weakly interacting bosons with singular three-body interactions

Jinyeop Lee

In this paper, we investigate the dynamics of a system of $N$ weakly interacting bosons with singular three-body interactions in three dimensions. By assuming factorized initial data $Ψ_{N,0}=φ_{0}^{\otimes N}$ and triple collisions, we prove that in the many-particle limit, its mean-field approximation converges to quintic Hartree dynamics. Moreover, we prove that the rate of convergence towards the mean-field quintic Hartree evolution is of $O(N^{-(1+4a)/(3+2a)})$ for $φ_{0}\in H^{(3/2)+a}(\mathbb{R}^{3})$ where $0\leq a<1/2$ and $O(N^{-1})$ for $a>1$. Our proof is based on and extends the Fock space approach.

arXiv:2006.13040•Jun 22, 2020•2 citations

Rate of convergence toward Hartree type equations for mixture condensates with factorized initial data

Jinyeop Lee

We consider a system of p species of bosons, each of which consists of N1, N2, …, Np particles. The bosons are in three dimensions with interactions via an interaction potential V such that V ≤ D(1 − Δ), which includes the Coulomb interaction. We set the initial condition to describe a mixture condensate, i.e., a tensor product of factorized states. We show that the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding p particle dynamics due to a system of coupled Hartree equations is O(N−1), where N=∑q=1pNq.

arXiv:1907.03388•Journal of Mathematical Physics•Sep 1, 2021•7 citations

Bogoliubov corrections and trace norm convergence for the Hartree dynamics

David Mitrouskas, Sören Petrat, Peter Pickl

We consider the dynamics of a large number [Formula: see text] of nonrelativistic bosons in the mean field limit for a class of interaction potentials that includes Coulomb interaction. In order to describe the fluctuations around the mean field Hartree state, we introduce an auxiliary Hamiltonian on the [Formula: see text]-particle space that is similar to the one obtained from Bogoliubov theory. We show convergence of the auxiliary time evolution to the fully interacting dynamics in the norm of the [Formula: see text]-particle space. This result allows us to prove several other results: convergence of reduced density matrices in trace norm with optimal rate, convergence in energy trace norm, and convergence to a time evolution obtained from the Bogoliubov Hamiltonian on Fock space with expected optimal rate. We thus extend and quantify several previous results, e.g., by providing the physically important convergence rates, including time-dependent external fields and singular interactions, and allowing for more general initial states, e.g., those that are expected to be ground states of interacting systems.

arXiv:1609.06264•Reviews in Mathematical Physics•Feb 8, 2019•41 citations

Quantum dynamics with mean field interactions: a new approach

László Erdős, Benjamin Schlein

We propose a new approach for the study of the time evolution of a factorized N-particle bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the many-particle Schrödinger dynamics and the one-particle nonlinear Hartree dynamics. In particular the one-particle density matrix associated with the solution to the N-particle Schrödinger equation is shown to converge to the projection onto the one-dimensional subspace spanned by the solution to the Hartree equation with a speed of convergence of order 1/N for all fixed times.

•Jan 1, 2008•32 citations

Mean-field quantum dynamics with magnetic fields

Jonas Lührmann

We consider a system of N bosons in three dimensions interacting through a mean-field Coulomb potential in an external magnetic field. For initially factorized states we show that the one-particle density matrix associated with the solution of the N-body Schrödinger equation converges to the projection onto the solution of the magnetic Hartree equation in trace norm and in energy as N → ∞. Estimates on the rate of convergence are provided.

arXiv:1202.1065•Journal of Mathematical Physics•Feb 1, 2012•18 citations
We consider a system of N-bosons with a two-body interaction potential V∈L2(R3)+L∞(R3), possibly more singular than the Coulomb interaction. We show that, with H1(R3) initial data, the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding Hartree dynamics is of order 1/N, for any fixed time. The N-dependence of the bound is optimal.
We consider the semi-relativistic system of N gravitating Bosons with gravitation constant G. The time evolution of the system is described by the relativistic dispersion law, and we assume the mean-field scaling of the interaction where N → ∞ and G → 0 while GN = λ fixed. In the super-critical regime of large λ, we introduce the regularized interaction where the cutoff vanishes as N → ∞. We show that the difference between the many-body semi-relativistic Schrödinger dynamics and the corresponding semi-relativistic Hartree dynamics is at most of order N −1 for all λ, i.e., the result covers the sub-critical regime and the super-critical regime. The N dependence of the bound is optimal.
In this paper, we investigate the dynamics of a system of NNN weakly interacting bosons with singular three-body interactions in three dimensions. By assuming factorized initial data ΨN,0=φ0⊗NΨ_{N,0}=φ_{0}^{\otimes N}ΨN,0​=φ0⊗N​ and triple collisions, we prove that in the many-particle limit, its mean-field approximation converges to quintic Hartree dynamics. Moreover, we prove that the rate of convergence towards the mean-field quintic Hartree evolution is of O(N−(1+4a)/(3+2a))O(N^{-(1+4a)/(3+2a)})O(N−(1+4a)/(3+2a)) for φ0∈H(3/2)+a(R3)φ_{0}\in H^{(3/2)+a}(\mathbb{R}^{3})φ0​∈H(3/2)+a(R3) where 0≤a<1/20\leq a<1/20≤a<1/2 and O(N−1)O(N^{-1})O(N−1) for a>1a>1a>1. Our proof is based on and extends the Fock space approach.
We consider a system of p species of bosons, each of which consists of N1, N2, …, Np particles. The bosons are in three dimensions with interactions via an interaction potential V such that V ≤ D(1 − Δ), which includes the Coulomb interaction. We set the initial condition to describe a mixture condensate, i.e., a tensor product of factorized states. We show that the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding p particle dynamics due to a system of coupled Hartree equations is O(N−1), where N=∑q=1pNq.
We consider the dynamics of a large number [Formula: see text] of nonrelativistic bosons in the mean field limit for a class of interaction potentials that includes Coulomb interaction. In order to describe the fluctuations around the mean field Hartree state, we introduce an auxiliary Hamiltonian on the [Formula: see text]-particle space that is similar to the one obtained from Bogoliubov theory. We show convergence of the auxiliary time evolution to the fully interacting dynamics in the norm of the [Formula: see text]-particle space. This result allows us to prove several other results: convergence of reduced density matrices in trace norm with optimal rate, convergence in energy trace norm, and convergence to a time evolution obtained from the Bogoliubov Hamiltonian on Fock space with expected optimal rate. We thus extend and quantify several previous results, e.g., by providing the physically important convergence rates, including time-dependent external fields and singular interactions, and allowing for more general initial states, e.g., those that are expected to be ground states of interacting systems.
We propose a new approach for the study of the time evolution of a factorized N-particle bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the many-particle Schrödinger dynamics and the one-particle nonlinear Hartree dynamics. In particular the one-particle density matrix associated with the solution to the N-particle Schrödinger equation is shown to converge to the projection onto the one-dimensional subspace spanned by the solution to the Hartree equation with a speed of convergence of order 1/N for all fixed times.