Rate of Convergence Towards Semi-Relativistic Hartree Dynamics

Abstract

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 $N\to \infty$ and $G\to 0$ while $GN=\lambda$ fixed. In the super-critical regime of large $\lambda$, we introduce the regularized interaction where the cutoff vanishes as $N\to \infty$. 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 $\lambda$, i.e., the result covers the sub-critical regime and the super-critical regime. The $N$ dependence of the bound is optimal.

29 pages

References in corpus (5)

• Rigorous Derivation of the Gross-Pitaevskii Equation
• Rate of Convergence Towards Hartree Dynamics
• Second order corrections to mean field evolution for weakly interacting Bosons in the case of 3-body interactions
• Quantum Dynamics with Mean Field Interactions: a New Approach
• Rate of Convergence in Nonlinear Hartree Dynamics with Factorized Initial Data

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• Randomization and the Gross-Pitaevskii hierarchy
• A simple proof of convergence to the Hartree dynamics in Sobolev trace norms
• Local existence of solutions to Randomized Gross-Pitaevskii hierarchies
• Rate of Convergence towards Hartree Dynamics with Singular Interaction Potential
• Rate of convergence towards equations of Hartree type for mixture condensates with factorized initial data
• Convergence rate towards the fractional Hartree-equation with singular potentials in higher Sobolev norms
• On the time dependence of the rate of convergence towards Hartree dynamics for interacting Bosons
• Microscopic derivation of a Schrödinger equation in dimension one with a nonlinear point interaction
• Semi-classical limit of the massive Klein-Gordon-Maxwell system toward the relativistic Euler-Maxwell system via an adapted modulated energy method
• Uniform in Time Convergence to Bose-Einstein Condensation for a Weakly Interacting Bose Gas with an External Potential