Mean-Field Dynamics: Singular Potentials and Rate of Convergence

By Antti Knowles, P. Pickl. 2009

210 citations recorded by the source corpus.

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 are in the same one-particle state. It is well known that in the mean-field limit N → ∞ the quantum N-body dynamics is governed by the nonlinear Hartree equation. Using a nonperturbative method, we extend previous results on the mean-field limit in two directions. First, we allow a large class of singular interaction potentials as well as strong, possibly time-dependent external potentials. Second, we derive bounds on the rate of convergence of the quantum N-body dynamics to the Hartree dynamics.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/46a39afb02598c54bce7ef8693512624caa40efa

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In Atomic and Molecular Physics, and Optics

In Statistical and Nonlinear Physics

Citations in the corpus

Quantitative Derivation of the Gross‐Pitaevskii Equation (2012, methodology): “Using techniques similar to those proposed in [21], bounds for the rate of convergence towards the nonlinear Hartree dynamics were also obtained in [17], allowing also for potential with strong singularities.”
Effective Evolution Equations from Quantum Dynamics (2015, background): “Other bounds on the rate of convergence towards the Hartree evolution have been established in [44] (based on ideas proposed in [54]) and recently in [3].”
Mean–Field Evolution of Fermionic Systems (2013, background): “In [32, 35], a different approach to obtain control of the rate of the convergence towards the Hartree evolution was proposed; its main advantage”
Derivation of Hartree's theory for generic mean-field Bose systems (2013, background): “There are many works on the related (but still different) derivation of the time-dependent Hartree theory from the time-dependent Schrödinger equation associated with the Hamiltonian HN , see for instance [29, 25, 63, 7, 20, 21, 3, 22, 24, 54, 34, 51].”
A Simple Derivation of Mean Field Limits for Quantum Systems (2009, background): “Before we prove this equivalence note first that the weights n(k) = k/N has the special property that n̂ϕ := N∑ k=0 k N PϕN,k = N −1 N∑ j=1 qϕj .”