Pickl's Proof of the Quantum Mean-Field Limit and Quantum Klimontovich Solutions
Abstract
This paper discusses the mean-field limit for the quantum dynamics of identical bosons in interacting via a binary potential with Coulomb type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in [F. Golse, T. Paul, Commun. Math. Phys. 369 (2019), 1021-1053]. Our first main result is a definition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials slightly less general than those considered in [T. Kato, Trans. Amer. Math. Soc. 70 (1951), 195-211]. Our second main result is a new operator inequality satisfied by the quantum Klimontovich solution in the case of an interaction potential with Coulomb type singularity. When evaluated on an initial bosonic pure state, this operator inequality reduces to a Gronwall inequality for a functional introduced in [P. Pickl, Lett. Math. Phys. 97 (2011), 151-164], resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation.
Keywords
Cite
@article{arxiv.2203.13373,
title = {Pickl's Proof of the Quantum Mean-Field Limit and Quantum Klimontovich Solutions},
author = {Immanuel Ben Porat and François Golse},
journal= {arXiv preprint arXiv:2203.13373},
year = {2024}
}
Comments
32 pages, no figure. In this version, the introduction has been slightly modified so as to better indicate the main results in the paper. Similarly, the beginning of section 8.2 has been slightly expanded so as to better explain the connection between quantum Klimontovich solutions and Pickl's functional (specifically, the equivalence between formulas (58) and (18))