Pickl's Proof of the Quantum Mean-Field Limit and Quantum Klimontovich Solutions

TL;DR

This paper proves the quantum mean-field limit for bosonic systems with Coulomb interactions using quantum Klimontovich solutions, providing new inequalities and convergence rate estimates for the Hartree equation.

Contribution

It introduces a new definition of the interaction nonlinearity for quantum Klimontovich solutions and establishes an operator inequality for Coulomb-type potentials.

Findings

Defined interaction nonlinearity for quantum Klimontovich solutions.

Established a new operator inequality for Coulomb singularities.

Provided convergence rate estimates for the quantum mean-field limit.

Abstract

This paper discusses the mean-field limit for the quantum dynamics of $N$ identical bosons in $\mathbf R^3$ 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.