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Journal of Mathematical Physics

Disproof of a conjecture of D. Muhlerin and I. Zinnes

John D. Dollard

Published Feb 1, 1975• 4 citations
Publisher

Abstract

It is shown that the operator U(±)c(t) defined by Muhlerin and Zinnes [J. Math.Phys.11,1402 (1970)] in their account of Coulomb potential scattering theory is not an isometry when the Coulomb potential is weak enough. It is opined that for this reason their account of Coulomb potential scattering theory does not have a straightforward physical interpretation.

DOI
10.1063/1.522513

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It is shown that the operator U(±)c(t) defined by Muhlerin and Zinnes [J. Math.Phys.11,1402 (1970)] in their account of Coulomb potential scattering theory is not an isometry when the Coulomb potential is weak enough. It is opined that for this reason their account of Coulomb potential scattering theory does not have a straightforward physical interpretation.
With the aid of a Coulomb ``free propagator'' Uc(±)(t) constructed from the asymptotic form of the conventional time-independent solution to the scattering problem in a pure Coulomb field, Mo/ller wave operators are shown to exist for general Coulomb-like (Coulomb + short-range) potentials. It is emphasized that when scattering occurs in such long-range fields, one should distinguish three classes of states: the prepared and detected states, the scattering states, and certain asymptotic states which are peculiar to long-range potentials. The existence of the third class as a class distinct from the first suggests a connection between Uc(±)(t) and the quantum theory of measurements.
Abstract This paper discusses the mean-field limit for the quantum dynamics of N identical bosons in R3{\textbf{R}}^3R3 R 3 interacting via a binary potential with Coulomb-type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in Golse and Paul (Commun Math Phys 369:1021–1053, 2019) . 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 Kato (Trans Am Math Soc 70:195–211, 1951). 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 Pickl (Lett Math Phys 97:151-164, 2011), resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation.