Stationary scattering theory, the $N$-body long-range case
Abstract
Within the class of Derezi{\'n}ski-Enss pair-potentials which includes Coulomb potentials and for which asymptotic completeness is known \cite{De}, we show that all entries of the -body quantum scattering matrix have a well-defined meaning at any given non-threshold energy. As a function of the energy parameter the scattering matrix is weakly continuous. This result generalizes a similar one obtained previously by Yafaev for systems of particles interacting by short-range potentials \cite{Ya1}. As for Yafaev's paper we do not make any assumption on the decay of channel eigenstates. The main part of the proof consists in establishing a number of Kato-smoothness bounds needed for justifying a new formula for the scattering matrix. Similarly we construct and show strong continuity of channel wave matrices for all non-threshold energies. Away from a set of measure zero we show that the scattering and channel wave matrices constitute a well-defined `scattering theory', in particular at such energies the scattering matrix is unitary, strongly continuous and characterized by asymptotics of minimum generalized eigenfunctions.
Keywords
Cite
@article{arxiv.2111.13077,
title = {Stationary scattering theory, the $N$-body long-range case},
author = {Erik Skibsted},
journal= {arXiv preprint arXiv:2111.13077},
year = {2021}
}