Stationary scattering theory, the $N$-body long-range case

TL;DR

This paper extends stationary scattering theory to the N-body long-range case, establishing well-defined, continuous scattering matrices at non-threshold energies for a broad class of potentials including Coulomb.

Contribution

It generalizes previous short-range results to long-range potentials, proving well-defined and continuous scattering matrices without decay assumptions on eigenstates.

Findings

Scattering matrix entries are well-defined at all non-threshold energies.

The scattering matrix is weakly continuous as a function of energy.

Constructs and proves strong continuity of channel wave matrices.

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 $N$-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.