2021/11/25 by Erik Skibsted, Skibsted, Erik
Mathematics · Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2111.13077
openalex publication_date 2021/11/25 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
Within the class of Dereziński-Enss pair-potentials which includes Coulomb potentials and for which asymptotic completeness is known \citeDe, 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 \citeYa1. 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.