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Stationary Scattering Theory: The N-Body Long-Range Case

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Within the class of Dereziński–Enss pair-potentials which includes Coulomb potentials and for which asymptotic completeness is known (Dereziński in Ann Math 38:427–476, 1993), 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 (Yafaev in Integr Equ Oper Theory 21:93–126, 1995). As for Yafaev’s paper we do not make any assumption on the decay of channel bound states. 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 generalized eigenfunctions of minimal growth.

Original languageEnglish
JournalCommunications in Mathematical Physics
Pages (from-to)2193-2267
Number of pages75
Publication statusPublished - Jul 2023

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© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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