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

Within the class of Dereziński-Enss pair-potentials which includes Coulomb potentials and for which asymptotic completeness is known [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 [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.

[1]  H. Isozaki,et al.  N-body resolvent estimates , 1996 .

[2]  Quantum scattering at low energies , 2007, 0712.0195.

[3]  H. Isozaki.  Eikonal equations and spectral representations for long-range Schrödinger Hamiltonians , 1980 .

[4]  Tosio Kato.  Smooth operators and commutators , 1968 .

[5]  L. Hörmander,et al.  The Analysis of Linear Partial Differential Operators IV , 1985 .

[6]  On solutions of the Schrödinger equation with radiation conditions at infinity : The long-range case , 1999 .

[7]  G. M. Graf.  Asymptotic completeness forN-body short-range quantum systems: A new proof , 1990 .

[8]  加藤 敏夫,et al.  The abstract theory of scattering , 1969 .

[9]  J. Dollard.  Asymptotic Convergence and the Coulomb Interaction , 1964 .

[10]  Time-dependent scattering theory on manifolds , 2019, Journal of Functional Analysis.

[11]  I. Herbst,et al.  Free channel fourier transform in the long-rangen-body problem , 1995 .

[12]  E. Skibsted,et al.  New methods in spectral theory of N-body Schrödinger operators , 2018, Reviews in Mathematical Physics.

[13]  Sohei Ashida.  Scattering matrices and generalized Fourier transforms in long-range $N$-body problems. , 2018, 1811.07527.

[14]  Stationary scattering theory on manifolds, II , 2016, 1602.07487.

[15]  E. Skibsted.  SMOOTHNESS OF N-BODY SCATTERING AMPLITUDES , 1992 .

[16]  D. Yafaev.  Radiation conditions and scattering theory forN-particle Hamiltonians , 1993 .

[17]  M. Schechter.  The existence of wave operators in scattering theory , 1977 .

[18]  D. Yafaev.  Scattering Theory: Some Old and New Problems , 2000 .

[19]  Radiation condition bounds on manifolds with ends , 2020 .