scholarly journals Propagation properties in scattering theory

1980 ◽  
Vol 21 (4) ◽  
pp. 474-485 ◽  
Author(s):  
Derek W. Robinson
Keyword(s):  
Ergodic Theorem ◽  
Scattering Theory ◽  
Bound States ◽  
Semi Group ◽  
Easy Consequence ◽  
Scattering States ◽  
Mean Ergodic Theorem ◽  
The Mean ◽  
Propagation Properties

AbstractGeneralizations of the Green-Lanford-Dollard theorem on scattering into cones and Ruelle-Amerin-Georgescu theorem characterizing bound states and scattering states are derived. The first is shown to be an easy consequence of the Kato-Trotter theorem on semi-group convergence whilst the latter is corollary of Wiener's version of the mean ergodic theorem.

ON THE SCATTERING THEORY OF MASSLESS NELSON MODELS

2002 ◽  
Vol 14 (11) ◽  
pp. 1165-1280 ◽  
Author(s):  
C. GÉRARD
Keyword(s):  
Scattering Theory ◽  
Bound States ◽  
Relativistic Quantum ◽  
Asymptotic Completeness ◽  
Induced Representations ◽  
Vacuum States ◽  
Relativistic Atom ◽  
Propagation Properties ◽  
Invariant Spaces

We study the scattering theory for a class of non-relativistic quantum field theory models describing a confined non-relativistic atom interacting with a massless relativistic bosonic field. We construct invariant spaces [Formula: see text] which are defined in terms of propagation properties for large times and which consist of states containing a finite number of bosons in the region {|x| ≥ ct} for t → ±∞. We show the existence of asymptotic fields and we prove that the associated asymptotic CCR representations preserve the spaces [Formula: see text] and induce on these spaces representations of Fock type. For these induced representations, we prove the property of geometric asymptotic completeness, which gives a characterization of the vacuum states in terms of propagation properties. Finally we show that a positive commutator estimate imply the asymptotic completeness property, i.e. the fact that the vacuum states of the induced representations coincide with the bound states of the Hamiltonian.

Uniform distribution and the mean ergodic theorem

1978 ◽  
Vol 50 (1) ◽  
pp. 65-74 ◽  
Author(s):  
V. Losert ◽  
H. Rindler
Keyword(s):  
Uniform Distribution ◽  
Ergodic Theorem ◽  
Mean Ergodic Theorem ◽  
The Mean

Computing Solutions of the Yang-Baxter-like Matrix Equation for Diagonalisable Matrices

2015 ◽  
Vol 5 (1) ◽  
pp. 75-84 ◽  
Author(s):  
J. Ding ◽  
Noah H. Rhee
Keyword(s):  
Numerical Methods ◽  
Ergodic Theorem ◽  
Matrix Equation ◽  
Mean Ergodic Theorem ◽  
The Mean ◽  
Square Matrix

AbstractThe Yang-Baxter-like matrix equation AXA = XAX is reconsidered, where A is any complex square matrix. A collection of spectral solutions for the unknown square matrix X were previously found. When A is diagonalisable, by applying the mean ergodic theorem we propose numerical methods to calculate those solutions.

scholarly journals TIME-DEPENDENT SCATTERING THEORY OF N-BODY QUANTUM SYSTEMS

2000 ◽  
Vol 12 (08) ◽  
pp. 1033-1084 ◽  
Author(s):  
W. HUNZIKER ◽  
I. M. SIGAL
Keyword(s):  
Phase Space ◽  
Scattering Theory ◽  
Asymptotic Completeness ◽  
Quantum Systems ◽  
Time Dependent ◽  
General Interest ◽  
Mean Square ◽  
Scattering States ◽  
New Ideas ◽  
The Mean

We give a full and self contained account of the basic results in N-body scattering theory which emerged over the last ten years: The existence and completeness of scattering states for potentials decreasing like r-μ, [Formula: see text]. Our approach is a synthesis of earlier work and of new ideas. Global conditions on the potentials are imposed only to define the dynamics. Asymptotic completeness is derived from the fact that the mean square diameter of the system diverges like t2 as t → ±∞ for any orbit ψt which is separated in energy from thresholds and eigenvalues (a generalized version of Mourre's theorem involving only the tails of the potentials at large distances). We introduce new propagation observables which considerably simplify the phase–space analysis. As a topic of general interest we describe a method of commutator expansions.

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