[ital N]-body quantum scattering theory in two Hilbert spaces. VII. Real-energy limits

A study is made of the real-energy limits of approximate solutions of the Chandler--Gibson equations, as well as the real-energy limits of the approximate equations themselves. It is proved that (1) the approximate time-independent transition operator [ital T][sup [pi]]([ital z]) and an auxiliary op...

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Bibliographic Details
Published in:Journal of mathematical physics 1994-04, Vol.35:4
Main Authors: Chandler, C., Gibson, A.G.
Format: Article
Language:English
Subjects:
661100 - Classical & Quantum Mechanics- (1992-)
663000 - Nuclear Physics- (1992-)
AMPLITUDES
BANACH SPACE
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
ENERGY
HAMILTONIANS
HILBERT SPACE
KERNELS
MANY-BODY PROBLEM
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MECHANICS
NUCLEAR PHYSICS AND RADIATION PHYSICS
NUCLEAR REACTIONS
QUANTUM MECHANICS
QUANTUM OPERATORS
SCATTERING
SCATTERING AMPLITUDES
SPACE
TRANSITION AMPLITUDES
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Summary:A study is made of the real-energy limits of approximate solutions of the Chandler--Gibson equations, as well as the real-energy limits of the approximate equations themselves. It is proved that (1) the approximate time-independent transition operator [ital T][sup [pi]]([ital z]) and an auxiliary operator [ital M][sup [pi]]([ital z]), when restricted to finite energy intervals, are trace class operators and have limits in trace norm for almost all values of the real energy; (2) the basic dynamical equation that determines the operator [ital M][sup [pi]]([ital z]), when restricted to the space of trace class operators, has a real-energy limit in trace norm for almost all values of the real energy; (3) the real-energy limit of [ital M][sup [pi]]([ital z]) is a solution of the real-energy limit equation; (4) the diagonal (on-shell) elements of the kernels of the real-energy limit of [ital T][sup [pi]]([ital z]) and of all solutions of the real-energy limit equation exactly equal the on-shell transition operator, implying that the real-energy limit equation uniquely determines the physical transition amplitude; and (5) a sequence of approximate on-shell transition operators converges strongly to the exact on-shell transition operator. These mathematically rigorous results are believed to be the most general of their type for nonrelativistic [ital N]-body quantum scattering theories.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.530603