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

• 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
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.530603

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.