The S-matrix in classical mechanics
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In the Hilbert-space version of classical mechanics, scattering theory forN-particle systems is developed in close analogy to the quantum case. Asymptotic completeness is proved for forces of finite range. Infinite-range forces lead to the problem of stability of bound states and can be dealt with only in some simple cases.
KeywordsNeural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing
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- 1.After this was written, we learned thatJ. M. Cook had already treated the caseN=2 in the same spirit (see 1965 Cargèse Lectures in Theoretical Physics, edited byF. Lurcat. New York: Gordon & Breach 1967).Google Scholar
- 2.Kato, T.: Trans. Am. Math. Soc.70, 195 (1951).Google Scholar
- 3.Nelson, E.: Operator differential equations, Lemma 12.1, mimeographed lecture notes. Princeton University 1964.Google Scholar
- 4.Ruelle, D., unpublished.Google Scholar
- 5.Essentially we followJauch, J. M., Helv. Physica Acta31, 661 (1958), but we prefer a different definition of theS-operator, due toBerezin, F. A., L. D. Faddeev, andR. A. Minlos, Proceedings of the Fourth All-Union Mathematical Conference, held in Leningrad 1961.Google Scholar
- 6.This is a classical result: seeSiegel, C. L.: Vorlesungen über Himmelsmechanik, § 30, Berlin, Göttingen, Heidelberg: Springer 1956. I am indebted toR. Jost for this remark.Google Scholar
- 7.For the quantum mechanical proof seeHack, M. N.: Nuovo Cimento13, 231 (1959).Google Scholar