Abstract
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.
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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).
Kato, T.: Trans. Am. Math. Soc.70, 195 (1951).
Nelson, E.: Operator differential equations, Lemma 12.1, mimeographed lecture notes. Princeton University 1964.
Ruelle, D., unpublished.
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.
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.
For the quantum mechanical proof seeHack, M. N.: Nuovo Cimento13, 231 (1959).
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It is a pleasure to thank Prof.L. Motchane for his kind hospitality at the I.H.E.S., where most of this work was done, and where the author profited from discussions withD. Ruelle andO. E. Lanford.
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Hunziker, W. The S-matrix in classical mechanics. Commun.Math. Phys. 8, 282–299 (1968). https://doi.org/10.1007/BF01646269
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DOI: https://doi.org/10.1007/BF01646269