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Long-range many-body scattering. Asymptotic clustering for Coulomb-type potentials. (English) Zbl 0702.35197
The authors study the scattering theory for quantum many-body systems with pair potentials of Coulomb type (i.e. \(O(| x|^{-1})\) at \(\infty)\). The main result of the paper is that the system is asymptotically clustering for any non-threshold energy. Roughly speaking, this means that starting at a state located near a non-threshold energy, as \(t\to \pm \infty\), the system disintegrates into non-interacting, freely moving subsystems. As the authors show, the asymptotic clustering does not imply in general the asymptotic completeness and one can not rule out the possibility that asymptotic completeness breaks down for longe-range systems. However, this can happen only for more than four particles. For three- and four-particle systems the asymptotic completeness has been proved by V. Enss [Lect. Notes Math. 1159, 39-176 (1985; Zbl 0585.35023)] and by the authors in a forthcoming paper, respectively.
Reviewer: P.Stefanov

MSC:
35P25 Scattering theory for PDEs
81U10 \(n\)-body potential quantum scattering theory
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References:
[1] [Comb] Combes, J.M.: Relatively compact interactions in many particle systems. Commun. Math. Phys.12, 283 (1969) · Zbl 0174.28304
[2] [CFKS] Cycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger operators. Berlin Heidelberg New York: Springer 1987 · Zbl 0619.47005
[3] [Der] Derezinski, J.: A new proof of propagation theorem forN-body quantum systems. Viginia Tech. 1988 (Preprint)
[4] [En] Enss, V.: Quantum scattering theory for two and three-body systems with potentials of short- and long-range. In: Graffi, S. (ed.) Schrödinger operators. (Lect. Notes Math., Vol. 1159) Berlin Heidelberg New York: Springer 1985 · Zbl 0585.35023
[5] [FH] Froese, R., Herbst, I.: A new proof of Mourre estimate. Duke Math. J.49, 1975 (1982) · Zbl 0514.35025
[6] [Ka1] Kato, T.: Fundamental properties of Hamiltonian operators of Schrödinger type. Trans. Am. Math. Soc.70, 195–211 (1951) · Zbl 0044.42701
[7] [Ka2] Kato, T.: Wave operators and similarity for some non-self-adjoint operators. Math. Ann.162, 258–269 (1966) · Zbl 0139.31203
[8] [Mo1] Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys.78, 391–408 (1981) · Zbl 0489.47010
[9] [Mo2] Mourre, E.: private communication
[10] [Pe] Perry, P.: Commun. Math. Phys.81, 243–259 (1981) · Zbl 0471.47007
[11] [PSS] Perry, P., Sigal, I.M., Simon, B.: Spectral analysis ofN-body Schrödinger operators. Ann. Math.114, 519–567 (1981) · Zbl 0477.35069
[12] [RS] Reed, M., Simon, B.: Methods of modern mathematical physics II, III, IV. New York: Academic Press
[13] [Sig1] Sigal, I.M.: On the long-range scattering. Toronto (Preprint)
[14] [Sig2] Sigal, I.M.: Lectures on scattering theory. Toronto: xxx 1987
[15] [SigSof1] Sigal, I.M., Soffer, A.: TheN-particle scattering problem: asymptotic completeness for short-range systems. Anal. Math.125, (1987) · Zbl 0646.47009
[16] [SigSof2] Sigal, I.M., Soffer, A.: Asymptotic completeness of multiparticle scattering. In: Knowles, Saito (eds.) Differential equations and mathematical physics. (Lect. Notes Math., Vol. 1285) Berlin Heidelberg New York: Springer 1987 · Zbl 0651.35071
[17] [SigSof3] Sigal, I.M. Soffer, A.: Local decay and velocity bounds. Princeton (1988) (Preprint)
[18] [SigSof4] Sigal, I.M., Soffer, A.: Asymptotic completeness for Coulomb-type 4-body systems. (In preparation)
[19] [SinMuth] Sinha, K., Muthuramalingam, P.L.: J. Funct. Anal.55, 323–343 (1984) · Zbl 0531.47008
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