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Asymptotic completeness for N-body short-range quantum systems: A new proof. (English) Zbl 0726.35096
The author gives a new and shorter proof of the asymptotic completeness for N-body short-range quantum systems. The main originality of this work is in the way of proving a crucial propagation estimate, using some boosted time-dependent Hamiltonian, involving a real-valued phase function: \[ K(t)=(p-v(x,t))^ 2+V(x) \] with v(x,t) close to x/2t.
This permits to show that the intercluster motion \(P_ a\) behaves asymptotically as \(x_ a/2t\) for a given cluster decomposition a.

35P25 Scattering theory for PDEs
81U10 \(n\)-body potential quantum scattering theory
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[1] Combes, J.M., Thomas, L.: Asymptotic behavior of eigenfunctions for multiparticle Schrödinger operators. Commun. Math. Phys.34, 251–270 (1973) · Zbl 0271.35062
[2] Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators. Berlin, Heidelberg, New York: Springer 1987
[3] Deift, P., Hunziker, W., Simon, B., Vock, E.: Pointwise bounds on eigenfunctions and wave packets inN-body quantum systems. IV. Commun. Math. Phys.64, 1–34 (1978) · Zbl 0419.35079
[4] Deift, P., Simon, B.: A time-dependent approach to the completeness of multiparticle quantum systems. Commun. Pure Appl. Math.30, 573–583 (1977) · Zbl 0354.47004
[5] Dereziński, J.: A new proof of the propagation theorem fonN-body quantum systems. Commun. Math. Phys.122, 203–231 (1989) · Zbl 0677.47006
[6] Enss, V.: Asymptotic completeness for quantum-mechanical potential scattering. I. Shortrange potentials. Commun. Math. Phys.61, 285–291 (1978) · Zbl 0389.47005
[7] Enss, V.: Asymptotic completeness for quantum-mechanical potential scattering. II. Singular and long-range potentials. Ann Phys.119, 117–132 (1979) · Zbl 0408.47009
[8] Enss, V.: Completeness of three-body quantum scattering. In: Dynamics and processes, Blanchard, P., Streit, L. (eds.). Lecture Notes in Mathematics, vol. 1031, pp. 62–88. Berlin, Heidelberg, New York: Springer 1983 · Zbl 0531.47009
[9] Enss, V.: Introduction to asymptotic observables for multi-particle quantum scattering. In: Schrödinger operators, Aarhus 1985. Balslev, E. (ed.). Lecture Notes in Mathematics, vol. 1218, pp. 61–92. Berlin, Heidelberg, New York: Springer 1986
[10] Froese, R.G., Herbst, I.: A new proof of the Mourre estimate. Duke Math. J.49, 1075–1085 (1982) · Zbl 0514.35025
[11] Graf, G.M.: Phase space analysis of the charge transfer model. Helv. Phys. Acta63, 107–138 (1990) · Zbl 0741.35050
[12] Hack, M.N.: Wave operators in multichannel scattering. Nuovo Cim. Ser.X 13, 231–236 (1959) · Zbl 0086.42804
[13] Hunziker, W.: On the space-time behavior of Schrödinger wavefunctions. J. Math. Phys.7, 300–304 (1966) · Zbl 0151.43801
[14] Hunziker, W.: Time dependent scattering theory for singular potentials. Helv. Phys. Acta40, 1052–1062 (1967) · Zbl 0152.46303
[15] Jauch, J.M.: Theory of the scattering operator. I, II. Helv. Phys. Acta31, 127–158, 661–684 (1958) · Zbl 0081.43304
[16] Mourre, E.: Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys.78, 391–408 (1981) · Zbl 0489.47010
[17] Perry, P., Sigal, I.M., Simon, B.: Spectral analysis ofN-body Schrödinger operators. Ann. Math.114, 519–567 (1981) · Zbl 0477.35069
[18] Radin, C., Simon, B.: Invariant domains for the time-dependent Schrödinger equation. J. Diff. Eq.29, 289–296 (1978) · Zbl 0378.34008
[19] Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. I–IV. New York: Academic Pres 1972–79
[20] Sigal, I.M.: Scattering theory for Many-Body quantum mechanical systems. Lecture Notes in Mathematics, Vol. 1011. Berlin, Heidelberg, New York: Springer 1983 · Zbl 0522.47006
[21] Sigal, I.M., Sigalov, A.G.: Description of the spectrum of the energy operator of quantum mechanical systems that is invariant with respect to permutations of identical particles. Theor. Math. Phys.5, 990–1005 (1970)
[22] Sigal, I.M., Soffer, A.: TheN-particle scattering problem: asymptotic completeness for shortrange systems. Ann. Math.126, 35–108 (1987) · Zbl 0646.47009
[23] Sigal, I.M., Soffer, A.: Long-range many-body scattering. Asymptotic clustering for Coulombtype potentials. Invent. Math.99, 115–143 (1990) · Zbl 0702.35197
[24] Signal, I.M., Soffer, A.: Local decay and propagation estimates for time-dependent and time-independent Hamiltonians, Princeton University preprint (1988), to appear in Acta Math.
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