# zbMATH — the first resource for mathematics

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.

##### MSC:
 35P25 Scattering theory for PDEs 81U10 $$n$$-body potential quantum scattering theory
##### Keywords:
asymptotic completeness; intercluster motion
Full Text:
##### References:
 [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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.