Communications in Mathematical Physics

, Volume 154, Issue 3, pp 523–554 | Cite as

Radiation conditions and scattering theory forN-particle Hamiltonians

  • D. Yafaev


The correct form of the angular part of radiation conditions is found in scattering problem forN-particle quantum systems. The estimates obtained allow us to give an elementary proof of asymptotic completeness for such systems in the framework of the theory of smooth perturbations.


Radiation Neural Network Statistical Physic Complex System Nonlinear Dynamics 
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  1. 1.
    Faddeev, L.D.: Mathematical Aspects of the Three Body Problem in Quantum Scattering Theory. Trudy MIAN69, 1963 (Russian)Google Scholar
  2. 2.
    Ginibre, J., Moulin, M.: Hilbert space approach to the quantum mechanical three body problem. Ann. Inst. H. PoincaréA21, 97–145 (1974)Google Scholar
  3. 3.
    Thomas, L.E.: Asymptotic completeness in two- and three-particle quantum mechanical scattering. Ann. Phys.90, 127–165 (1975)CrossRefGoogle Scholar
  4. 4.
    Hepp, K.: On the quantum-mechanicalN-body problem. Helv. Phys. Acta42, 425–458 (1969)Google Scholar
  5. 5.
    Sigal, I.M.: Scattering Theory for Many-Body Quantum Mechanical Systems. Springer Lecture Notes in Math.1011, 1983Google Scholar
  6. 6.
    Iorio, R.J., O'Carrol, M.: Asymptotic completeness for multi-particle Schrödinger Hamiltonians with weak potentials. Commun. Math. Phys.27, 137–145 (1972)CrossRefGoogle Scholar
  7. 7.
    Kato, T.: Smooth operators and commutators. Studia Math.31, 535–546 (1968)Google Scholar
  8. 8.
    Lavine, R.: Commutators and scattering theory I: Repulsive interactions. Commun. Math. Phys.20, 301–323 (1971)CrossRefGoogle Scholar
  9. 9.
    Lavine, R.: Completeness of the wave operators in the repulsiveN-body problem. J. Math. Phys.14, 376–379 (1973)CrossRefGoogle Scholar
  10. 10.
    Sigal, I.M., Soffer, A.: TheN-particle scattering problem: Asymptotic completeness for short-range systems. Ann. Math.126, 35–108 (1987)Google Scholar
  11. 11.
    Derezinski, J.: A new proof of the propagation theorem forN-body quantum systems. Commun. Math. Phys.122, 203–231 (1989)CrossRefGoogle Scholar
  12. 12.
    Graf, G.M.: Asymptotic completeness forN-body short-range quantum systems: A new proof. Commun. Math. Phys.132, 73–101 (1990)Google Scholar
  13. 13.
    Enss, V.: Completeness of three-body quantum scattering. In: Dynamics and processes, Blanchard, P., Streit, L. (eds.), Springer Lecture Notes in Math.103, 62–88 (1983)Google Scholar
  14. 14.
    Kato, T.: Wave operators and similarity for some non-self-adjoint operators. Math. Ann.162, 258–279 (1966)CrossRefGoogle Scholar
  15. 15.
    Yafaev, D.R.: Radiation conditions and scattering theory for three-particle Hamiltonians. Preprint 91-01, Nantes University, 1991Google Scholar
  16. 16.
    Yafaev, D.R.: Mathematical Scattering Theory. Providence, RI: Am. Math. Soc., 1992Google Scholar
  17. 17.
    Saito, Y.: Spectral Representation for Schrödinger Operators with Long-Range Potentials. Springer Lecture Notes in Math.727, 1979Google Scholar
  18. 18.
    Constantin, P.: Scattering for Schrödinger operators in a class of domains with noncompact boundaries. J. Funct. Anal.44, 87–119 (1981)CrossRefGoogle Scholar
  19. 19.
    Il'in, E.M.: Scattering by unbounded obstacles for elliptic operators of second order. Proc. of the Steklov Inst. of Math.179, 85–107 (1989)Google Scholar
  20. 20.
    Yafaev, D.R.: Remarks on the spectral theory for the multiparticle type Schrödinger operator. J. Soviet Math.31, 3445–3459 (1985), translated from Zap. Nauchn. Sem. LOMI133, 277–298 (1984)Google Scholar
  21. 21.
    Combes, J.M.: Stationary scattering theory. In: Rigorous atomic and molecular physics, Velo, G., Wightmam, A. (eds.), Plenum Press 71–98 (1981)Google Scholar
  22. 22.
    Agmon, S.: Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations. Math. Notes, Princeton, NJ: Princeton Univ. Press 1982Google Scholar
  23. 23.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics III. New York: Academic Press 1979Google Scholar
  24. 24.
    Mourre, E.: Absence of singular spectrum for certain self-adjoint operators. Commun. Math. Phys.78, 391–400 (1981)CrossRefGoogle Scholar
  25. 25.
    Perry, P., Sigal, I.M., Simon, B.: Spectral analysis ofN-body Schrödinger operators. Ann. Math.144, 519–567 (1981)Google Scholar
  26. 26.
    Froese, R., Herbst, I.: A new proof of the Mourre estimate. Duke Math. J.49, 1075–1085 (1982)CrossRefGoogle Scholar
  27. 27.
    Lavine, R.: Commutators and scattering theory II: A class of one-body problems. Indiana Univ. Math. J.21, 643–656 (1972)CrossRefGoogle Scholar
  28. 28.
    Deift, P., Simon, B.: A time-dependent approach to the completeness of multiparticle quantum systems. Commun. Pure Appl. Math.30, 573–583 (1977)Google Scholar
  29. 29.
    Tamura, H.: Asymptotic completeness forN-body Schrödinger operators with short-range interactions. Commun. Part. Diff Eq.16, 1129–1154 (1991)Google Scholar
  30. 30.
    Iftimovici, A.: On asymptotic completeness for Agmon type Hamiltonians. C.R. Acad. Sci. Paris, S.I.314, 337–342 (1992)Google Scholar
  31. 31.
    Herbst, I., Skibsted, E.: Radiation conditions—The free channelN-body case. Aarhus University preprint, 1991Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • D. Yafaev
    • 1
    • 2
  1. 1.Université de NantesNantesFrance
  2. 2.Math. Inst.St. PetersburgRussia

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