Asymptotic Completeness for N-Body Quantum Systems

  • Gian Michele Graf
Conference paper

Abstract

We give a sketch of a geometrical proof of asymptotic completeness for an arbitrary number of quantum particles interacting through short-range pair potentials.

Keywords

Kato 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators, Springer (1987)MATHGoogle Scholar
  2. 2.
    Deift, P., Simon, B.: A time-dependent approach to the completeness of multiparticle quantum systems. Comm.Pure Appl.Math. 30, 573–583 (1977)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Derezinski, J.: A new proof of the propagation theorem for N-body quantum systems. Comm.Math.Phys. 122, 203–231 (1989)MathSciNetCrossRefMATHADSGoogle Scholar
  4. 4.
    Derezinski, J.: Algebraic approach to the N-body quantum long range scattering, Warsaw University preprint (1990)Google Scholar
  5. 5.
    Enns, V.: Asymptotic completeness for quantum-mechanical potential scattering, I. Short-range potentials. Comm.Math.Phys. 61, 285–291 (1978)MathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Enns, V.: Completeness of Three-Body Quantum Scattering. In: Dynamics and Processes. Ed. by P. Blanchard, L. Streit, Lecture Notes in Mathematics, Vol. 1031, pp. 62–88, Springer (1983)CrossRefGoogle Scholar
  7. 7.
    Enns, V.: Introduction to asymptotic observables for multi-particle quantum scattering. In: Schrödinger Operators. Aarhus 1985. Ed. by E. Balslev, Lecture Notes in Mathematics, Vol. 1218, pp. 61–92, Springer (1986)CrossRefGoogle Scholar
  8. 8.
    Enns, V.: Two- and three-body quantum scattering: Completeness revisited. In: Symposium “Partial Differential Equations” Holzhau 1988. Ed. by B.-W. Schulze and H. Triebel, Texte zur Mathematik, Vol. 112, pp. 108–120, Teubner (1989)Google Scholar
  9. 9.
    Faddeev, L.: Mathematical Aspects of the Three Body Problem in Quantum Scattering Theory (Steklov Institute 1963), Israel Program for Scientific Translation (1965)MATHGoogle Scholar
  10. 10.
    Froese, R., Herbst, I.: A new proof of the Mourre estimate. Duke Math.J. 49, 1075–1085 (1982)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Graf, G.M.: Asymptotic completeness for N-body short-range quantum systems: A new proof. Comm.Math.Phys. 132, 73–101 (1990)MathSciNetCrossRefMATHADSGoogle Scholar
  12. 12.
    Kato, T.: Wave operators and similarity for some non-self-adjoint operators. Math.Ann. 162, 258–279 (1966)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kato, T.: Smooth operators and commutators. Stud.Math. 31, 535–546 (1968)MATHGoogle Scholar
  14. 14.
    Lavine, R.: Absolute continuity of Hamiltonian operators with repulsive potentials. Proc.Amer.Math.Soc. 22, 55–60 (1969)MathSciNetMATHGoogle Scholar
  15. 15.
    Mourre, E.: Link between the geometrical and the spectral transformation approaches in scattering theory. Comm.Math.Phys. 68, 91–94 (1979)MathSciNetCrossRefMATHADSGoogle Scholar
  16. 16.
    Mourre, E.: Absence of singular continuous spectrum for certain selfadjoint operators. Comm.Math.Phys. 78, 391–408 (1981)MathSciNetCrossRefMATHADSGoogle Scholar
  17. 17.
    Mourre, E.: Operateurs conjugés et propriété de propagation. Comm.Math. Phys. 91, 279–300 (1983)MathSciNetCrossRefMATHADSGoogle Scholar
  18. 18.
    Perry, P., Sigal, I.M., Simon, B.: Spectral analysis of N-body Schrödinger operators. Ann.Math. 114, 519–567 (1981)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Povzner, A.J.: On eigenfunction expansions in terms of scattering solutions. Dokl.Akad.Nauk SSSR 104, (1955)Google Scholar
  20. 20.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III. Scattering Theory. Academic Press (1979)MATHGoogle Scholar
  21. 21.
    Sigal, I.M.: Scattering Theory for Many-Body Quantum Mechanical Systems. Lecture Notes in Mathematics, Vol. 1011, Springer Verlag (1983)MATHGoogle Scholar
  22. 22.
    Sigal, I.M., Soffer, A.: The N-particle scattering problem: asymptotic completeness for short-range systems. Ann.Math. 126, 35–108 (1987)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Sigal, I.M., Soffer, A.: Local decay and propagation estimates for time-dependent and time-independent Hamiltonians. Princeton University preprint (1988)Google Scholar
  24. 24.
    Sigal, I.M., Soffer, A.: Long-range many-body scattering. Asymptotic clustering for Coulomb-type potentials. Invent.Math. 99, 115–143 (1990)MathSciNetCrossRefMATHADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Gian Michele Graf
    • 1
  1. 1.Division of Physics, Mathematics and AstronomyCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations