Advertisement

Asymptotic Observables in the N-Body Quantum Long Range Scattering

  • Jan Dereziński
Chapter
Part of the Mathematical Physics Studies book series (MPST, volume 12)

Abstract

Certain observables converge in the Heisenberg picture to a limit as time goes to ±∞. These limits, which may be called asymptotic observables, are especially interesting in the context of long range N-body Schrödinger operators. By studying certain natural classes of asymptotic observables one can get a lot of insight in the quantum N-body scattering.

This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A]
    Agmon, S.: Lectures on the exponential decay of solutions of second order elliptic equations, Princeton University Press, 1982.Google Scholar
  2. [ABG]
    Amrein, W.O., Boutet de Monvel-Berthier,A.M. and Georgescu,V.: Notes on the N-body problem, preprint, Génève, 1989.Google Scholar
  3. [AGM]
    Amrein, E.O., Georgescu, V. and Martin, Ph.A.: Approche algébrique de la théorie non-relativiste de la diffusion aux canaux multiples, in: Physical Reality and Mathematical Description, Enz/Mehra (eds), 255–276 (1974).CrossRefGoogle Scholar
  4. [AMM]
    Amrein, W.O., Martin, Ph.A. and Misra, B.: On the asymptotic condition of scattering theory, Helv.Phys.Acta 43, 313–344 (1970).MathSciNetzbMATHGoogle Scholar
  5. [CFKS]
    Cycon, H.L., Froese, R., Kirsch, W. and Simon, B.: Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Springer, Berlin, Heidelberg, New York, 1987.Google Scholar
  6. [Del]
    Derezinski, J.: A new proof of the propagation theorem for N-body quantum systems, Comm.Math.Phys. 122, 203–231 (1989).MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. [De2]
    Derezinski, J.: Algebraic Approach to the N-body Quantum Long Range Scattering, Preprint 1990.Google Scholar
  8. [Do]
    Dollard, J.: Asymptotic convergence and Coulomb interaction, Journ.Math.Phys. 5, 729–738 (1964).MathSciNetADSCrossRefGoogle Scholar
  9. [El]
    Enss, V.: Asymptotic observables on scattering states, Comm. Math.Phys. 89, 245–268 (1983).MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. [E2]
    Enss, V.: Quantum scattering theory of two-and three-body systems with potentials of short and long range, in: Schrödinger Operators, ed. by S.Graffi, Lecture Notes in Mathematics, vol. 1159, ( Springer, Berlin, Heidelberg, New York 1985 ).Google Scholar
  11. [IsoKi]
    Isozaki, H. and Kitada, H.: Scattering matrices for two-body Schrödinger operators, Scientific Papers of the College of Arts and Sciences, Tokyo Univ. 35, 81–107 (1985).MathSciNetGoogle Scholar
  12. [La]
    Lavine, R.: Scattering theory for long range potentials, Journ.Func.Anal. 5, 368–382, (1970).MathSciNetzbMATHCrossRefGoogle Scholar
  13. [Ml]
    Mourre, E.: Absence of singular continuous spectrum for certain self adjoint operators, Comm.Math.Phys. 78, 391–408 (1981)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. [M2]
    Mourre, E.: Opérateurs conjugués et propriétés de propagations, Comm.Math.Phys. 91, 279–300 (1983).MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. [Pe]
    Perry, P.: Scattering Theory by the Enss Method(Harwood Academic London 1983).zbMATHGoogle Scholar
  16. [RS]
    Reed, M. and Simon B.: Methods of Modern Mathematical Physics, III: Scattering Theory (Academic Press, New York, 1979 ).zbMATHGoogle Scholar
  17. [Sig]
    Sigal, I.M.: Geometric methods in the quantum many-body problem. Nonexistence of very negative ions., Comm.Math.Phys. 85, 309–324 (1982).MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. [SigSofl]
    Sigal, I.M. and Soffer, A.: The N-particle scattering problem: asymptotic completeness for short range systems, Anal.Math. 125, 35–108, (1987).MathSciNetGoogle Scholar
  19. [SigSof2]
    Sigal, I.M. and Soffer, A.: Local decay and velocity bounds, preprint, Princeton (1988).Google Scholar
  20. [SigSof3]
    Sigal, I.M. and Soffer, A.: Long range many body scattering, Asymptotic clustering for Coulomb type potentials, preprint, Toronto (1988).Google Scholar
  21. [Sim]
    Simon, B.: Geometric methods in multiparticle quantum systems, Comm.Math.Phys. 55, 259–274, (1977).MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. [Ta]
    Takesaki, M.: Theory of Operator Algebras I. Springer Berlin, Heidelberg, New York 1979.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1991

Authors and Affiliations

  • Jan Dereziński
    • 1
  1. 1.Department of Mathematical Methods in PhysicsWarsaw UniversityWarszawaPoland

Personalised recommendations

Cite chapter

Buy options