Advertisement

Unstable quantum states and rigged Hilbert spaces

  • V. Gorini
  • G. Parravicini
Groups and Semigroups in the Description of Decaying Systems
Part of the Lecture Notes in Physics book series (LNP, volume 94)

Abstract

We apply rigged Hilbert space techniques to the quantum mechanical treatment of unstable states in nonrelativistic scattering theory.We discuss a method which is based on representations of decay amplitudes in terms of expansions over complete sets of generalized eigenvectors of the interacting Hamiltonian, corresponding to complex eigenvalues.These expansions contain both a “discrete” and a “continuum” contribution. The former corresponds to eigenvalues located at the second sheet poles of the S matrix, and yields the exponential terms in the survival amplitude. The latter arises from generalized eigenvectors associated to complex eigenvalues on background contours in the complex plane, and gives the corrections to the exponential law.

Keywords

Analytic Continuation Unstable State Complex Eigenvalue Analytic Family Laurent Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    L. Fonda, G. C. Ghirardi, A. Rimini: Rep. Progr. Phys. 41, 587 (1978).Google Scholar
  2. [2]
    J. Schwinger: Ann. Phys. (N.Y.) 9, 169 (1960).Google Scholar
  3. [3]
    A. Grossmann: J. Math. Phys. 5, 1025 (1964).Google Scholar
  4. [4]
    J. S. Howland:Amer. J. Math. 91, 1106 (1969); Bull. Am. Math. Soc. 78, 280 (1972); Trans. Am. Math. Soc. 162, 141 (1971); J. Math. Anal. and Appl. 50, 415 (1975).Google Scholar
  5. [5]
    J. Aguilar, J. M. Combes; Commun. Math. Phys. 22, 269 (1961). E. Balslev, J. M. Combes: Commun. Math. Phys. 22, 280 (1971). R. A. Weder: J. Math. Phys. 15, 20 (1974).Google Scholar
  6. [6]
    M. Reed, B. Simon: Methods of Modern Mathematical Physics (Academic Press New York).Google Scholar
  7. [7]
    L. P. Horwitz, J. P. Marchand: Rocky Mountain J. Math. 1, 225 (1971).Google Scholar
  8. [8]
    H. Baumgdrtel: Math. Nachr. 75, 133 (1976).Google Scholar
  9. [9]
    L. P. Horwitz, I. M. Sigal: Tel Aviv University Preprint, 1976.Google Scholar
  10. [10]
    I. M. Gel'fand, et al.: Generalized Functions (Academic Press, New York, 1964).Google Scholar
  11. [11]
    K. Maurin: General Ei en coon Expansions and Unitary Representations of Topological Groups PWN Warsaw, 1968).Google Scholar
  12. [12]
    J. E. Roberts: Commun. Math. Phys. 3, 98 (1966); J. P. Antoine: J. Math. Phys. 10, 53 (1969); 0. Melsheimer: I-II, J. Math. Phys. 15, 902 (1974); A. Böhm: Boulder Lectures in Theoretical Physics, Vol. 9A, 255 (1966); The Rigged Hilbert Space and Quantum Mechanics, Lecture Notes in Physics, vol. 78 pringer, New York, 19Y8).Google Scholar
  13. [13]
    J. G. Kuriyan, N. Mukunda, E.C.G. Sudarshan: J. Math. Phys. 9, 2100 (1968).Google Scholar
  14. [14]
    G. J. Iverson: Unita Adjoint Representations of the Lorentz Groups I, II, University of Adelaide (1967); Phys. Lett. 26B, 229 (1968).Google Scholar
  15. [15]
    E.C.G. Sudarshan, C. B. Chiu, V. Gorini: Phys. Rev. D, in press.Google Scholar
  16. [16]
    K. 0. Friedrichs: Commun. Pure and Appl. Math. 1, 361 (1948).Google Scholar
  17. [17]
    G. Parravicini, V. Gorini, E.C.G. Sudarshan, C. B. Chiu:in preparation.Google Scholar
  18. [18]
    E. B. Davies: Lett. Math. Phys. 1, 31 (1975).Google Scholar
  19. [19]
    T. K. Bailey, W. C. Schieve: Complex Energy Eigenstates in Quantum Decay Models, Nuovo Cimento, in press.Google Scholar
  20. [20]
    A. P. Grecos, I. Prigogine: Irreversible Processes in Quantum Theory, Contribution in these Proceedings, an references quoted therein.Google Scholar
  21. [21]
    G. Lindblad, B. Nagel: Ann. Inst. H. Poincaré, 13A, 27 (1970).Google Scholar
  22. [22]
    A. Böhm: Quantum Mechanics (Springer, New York, 1979).Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • V. Gorini
    • 1
    • 2
  • G. Parravicini
    • 1
  1. 1.Physics Department, CPTThe University of Texas at AustinAustin
  2. 2.Istituto di Fisica dell 'UniversitáMilanoItaly

Personalised recommendations