Il Nuovo Cimento (1955-1965)

, Volume 17, Issue 4, pp 558–579 | Cite as

Dispersion relations for scattering wave functions in potential theory

  • B. Bosco
Article
First Online:
Received:

Summary

Using the well known analytic properties of partial wave functions, dispersion relations are derived which can be transformed by unitarity into singular integral equations. In absence of bound-states a procedure is given which allows to determine, order by order, the wave function when theS-matrix is known at every order. A quite intuitive deduction of dispersion relations for the case of finite range potential based on the completeness relation is also reported. This result allows the derivation of dispersion relations for the transition matrix elements.

Riassunto

Facendo uso delle note proprietà analitiche delle funzioni d’onda parziali vengono derivate relazioni di dispersione che possono essere trasformate in equazioni integrali singolari mediante la unitarietà. In assenza di stati legati, viene descritto un procedimento atto a determinare, ordine per ordine, la funzione d’onda quando ia matriceS è nota a tutti gli ordini. È anche descritto un metodo intuitivo per dedurre le relazioni di dispersione nel caso di un potenziale a range finito. Questo risultato permette una deduzione delle relazioni di dispersione per gli elementi di matrice di transizione.

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. (1).
    J. Bowcock andA. Martin:Nuovo Cimento,14, 516 (1959).MathSciNetCrossRefGoogle Scholar
  2. (2).
    T. Regge:Nuovo Cimento,14, 951 (1959).MathSciNetCrossRefMATHGoogle Scholar
  3. (3).
    R. Blankenbecler, M. L. Goldberger, N. N. Khuri andS. B. Treiman:Ann. Phys. (to be published).Google Scholar
  4. (4).
    S. Mandelstam:Phys. Rev.,112, 1344 (1958).MathSciNetADSCrossRefGoogle Scholar
  5. (5).
    M. Gell-Mann:High Energy Nuclear Physics: Proceedings of the Sixth Annual Rochester Conference (1956), III p. 30. This program has been applied to the potential theory in reference (3) and to quantize theories byS. Mandelstam.Google Scholar
  6. (6).
    L. Castillejo, R. W. Dalitz andF. J. Dyson:Phys. Rev.,101, 453 (1956).ADSCrossRefMATHGoogle Scholar
  7. (8).
    R. Jost andW. Kohn:Dan. Mat. Fys. Medd.,27, 9 (1953).MathSciNetGoogle Scholar
  8. (9).
    N. I. Muskhelishvili:Singular Integral Equations (Groningen, 1953).Google Scholar
  9. (10).
    R. Jost andA. Pais:Phys. Rev.,82, 840 (1951).MathSciNetADSCrossRefGoogle Scholar
  10. (11).
    N. I. Muskhelishvili:Singular Integral Equations (Groningen, 1953).Google Scholar
  11. (12).
    R. Omnès:Nuovo Cimento,8, 316 (1958).CrossRefMATHGoogle Scholar
  12. (13).
    See for example the paper byR. Jost andW. Kohn quoted in ref. (8) footnote (6).MathSciNetGoogle Scholar
  13. (14).
    See the paper byR. Jost andA. Pais quoted in reference (10).MathSciNetADSCrossRefMATHGoogle Scholar
  14. (15).
    M. Verde:Nuovo Cimento,6, 340 (1957).MathSciNetCrossRefMATHGoogle Scholar
  15. (16).
    T. Regge:Nuovo Cimento,8, 671 (1958).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Società Italiana di Fisica 1960

Authors and Affiliations

  • B. Bosco
    • 1
    • 2
  1. 1.Istituto di Fisica dell’UniversitàTorino
  2. 2.Istituto Nazionale di Fisica NucleareSezione di TorinoItaly

Personalised recommendations