Il Nuovo Cimento A (1965-1970)

, Volume 50, Issue 4, pp 894–922 | Cite as

Existence of the many-channelN/D representation

  • R. L. Warnock
Article
First Online:
Received:

Summary

Existence of theN/D representation is proved for an appropriate class of scattering matrices. The proof refers to a finite number of two-body channels. Alternatively, it refers to theN/D method with absorption in whichn two-body channels appear explicitly, and all other channels are taken into account by absorption parameters. The existence theorem requires asymptotic conditions on the scattering matrix weaker than those needed in earlier work on the subject. The conditions are compatible with the Regge-pole model of asymptotic behaviour provided theN/D method with absorption is used. An important factorization of the generalD-matrix is obtained in the formD=DR, whereD is a certain matrix which is nonsingular and free of poles at all finite points of the plane, andR is a matrix of rational functions. An analogous factorization is familiar in the single-channel problem, whereD is an exponential function of a phase-shift integral. By means of the formulaD=DR, generalized Levinson relations are formulated. One such relation is new; it has to do withn two-body channels, but does not involve neglect of the other channels.

Keywords

Asymptotic Behaviour Hilbert Problem Fredholm Equation Finite Plane Asymptotic Magnitude 
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.

Существование многоканальногоN/D представления

Резюме

Доказывается существованиеN/D представления для соответствующего класса матриц рассеяния. Доказательство учитывает конечное число двухчастичных каналов. С другой стороны доказательство ссылается наN/D метод с поглощением, в котором оказывается ровноn двухчастичных каналов; и все другие каналы учитываются при помощи параметров поглощения. Теорема существования требует асимптотических условий на матрицу рассеяния слабее, чем условия необходимые в более ранней работе на эту тему. Эти условия совместимы с моделью Редже полюсов для асимптотического проведения, при условии, что используетсяN/D метод с поглощением. Получается факторизация для общейD матрицы в формеD=DR, гдеD— определенная матрица, которая несингулярна и свободна от полюсов во всех конечных точках плоскостиR— матрица от рациональных функций. Аналогичная факторизация является хорошо известной в проблеме единственного канала, гдеD есть экспоненциальная функция для интеграла фазового сдвига. С помощью формулыD=DR формулируются обобщенные соотношения Левинсона. Одно из них является новым; оно связано сn двухчастичными каналами, но не затрагивает пренебрежение другими каналами.

Riassunto

Si dimostra l'esistenza della rappresentazioneN/D in un'opportuna classe di matrici di scattering. La prova si riferisce ad un numero finito di canali di due corpi. Alternativamente esso si riferisce al metodoN/D con assorbimento in cui appaiono esplicitamenten canali di due corpi, e si tiene conto di tutti gli altri canali con parametri di assorbimento. Il teorema dell'esistenza richiede per la matrice di scattering condizioni più deboli di quelle richieste in lavori precedenti su questo soggetto. Le condizioni sono compatibili col modello a poli di Regge del, comportamento asintotico purché si usi il metodoN/D con assorbimento. Si ottiene un'importante fattorizzazione della matriceD generale nella formaD=DR, in cuiD è una certa matrice non singolare e priva di poli in tutti i punti finiti del piano, edR è una matrice di funzioni razionali. Una fattorizzazione analoga è usuale nel problema di un solo canale, doveD è una funzione esponenziale di un integrale degli spostamenti di fase. Con la formulaD=DR si serivono relazioni di Levinson generalizzate. Una di queste relazioni è nuova; essa si riferisce adn canali di due corpi, ma non implica che si trascurino gli altri canali.

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.

Literatur

  1. (1).
    G. Frye andR. L. Warnock:Phys. Rev.,130, 478 (1963).ADSMathSciNetCrossRefGoogle Scholar
  2. (2).
    C. Lovelace:Tests for the Uniqueness of the Physical Universe: an Application of Nonlinear Functional Analysis to S Matrix Theory, CERN preprint TH. 689 (July, 1966).Google Scholar
  3. (3).
    M. Froissart:Nuovo Cimento,22, 191 (1961).MathSciNetCrossRefGoogle Scholar
  4. (4).
    R. L. Warnock:Phys. Rev.,146, 1109 (1966).ADSMathSciNetCrossRefGoogle Scholar
  5. (5).
    R. L. Warnock:Phys. Rev.,131, 1320 (1963); Trieste preprint IC/67/34; see also ref. (5a)R. L. Warnock:Lectures at the American University of Beirut, to be published.ADSMathSciNetCrossRefGoogle Scholar
  6. (5a).
    R. L. Warnock:Lectures at the American University of Beirut, to be published.Google Scholar
  7. (6).
    A. P. Balachandran andF. Von Hippel:Ann. of Phys.,30, 446 (1964);Y. S. Jin andA. Martin:Phys. Rev.,135, B 1369 (1964).ADSCrossRefGoogle Scholar
  8. (7).
    A. Martin:Nuovo Cimento,38, 1326 (1965).CrossRefGoogle Scholar
  9. (8).
    D. Atkinson, K. Dietz andD. Morgan:Ann. of Phys.,37, 77 (1966).ADSMathSciNetCrossRefGoogle Scholar
  10. (9).
    T. Kinoshita:Phys. Rev. Lett.,16, 869 (1966);Phys. Rev.,154, 1438 (1967).ADSCrossRefGoogle Scholar
  11. (10).
    A. P. Contogouris andA. Martin:Nuovo Cimento,49A, 61 (1967). This paper includes several references to earlier work.ADSMathSciNetCrossRefGoogle Scholar
  12. (11).
    S. Mandelstam:Ann. of Phys.,21, 302 (1963).ADSMathSciNetCrossRefGoogle Scholar
  13. (12).
    R. Jost andR. Newton:Nuovo Cimento,1, 590 (1955).MathSciNetCrossRefGoogle Scholar
  14. (13).
    J. B. Hartle andJ. R. Taylor:Journ. Math. Phys.,8, 651 (1967).ADSCrossRefGoogle Scholar
  15. (14).
    R. G. Newton:A Scattering Theory of Waves and Particles (New York, 1966).Google Scholar
  16. (15).
    J. Plemelj:Problems in the Sense of Riemann and Klein (New York, 1964).Google Scholar
  17. (16).
    N. I. Muskhelishvili:Singular Integral Equations (Groningen, 1963). See especially Chapter 18.Google Scholar
  18. (17).
    J. Plemelj:Monatshefte für Math. u. Phys., Jahrgang,19, 211 (1908).MathSciNetCrossRefGoogle Scholar
  19. (19).
    S. Mikhlin:Integral Equations (Oxford, 1964).Google Scholar
  20. (21).
    The best discussions of the Levinson relation are in the second paper mentioned in ref. (5); Trieste preprint IC/67/34; and in ref. (5a).R. L. Warnock:Lectures at the American University of Beirut, to be published. In the single-channel case, (5.8) may be replaced by the assumption of a dispersion relation withn subtractions, and (5.7) need not hold att=0. Instead of (5.7) att=0 one requires that δ approach a limit δ(∞).ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica 1967

Authors and Affiliations

  • R. L. Warnock
    • 1
  1. 1.International Atomic Energy AgencyInternational Centre for Theoretical PhysicsTrieste

Personalised recommendations