Il Nuovo Cimento (1955-1965)

, Volume 23, Issue 6, pp 954–1004 | Cite as

Potential scattering for complex energy and angular momentum

  • A. Bottino
  • A. M. Longoni
  • T. Regge


The analytic properties of the partial wave scattering amplitude for potential scattering in the pair of variablesk (wave number) andλ=l+1/2 have been derived when both variables are complex. Several results on the location of the poles of theS-matrix follow from a procedure of analytic completion. The scattering process is then considered as described by the variablesλ andk, instead ofs andt, as in Mandelstam work. The set of properties ofS(λ,k)=exp [2(λ,k)] here derived is exactly equivalent to the double dispersion formulas for energy and momentum transfer.


Vengono ricavate le proprietà analitiche dell’ampiezza di diffusione in onde parziali, per diffusione da potenziale, quando le variabilik (numero d’onda) eλ=l+1/2 sono simultaneamente complesse. Alcuni risultati sulla posizione dei poli della matriceS derivano da un procedimento di completamento analitico. Le proprietà qui derivate della matriceS sono esattamente equivalenti alle formule di dispersione doppie dell’ampiezza di diffusione nell’energia e nel momento trasferito.


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  1. (1).
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  2. (*).
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    A more complete investigation on this subject has been carried out in a forthcoming paper by two of us (A.B.-A.M.L.): Holomorphy domain of theS-matrix in potential scattering (submitted toNuovo Cimento).Google Scholar
  15. (*).
    We are indebted to Prof.R. Ascoli for very interesting discussions on this matter and to Prof.V. Glaser for his kind permission to use his results in this paper.Google Scholar
  16. (*).
    An accurate discussion about poles in the case of complex values ofk (λ real) has been carried out byR. G. Newton (9).ADSCrossRefzbMATHGoogle Scholar
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  18. (*).
    This property off(E, cosθ) is true only for potentials which are of the Yukawa type. In (2–4) are reported proofs which are essentially based on the perturbative expansion of the scattering amplitude in momentum space. In (1) this analyticity is derived with the help of the present W.K.B. approximation and of the Watson transform (2.13). This proof is the one reported here.MathSciNetCrossRefzbMATHGoogle Scholar
  19. (*).
    For a complete discussion of this point, see Lectures deliverel by one of us (T.R.) at the Summer School on High Energy Physics, in Hercegnovi, Yugoslavia (1961).Google Scholar
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  23. (*).
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  24. (**).
    Bateman Project Staff:Higher Trascendental Functions, vol.2 (1953), p. 21, eq. (27).Google Scholar

Copyright information

© Società Italiana di Fisica 1962

Authors and Affiliations

  • A. Bottino
    • 1
    • 2
  • A. M. Longoni
    • 1
    • 2
  • T. Regge
    • 1
    • 2
  1. 1.Istituto di Fisica dell’UniversitàTorino
  2. 2.Istituto Nazionale di Fisica NucleareSezione di TorinoItaly

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