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High-momentum transfers and completeness of the Mandelstam program in potential scattering

Больщие передаваемые импульсы и полнота программы Манделстама в потенциальном рассеянии

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Il Nuovo Cimento A (1971-1996)

Summary

The Lippmann-Schwinger series for regular potentials is transformed into a representation containing only on-shell matrix elements. It proves that the Mandelstam program can be carried on without supplementary conditions even for non-Yukawa potentials, if the unitarity iterations are performed for the potential itself instead of itst-channel absorptive part. The analysis clarifies the relations between the Schrödinger andS-matrix descriptions of potential scattering and also the algebraic properties which turn the nonlinear on-shell iterations into the linear off-shell ones defined by the Lippmann-Schwinger equation. We also discuss the applicability of fixed-point techniques to the on-shell iterations and their relevance to the regularization of the perturbation series generated by singular potentials.

Riassunto

La serie di Lippmann e Schwinger per potenziali regolari è transformata in una rappresentazione contenente solo elementi di matrice sullo strato. Si prova che il programma di Mandelstam può essere portato avanti senza condizioni supplementari anche per potenziali non del tipo Yukawa se le iterazioni di unitarietà sono compiute per il potenziale stesso invece che per la sua parte assorbente del canalet. L’analisi chiarisce le relazioni fra le descrizioni di Schrödinger e della matriceS dello scattering di potenziale e anche le proprietà algebriche che cambiano le iterazioni non lineari sullo strato in quelle lineari fuori dallo strato definite dall’equazione di Lippmann e Schwinger. Si discute anche l’applicabilità delle tecniche del punto fisso a iterazioni sullo strato e la loro rilevanza per la regolarizzazione delle serie di perturbazioni generata da potenziali singolari.

Реэюме

Ряд Липмана-Щвингера для регулярных потенциалов трансформируется в представление, содержашее только оболочечные матричные злементы. Докаэывается, что программа Манделстама может быть осушествлена беэ дополнительных условий даже для потенциалов неюкавовского типа, если унитарные итерации проводятся для потенциала, а не дляt-канальной абсорбционной части. Предложенный аналиэ раэщясняет свяэь между описаниями Щвингера и с помошьюS-матрицы потенциального рассеяния. Иэ предложенного аналиэа следуют алгебраические свойства, которые преврашают нелинейные итерации на массовой поверхности в линейные итерации вне массовой поверхности, которые определяются уравнением Липмана-Щвингера. Мы также обсуждаем применимость техники с фиксированной точкой для итераций на массовой поверхности и уместность зтих итераций при регуляриэации рядов теории воэмушений для сингулярных потенциалов.

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References

  1. S. Mandelstam:Phys. Rev.,112, 1344 (1958).

    Article  ADS  MathSciNet  Google Scholar 

  2. R. Blankenbecler, M. L. Goldberger, N. N. Khuri andS. B. Treiman:Ann. Phys. (N. Y.),10, 62 (1960).

    Article  ADS  Google Scholar 

  3. H. M. Nussenzveig:Causality and Dispersion Relations (New York, N. Y., and London, 1972).

  4. S. Gasiorowicz andM. A. Ruderman:Phys. Rev.,107, 868 (1957).

    Article  ADS  Google Scholar 

  5. N. N. Khuri:Phys. Rev.,107, 1148 (1957).

    Article  ADS  MathSciNet  Google Scholar 

  6. R. J. Eden, P. V. Landshoff, D. I. Olive andJ. C. Polkinghorne:The Analytic S-Matrix (Cambridge, 1966).

  7. N. I. Muskhelishvili:Singular Integral Equations (Groningen, 1972).

  8. H. J. Bremermann andL. Durand:J. Math. Phys. (N. Y.),2, 240 (1961).

    Article  ADS  Google Scholar 

  9. W. Langbein: Kaiserslautern preprint (November 1979).

  10. B. Simon:Quantum Mechanics for Hamiltonians Defined as Quadratic Forms (Princeton, N. J., 1971);M. Reed andB. Simon:Methods of Modern Mathematical Physics, Vol.2, 3 (New York, N. Y., 1975).

  11. M. L. Goldberger andK. M. Watson:Collision Theory (New York, N. Y., 1965).

  12. For singular potentials repulsive near tor=0F(s, t) is invariant against contributions (d2j/dr2j)δ(r),j ⩾ 0, toV(r). This can be used to rewriteF(s, t) as a regularized Lippmann-Schwinger series with finite L(n)(s, t) (12, 13).

  13. N. N. Khuri andA. Pais:Rev. Mod. Phys.,36, 590 (1965).

    Article  ADS  Google Scholar 

  14. W. Güttinger andE. Pfaffelhuber:Nuovo Cimento A,43, 423 (1966).

    Article  ADS  Google Scholar 

  15. A. Martin:Preludes in Theoretical Physics in Honor of V. F. Weisskopf, edited byA. De-Shalit, H. Feshbach andL. van Hove (Amsterdam, 1966).

  16. D. Atkinson andJ. S. Frederisken:Commun. Math. Phys.,40, 55 (1975).

    Article  ADS  Google Scholar 

  17. J. Kupsch:Fortschr. Phys.,19, 783 (1971).

    Article  Google Scholar 

  18. D. Atkinson:Nucl. Phys. B,7, 375 (1968);8, 377 (1968);13, 415 (1969).

    Article  ADS  Google Scholar 

  19. J. Kupsch:Unitarity integral and Mellin transformation, preprint (November 1975).

  20. E. Predazzi andT. Regge:Nuovo Cimento,24, 518 (1962).

    Article  Google Scholar 

  21. N. Limić:Nuovo Cimento,26, 581 (1962).

    Article  MathSciNet  Google Scholar 

  22. G. H. Hardy:Divergent Series (Oxford, 1973).

  23. W. Becker andD. Grosser:Nuovo Cimento A,10, 343 (1972).

    Article  ADS  Google Scholar 

  24. W. Zimmermann:Nuovo Cimento,21, 249 (1961).

    Article  Google Scholar 

  25. L. D. Faddeev:Mathematical aspects of the three-body problem in the quantum scattering theory, inWorks of the Steklov Mathematical Institute, Vol.69 (Jerusalem, 1965).

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  1. Fachbereich Physik, Universität Kaiserslautern, Kaiserslautern

    W. Langbein

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Langbein, W. High-momentum transfers and completeness of the Mandelstam program in potential scattering. Nuov Cim A 56, 445–469 (1980). https://doi.org/10.1007/BF02732096

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