Summary
Recent techniques on the asymptotic expansion of Mellin convolutions allow the evaluation of the interference of waves with discontinuous asymptotic behaviour. As an application, I consider the flux of representative points in theR 6 co-ordinate space for 3-3 collision and address myself toO(1) self-interferences of the anomalous double scattering. It is shown how the contribution from the classically forbidden region can be summed up to the ridge with a quite simple (and mandatory) regularization procedure. In addition, the matching across the ridge, where the wave has no uniform asymptotic behaviour, contributes noO(1) flux.
Riassunto
Recenti tecniche per trattare convoluzioni di Mellin rendono possibile il calcolo dell’interferenza di onde con comportamento asintotico discontinuo. Come applicazione, si considere il flusso dei punti rappresentativi nello spazioR 6 delle coordinate relative per lo scattering elastico 3-3 e si valutano le interferenze di ordineO(1) del doppio scattering. Si dimostra che il contributo dalle regioni che non sono permesse classicamente deve essere sommato fino alla zona limite («ridge») dello scattering classico mediante una ben precisa regolarizzazione. Inoltre, la zona intermedia in cui si verifica il cambiamento del comportamento asintotico non fornisce termini di ordineO(1).
Резюме
Недавно предложепные методы асимптотического разложения сверток Меллина позволяют определить интерференцию волн с разрывным асимптотическим поведением. Как приложение, рассматривается поток представительных точек вR 6 координатном пространстве для упругого рассеяния 3-3. Вычисляется интерференция порядкаO(1) для двукратного рассеяния. Показывается, как вклад от классически запрещенной области может быт просуммирован вплоть до «гребня», используя простую процедуры регуляизации. Кроме того, промежуточная зона, в которой происходит изменение асимптотического поведения, не дает вклада в члены порядкаO(1).
Similar content being viewed by others
References
L. D. Faddeev:Mathematical Aspects of the Three-Body Problem in the Quantum Scattering, Israel Program for Scientific Translations (Jerusalem, 1965) (English translation).
D. Jagolnitzer:J. Math. Phys. (N. Y.),6, 1576 (1965).
E. Gerjouy:Philos. Trans. R. Soc. London,270, 197 (1971).
S. P. Merkur’ev:Teor. Mat. Fiz.,8, 235 (1971).
J. Nuttall:J. Math. Phys. (N. Y.),12, 1896 (1971).
R. G. Newton:Ann. Phys. (N. Y.),74, 324 (1972);78, 561 (1973).
R. G. Newton andR. Shtockhamer:Phys. Rev. A,14, 642 (1976).
V. S. Potapov andJ. R. Taylor:Phys. Rev. A,16, 2264, 2276 (1977).
J.-M. Combes, R. G. Newton andR. Shtockhamer:Phys. Rev. D,11, 366 (1975).
S. Servadio:Nuovo Cimento B,65, 57 (1981).
S. Servadio:Phys. Rev. A 4, 1256 (1971).
V. S. Buslaev, S. P. Merkur’ev:Teor. Mat. Fiz.,5, 1216 (1970).
S. P. Merkur’ev:Zap. Nauchn. Semin. LOMI, 95 (1976).
R. Dashen, S.-Keng Ma andH. Bernstein:Phys. Rev.,187, 345 (1969).
S. Servadio:Optical theorem for three-three scattering, inProceedings of the XX Internationale Universitatswochen für Kernphysik, Schladming, 1981, Acta Phys. Austriaca, Suppl.23, 689 (1981).
F. W. J. Olver:Asymptotics and Special Functions (New York, N. Y., 1974).
R. Wong:J. Math. Anal. Appl.,72, 740 (1979);Error bounds for asymptotic expansions for integrals, SIAM Rev. (in press).
N. Bleistein andR. A. Handelsman:Asymptotic Expansions of Integrals (New York, N. Y., 1975).
S. Servadio:Phys. Rev. A,24, 793 (1981).
In these flux integrals the ω range is meant to be (0, π/2). Strictly speaking, the ω∼0 neighbourhood should be removed (10). since Ф13 has a different asymptotic behaviour where pair (2, 3) is still interacting. However, since noO(1) contribution comes from extending the integrals to ω=0, for the present purpose I can ignore that restriction.
Author information
Authors and Affiliations
Additional information
Переведено редакцуей.
Rights and permissions
About this article
Cite this article
Servadio, S. Calculation ofO(1) interferences of double-scattering waves in 3-3 collision. Nuov Cim B 69, 1–22 (1982). https://doi.org/10.1007/BF02721238
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02721238