LETTERE AL NUOVO CIMENTO VOL. 16, N. 16 14 Agosto 1976 Eikonal, Rytov, High-Energy Expansions. R. C*R Istituto di Fisiea del Politeenico - Milano Istituto Nazionale Fisica Nucleate, Sezione di Milano G. M. CICtITA Istituto di Fisica dell' Universith - Milano Istiluto Nazionale Fisiea Nucleate, Sezione di Milano ricevuto il 7 Giugno 1976) In the study of the high-energy limit of the scattering amplitude in potential theory it was often found the expedient to replace the free-particle Green function Gk(x ) with an eikonal Green function Gl,(x), see for instance (,-a) (1) G~:(x)-- 1 cxp[ikr]_f daUexp[ik'.x] = ( d3qexp[i(q+k).x] , 4~ r (2n)a[k '2 (k 2 + i~:)] J (2u)a(q'z + 2k-q--ie) co (2) (~,~(x) = i2~)~(2k.qY ie) = i exp [ik.x] dt ~3(x-- 2kt). 0 As it is well known, by inspecting the generic term of the Born series of the scat- tering amplitude in Inomcntunl space, one sees that this replacement amounts to replace the propagators of the virtual particle with lincarized propagators, i.e. the square of the momentum transfer contributed in each of the multiple interactions with the external potential is neglected. In fact, at high energy, scattering occurs mostly in the forward direction and, for smooth potentials, it is achieved by a succession of virtual scattering, each mostly in the forward direction ('). The linearization of the propagator of a fast virtual particle is an important approxi- mation also in the scattering of a high-energy projectile off a composite target (4.6) to obtain Glauber theory and in quantuin field theory models (7). (1) h. I. SCIIIFF: Phy.u. Rev., 103, 443 (1956). (2) H,. L. SUOAIr &Ild 1~. BLANKI.]NBECLEIr Phys. Rev., 183, 1387 (1969). (a) M. LEVY ~tnd J. S[TCltER: Phys. l~ev., 186, 1656 (1969). (4) D. R. HAICRINGTON: Phys. Rev., 184, 1745 (1969). (5) T.A. OSm)RN: Ann. o] Phys., 58, 417 (1970). (a) J. M. ElSI~NBERC-: Anat. o] Phys., 71, 542 (1972). (7) One may look at some reviews like if. AB~kRBANEL: Lectures at Kaiserslautern (1972); Lecture Notes in Physics, Vol. 17 (Berlin, 1.(}73); ~vV. DITTRICIt: Fortsehr. Phys., 22, 539 (1974). 481