coulomb - I L NUOVO C I M E N T 0 VOL. 23 B, 1~. 2 I1...

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IL NUOVO CIMENT0 VOL. 23 B, 1~. 2 I1 Ottobre 1974 A New Rigorous Approach to Coulomb Scattering. J. R. TA~o~ Department o/ Physics, Zmperial College - ~ondon Department o/ Physics and Astrophysics, University o] Colorado - Boulder, Colo. (*) (rieevuto il 23 Gennaio 1974) Summary. -- This paper establishes two properties of nonrelativistie Coulomb scattering. The first is that, when considered as a distribution, the Coulomb partial-wave series is convergent (even though divergent as a function) and eonverges to the Coulomb amplitude. The second property, the proof of which uses the first, is that the amplitude for any screened Coulomb potential converges as a distribution to the Coulomb amplitude (times an overall phase factor) when the screening radius tends to infinity. It is argued that this second property can be made the basis of an economical but rigorous theory of Coulomb scattering. 1. - Introduction. One of the most famous and frustrating anomalies in scattering theory is Coulomb scattering. Because the Coulomb potential V(r) =~ falls off so slowly when r -+ c~, almost none of the standard results of ordinary nonrelativistic scattering can be applied in the case of Coulomb scattering. To mention just four such results, we remind the reader that 1) the asymptotic (*) Permanent address. Work supported by a FacuLty Fellowship from the Univer- sity of Colorado. 21 - I1 Nuovo Cimento B. 313
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314 J.R. TAYLOR condition--the cornerstone of conventional scattering theory-- is not satisfied when Coulomb forces are present; 2) with a Coulomb potential, the standard definitions of the scattering amplitude and phase shifts are meaningless; 3) the Coulomb partial-wave series is divergent; 4) the amplitude for a screened Coulomb potential has no limit when the screening radius goes to infinity. One way of avoiding some of the difficulties has been to broaden the normal framework of scattering theory so as to include the anomalous case of the Coulomb potential. This approach has been pursued by DOLLARD (1) and by -AJVIREISr, MAI~TI>; and MISRA (3), who have been able to replace the usual asymptotic condition with a weaker, more general condition that is satisfied by the Coulomb potential. Since their generalized asymptotic condition is still strong enough to describe the essential observed features of scattering experiments, their work provides a rigorous and physically satisfactory basis for a theory that can handle problems involving Coulomb forces. lqevertheless, one cannot help feeling that the labour of constructing the theory of ])OLLARD et al. is, in a certain sense, unnecessary. This feelingis based on the fact that in practice Coulomb potentials are always screened. (For example, the Coulomb potential of a nucleus is screened by the atomic electrons~ and even in the best available vacuum the potential of an isolated charge would be screened by polarization of the residual particles.) This fact suggests that one ought to be able to build a theory of Coulomb scattering which discusses only screened Coulomb potentials. Since these screened potentials are (~ well
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coulomb - I L NUOVO C I M E N T 0 VOL. 23 B, 1~. 2 I1...

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