Derivative Matrix and Scattering Matrix
R.Kronig  was the first to point out, on the basis of his and Kramers’  work, that the principle of causality must entail some properties of the collision and scattering matrices. Starting from Kronig’s suggestion, Schutzer and Tiomno  gave, for non relativistic particles, a derivation of the well known theorem  that the poles of the scattering function S(k) lie either in the lower half plane or on the imaginary axis of k. Schutzer and Tiomno’s work has been extended, since, by Toll and by Van Kampen  to the case of relativistic particles with zero rest mass which formed also the subject of Kronig’s and Kramers’ early considerations . Results similar to these were obtained in the course of the last years also in communication engineering, following the pioneering work of Campbell, Zobel and Foster , and of Cauer , by Brune , by Fränz  and, particularly, by Richards .
KeywordsHalf Plane Imaginary Axis Relativistic Particle Lower Half Plane Left Half Plane
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- R.Kronig, Physica 12,543 (1946). This remark gave the original stimulus to the article of reference 3 and hence also to the present paperGoogle Scholar
- H. A. Kramers, Atti Cong. di Fisica Como 1927, p.545, R. de. L. Kronig, Ned. T. Natuurk 9,402 (1942)Google Scholar
- For the older literature of Chr. Moeller, K. Danske Vidensk. Selsk. 23, no. 1 (1945); 22, no. 19 (1946)Google Scholar
- Personal communicationGoogle Scholar
- G.A. Campbell, Bell System Tech. Journ. 1, no.2 (1922), O. J. Zobel, ibid. 2, 1 (1923), R. M. Foster, ibid. 3, 259 (1924)Google Scholar
- W. Cauer, Arch. Elektrotechnik 17, Chapter II (1926), Sitzungsber. Preuss. Akad. 1927, 1931, Math. Ann. 105, 101 (1931), 106, 369 (1932), Elektrische Nachrichten Technik 9, 157 (1932)Google Scholar
- O. Brune, Journ. of Math. and Phys. 10,191 (1931). This article contains numerous errorsGoogle Scholar
- P.I. Richards, Duke Math. Journ. 14, 777 (1947). I am much indebted to Dr. N. Greenspan for the last three references.Google Scholar
- and in a forthcoming article Amer. Math. Monthly. The concept of the derivative matrix (and derivative function) was used by E. P. Wigner and L. E.senbud, Phys. Rev. 72, 29 (1947)Google Scholar
- S,as function of E,has a branch point at E = 0 but is a single valued function of k in the case of pure scattering and will therefore always be considered to be a function of k. On the other hand, R is a single valued function of both E = k2and k in the non-relativistic case but its properties are simpler if it is considered to be a function of E and will always be considered to be a function of E. In the relativistic case, or if, in addition to scattering, reactions are also possible, (i.e. if S and R become matrices), S has in general several branch points and R is single valued only as a function of EGoogle Scholar
- Cf. e.g. the first article of reference 11Google Scholar
- The product expansion (10) for somewhat specialized R was given in references 9 and 10. A general proof is given in the second reference of 11Google Scholar