The Collected Works of Eugene Paul Wigner pp 453-458 | Cite as

# Derivative Matrix and Scattering Matrix

## Abstract

R.Kronig [1] was the first to point out, on the basis of his and Kramers’ [2] work, that the principle of causality must entail some properties of the collision and scattering matrices. Starting from Kronig’s suggestion, Schutzer and Tiomno [3] gave, for non relativistic particles, a derivation of the well known theorem [4] 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 [5] to the case of relativistic particles with zero rest mass which formed also the subject of Kronig’s and Kramers’ early considerations [2]. 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 [6], and of Cauer [7], by Brune [8], by Fränz [9] and, particularly, by Richards [10].

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### References

- [1]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 - [2]H. A. Kramers, Atti Cong. di Fisica Como 1927, p.545, R. de. L. Kronig, Ned. T. Natuurk
*9*,402 (1942)Google Scholar - [3]W. Schutzer and J. Tiomno. Phys. Rev.
*83*, 249 (1951)MathSciNetADSCrossRefGoogle Scholar - [4]For the older literature of Chr. Moeller, K. Danske Vidensk. Selsk.
*23*, no. 1 (1945);*22*, no. 19 (1946)Google Scholar - [5]Personal communicationGoogle Scholar
- [6]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 - [7]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 - [8]O. Brune, Journ. of Math. and Phys.
*10*,191 (1931). This article contains numerous errorsGoogle Scholar - [9]K. Franz, Elektrische Nachrichten Technik
*21*, 8 (1944)MathSciNetGoogle Scholar - [10]P.I. Richards, Duke Math. Journ.
*14*, 777 (1947). I am much indebted to Dr. N. Greenspan for the last three references.Google Scholar - [11]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
- [12]
*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 = k*^{2}and*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*E*Google Scholar - [13]Cf. e.g. the first article of reference 11Google Scholar
- [14]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