Abstract
We extend previous work concerning the construction of unitary scattering amplitudes that correspond to the scattering data at a given energy. The dispersive and absorptive parts are by construction analytic in cosϑ in the small and large Lehmann ellipses, respectively. The dispersive and absorptive parts obtained here, in contrast to those obtained before, are shown to have continuous derivatives on the boundary of their domains of analyticity. The continuum ambiguity in the determination of the scattering amplitude, which is associated with a lack of experimental information on the inelastic contribution to unitarity, is present here as well.
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Atkinson, D., Johnson, P., Warnock, R.: Commun. math. Phys.28, 133 (1972). References to previous work are given there. See also Ref. 6
Atkinson, D., Mahoux, G., Ynduräin, F.: Nucl. Phys. B54, 263 (1973)
Schauder's theorem is discussed in, for example, Kantorowitsch, L. V., Akilow, G. V.: Functional analyse in Normierten Räumen (German translation by Berlin: Akademie Verlag 1964)
Bart, G., Johnson, P., Warnock, R.: “Continuum Ambiguity in the Construction of preprint, Unitary Analytic Amplitudes from Fixed Energy Scattering Data”, IIT December, 1972
Liusternik, L., Sobolev, V.: Elements of Functional Analysis, pp. 138–141. English translation published by New York: F. Ungar, 1961
These integral representations are given explicity in Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products, p. 1001. New York: Academic Press, 1965
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Work supported in part by the National Science Foundation and a NATO Research Grant.
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Atkinson, D., Johnson, P.W. & Warnock, R.L. Construction of analytic, unitary scattering amplitudes from a given differential cross-section: A refined analysis. Commun.Math. Phys. 33, 221–242 (1973). https://doi.org/10.1007/BF01667919
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DOI: https://doi.org/10.1007/BF01667919