Dispersion relations and resonance reactions

Abstract

For one-channel scattering, a resonance formalism is developed which avoids continuing the scattering function into a region where it has resonance poles. In a non-relativistic terminology, it is assumed that for any particular partial wave the scattering function S is analytic in the energy plane cut from −∞ to −1 and from 0 to ∞. It is further assumed that S satisfies a dispersion relation, and that it takes complex conjugate values at complex conjugate points. Under these assumptions, S can be written as a Blaschke product times an exponential function. The Blaschke product contains the zeros of S. At energies below the threshold for inelastic processes, it describes the resonances. If in this energy region S has isolated zeros close to the real axis, these give rise to Breit-Wigner peaks in the cross section. The exponential function corresponds to the potential scattering. If refers to a scattering process in which the real part of the phase shift is connected with the absorption at higher energies through a dispersion relation. The formalism can be extended to energies at which more than one channel is involved. This generalization is, however, not discussed.