Dispersion relations and resonance reactions in multi-channel scattering

Abstract

A parameter expansion is presented for a multi-channel scattering matrix S( z) the elements of which are analytic functions of the energy z. Under the assumption that each matrix element satisfies a dispersion relation, it is shown that S( z) can be written as a Blaschke product times a matrix V( z) the determinant of which does not vanish. The Blaschke product contains the zeros of det S( z), a typical factor being characterized by a zero E n+ 1 2 iΛ n and a constant unitary matrix u n . An isolated zero close to the real axis gives rise to a resonance with energy E n and width Λ n . Given Λ n , the partial widths are determined by u n . The analytic properties of a matrix with non-vanishing determinant are discussed in some detail. In particular, it is pointed out that V( z) can be represented by a multiplicative integral. With the help of this representation it is shown that, in the interval in which S( z) is unitary, V( z) is a slowly varying function of z. From this it follows that V( z) refers to a background scattering.