Sommerfeld-Watson representation for double-spectral functions - II. Crossing symmetric pion-pion scattering amplitude without Regge poles

We discuss, for the case of pion-pion scattering, a closed system of equations which may be used for a self-consistent calculation of partial-wave amplitudes. It is shown that, for a given sufficiently small input function, the equations have a locally unique solution in a particular Banach space of doubly Hölder continuous partial wave amplitudes. At a fixed point, the scattering amplitude is shown to satisfy both a crossing symmetric unsubtracted Mandelstam representation and the elastic unitarity condition. In this initial study the partial-wave amplitudes are holomorphic in the right half complex angular-momentum plane.