General Solution of the One-Channel-Scattering Singular Integral Equations

Abstract

Integral equations for one-channel scattering are written and solved, starting from dispersion relations for the generalized Jost function in the momentum $k$ plane. This method is an alternative to the conventional $\frac{N}{D}$ method, but it allows a simple, physically meaningful generalization to the many-channel case, where dispersion relations and integral equations can be written for a unique generalized Jost function in the complex plane of a suitable variable which uniformizes all the right-hand cuts of the scattering amplitude. Even in the pure elastic-scattering case, a unified treatment is possible, whether the phase shift at infinity is or is not an integral multiple of $\ensuremath{\pi}$. In all cases, our singular integral equations are reduced to a Fredholm-type integral equation with a Hilbert-Schmidt kernel.