On the Riccati equations for the scattering matrices in two dimensions

We introduce the scattering matrices for the two-dimensional scattering problem for the Helmholtz equation. Naturally connected with the far-field scattering amplitude, the scattering matrices provide a forward model which governs the behaviour of the scattering process at any given frequency, and which is in turn described by a system of ordinary differential equations. The latter can be solved numerically in a stable manner and with arbitrary precision. The scattering matrices possess a rich analytical structure, which makes them an effective tool for the inverse scattering both analytically and numerically.