The scattering phase: seen at last

The scattering phase, defined as $\log \det S ( \lambda ) / 2\pi i$ where $S ( \lambda )$ is the (unitary) scattering matrix, is the analogue of the counting function for eigenvalues when dealing with exterior domains and is closely related to Krein's spectral shift function. We revisit classical results on asymptotics of the scattering phase and point out that it is never monotone in the case of strong trapping of waves. Perhaps more importantly, we provide the first numerical calculations of scattering phases for non-radial scatterers. They show that the asymptotic Weyl law is accurate even at low frequencies and reveal effects of trapping such as lack of monotonicity. This is achieved by using the recent high level multiphysics finite element software FreeFEM.