We give an overview of wave scattering in complex geometries, where the corresponding rays are typically chaotic. In the high-frequency regime, a number of universal (geometry-independent) properties that are described by random matrix theory emerge. Asymptotic methods based on the underlaying rays explain this universality and are able to go beyond it to account for geometry-specific effects. We discuss in this context statistics of the scattering matrix, scattering states, the fractal Weyl law for resonances, and fractal resonance wavefunctions.

Introduction

Our purpose here is to give an introductory overview of wave scattering in complex geometries, where the corresponding rays are typically chaotic. For simplicity, we focus our discussion on domains with lossless walls, inside which the wave speed is constant. The rays then are straight, with specular reflections at the boundaries. This situation is often encountered in experiments (Stöckmann 1999, Kuhl et al. 2005, Tanner & Søndergaard 2007) and in acoustic applications. However, many of the features we shall identify occur much more generally. Indeed, most recent developments in the subject have taken place in the context of quantum wave scattering, where the underlying rays are the classical trajectories of Newtonian mechanics. Much of this review will be devoted to translating quantum results into the language of classical wave scattering.

One of the main observations we wish to make is that many of the essential mathematical features of wave scattering in complex geometries can be found in certain very simple discrete models, which we here call wave maps.