This chapter is an introduction to the semiclassical approach for the Helmholtz equation in complex systems originating in the field of quantum chaos. A particular emphasis will be made on the applications of trace formulae in paradigmatic wave cavities known as wave billiards. Its connection with random matrix theory (RMT) and disordered scattering systems will be illustrated through spectral statistics.

Introduction

The study of wave propagation in complicated structures can be achieved in the high-frequency (or small-wavelength) limit by considering the dynamics of rays. The complexity of wave media can be due either to the presence of inhomogeneities (scattering centers) of the wave velocity or to the geometry of boundaries enclosing a homogeneous medium. It is the latter case that was originally addressed by the field of quantum chaos to describe solutions of the Schrödinger equation when the classical limit displays chaos. The Helmholtz equation is the strict formal analog of the Schrödinger equation for electromagnetic or acoustic waves, the geometrical limit of rays being equivalent to the classical limit of particle motion. To qualify this context, the new expression wave chaos has naturally emerged. Accordingly, billiards have become geometrical paradigms of wave cavities.

In this chapter we will particularly discuss how the global knowledge about ray dynamics in a chaotic billiard may be used to explain universal statistical features of the corresponding wave cavity, concerning spatial wave patterns of modes, as well as frequency spectra.