Wave chaos for the Helmholtz equation

Abstract : 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.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. These features are for instance embodied in notions such as the spatial ergodicity of modes and the spectral rigidity, which are indicators of particular spatial and spectral correlations. The spectral study can be done through the so-called trace formula based on the periodic orbits of chaotic billiards. From the latter we will derive universal spatial and spectral features in agreement with predictions of Random Matrix Theories, but also see that actual deviations from a universal behavior can be found, which carry information about the specific geometry of the cavity. Finally, as a first step towards disordered scattering systems, a description of spectra of cavities dressed with a point scatterer, will be given in terms of diffractive orbits.
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Olivier Legrand, Fabrice Mortessagne. Wave chaos for the Helmholtz equation. Matthew Wright & Richard Weaver. New Directions in Linear Acoustics and Vibration, Cambridge University Press, pp.24-41, 2010, 978-0-521-88508-9. ⟨hal-00847896⟩

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