Resonance Expansions and Rayleigh Waves

In this paper we study expansions of solutions of the wave equation in a compact set with initial data supported in the same set. We consider the general framework of the “black box scattering” introduced by Sjöstrand and Zworski [SjZ] (see sec. 2). In particular, this includes the clasical case of scattering by obstacle with Dirichlet or Neumann boundary conditions and metric perturbations of the Laplacian with a metric equal to the Euclidean one outside a large ball. Denote by U(t) the solution group corresponding to the wave equation in the energy space and let χ be the multiplication with a compactly supported function χ(x) equal to 1 on some compact set containing the “black box” (the scatterer). Then we are interested in asymptotic expansions of χU(t)χ, as t → ∞. If we study the wave equation in a bounded domain, then one can use the Fourier method to get expansion of U(t) in terms of the eigenvalues and eigenfunctions of the corresponding Laplacian (with self-adjoint boundary conditions). In the case under consideration, one gets expansions in terms of the resonances and resonance states. This has been confirmed in the non-trapping case by Lax-Phillips [LP] and Vainberg [Va1] in odd dimensions (see also [Va2]) and in the black box setting by Tang and Zworski [TZ2]. In this case,