Asymptotic behavior of stationary solutions to elastic wave equations in a perturbed half-space (Spectral and Scattering Theory and Related Topics)

We deal with stationary scattering problems for elastic wave equation with a free boundary condition in a local perturbation of the three dimensional half-space. Applying the method of Agmon-Hörmander, we show that all solutions to the stationary elastic wave equations in the Agmon-Hörmander space are characterized in terms of the generalized Fourier transform associated with the elastic operator. Moreover, we investigate asymptotic properties: a uniform asymptotic expansion of the solutions in the averaged sense and representation of the S-matrix. Our expansion describes the behavior of body waves and Rayleigh surface waves in the same topology