Singular Support of the Scattering Kernel for the Rayleigh Wave in Perturbed Half-Spaces

Abstract

This paper deals with the Rayleigh wave scattering on perturbed half-spaces in the framework of the Lax-Phillips type. Singular parts of the scattering kernel for this scattering are closely connected with singularities of the Rayleigh wave passing through the perturbation on the boundary. This can be described by estimating the singular support of the scattering kernel on the Rayleigh wave channel. The proof is based on a representation formula of the scattering kernel that was obtained in the previous work. However, the formula does not suit the situation of the Rayleigh wave, even though it is a natural extension of Majda’s formula for the usual wave equation. Hence, the formula needs to be reformed, and the problem needs to be reduced to a pseudo-differential equation on the boundary governing the Rayleigh wave. Key methods for the reduced problem are construction of an approximate solution for the Rayleigh wave and analysis of an oscillatory integral distilled by using the solution. The phase function of the oscillatory integral is always degenerate along the characteristic curve of the Rayleigh wave. This degeneracy is handled by introducing a certain criterion for the regularity of the distribution defined by the oscillatory integral.