Scattering of elastic waves in a perturbed isotropic half space with a free boundary. The limiting absorption principle

Abstract

In this article we consider the self‐adjoint operator governing the propagation of elastic waves in a perturbed isotropic half space with a free boundary condition. We prove the limiting absorption principle in appropriate Hilbert spaces for this operator. We also prove decreasing properties for the eigenfunctions associated with strictly positive eigenvalues of this operator. The proofs are based on the limiting absorption principle for the self‐adjoint operator governing the propagation of elastic waves in a homogeneous isotropic half space with a free boundary and on the so called division theorem for it. Both perturbations of R+2 ={(x1, x2) ϵ R2; x2 > 0} and R+2 = {(x1, x2, x3) ϵ R3; x3 > 0} are considered.

Generalized eigenfunctions of layered elastic media and application to diffuse fields

“…Once the eigenvectors of the transformed operator have been obtained, those pertaining to the original operator are recovered by application of the inverse operation U † . As shown previously 11,12 , the elastodynamic operator is positive definite and self-adjoint in the Hilbert space with scalar product…”

Scattering theory for the elastic wave equation in perturbed half-spaces

“…[1] or [2]). Note that 0 < c R < c S < c P , and the Rayleigh waves of the time harmonic type e −iσt φ R 0 (x; σ, ω) are given by…”

“…We have the generalized eigenfunction expansion with respect to {φ α 0 } (cf. §6 in M. Kawashita, W. Kawashita and Soga [7] or Dermenjan and Guillot [2]). (i) follows from Proposition 5.1 in [7] immediately.…”

Resolvent Estimates and Scattering States for Dissipative Systems

“…However present framework can be applied to other wave equations with dissipative terms, for instance acoustic wave in two unbounded media (cf. Eidus [7] and Kadowaki [10]) and elastic wave in a half space (DermenjianGuillot [4]) or stratified (two-layered) media (cf. Shimizu [22]).…”

“…Because for other cases, n = 1, 2 or 1 < θ < 2, it is difficult to be held (1.8) with A 0 = − , B = (1 + |x| 2 ) −θ/2 (for detail, see [16] Theorem 4.4.1 (4) and Corollaries 4.4.3, 4.4.5 and those proof). For the cases n 3 and θ 2 we note the following proposition which is concerning the low energy part (for the high energy part, see Proposition 1.0).…”