Spectral analysis of an isotropic stratified elastic strip and applications
Tark Bouhennache · Osaka City University (Osaka City University) · 2000
The characteristics of an elastic isotropic stratified strip Ω = {x = (x\,X2)', X2 Ξ (0, L)} C M are the density p and Lame coefficients λ and μ, that we assume to be measurable functions, depending on X2 only, and bounded from below and above by two positive constants. We shall derive a limiting absorption principle (LAP) and a division theorem for the selfadjoint operator (D(A), A) (see (2)) associated with Ω with Dirichlet and free surface conditions on [xj = L] and {x2 = 0}, respectively. These boundary conditions come from a model of a seismic problem. For other studies dealing with elasticity in different situations see for instance [7] and [12]. Roughly speaking, a LAP means that the resolvent operator z —> RA(Z) •= (A — zld)~ can be extended continuously to the essential spectrum (a part of the real axis) in suitable topologies. It is an important stage in scattering theory (cf. [1]). A division theorem enables to deal with a perturbed or a multistratified strip (cf. [3]). A multiplication operator in Θ°°L(1R) := {(/)„>!;Σ°° l l/Ίl£ 2 ( R ) 1, is defined by j V{M) = {(/)„>! € θ°°L (R);(μ I I/ Λ)π> 1 e Θ°°L (M)} [M(r)n>! =(μnf n)n>ι.