Spectral analysis of a perturbed multistratified isotropic elastic strip: new method

We study the self‐adjoint operator (( A ), A ) associated with an elastic isotropic and multistratified strip Ω = {( x 1 , x 2 ) ∈ ℝ 2 ; 0 < x 2 < L }, which means that there exists a constant a > 0 such that the density ρ and Lamé coefficients λ and μ are, for (−1) k x 1 ⩾ a , k = 1, 2, respectively, equal to functions ρ k , λ k and μ k , depending only on x 2 . Thanks to [4] the properties of the free operators A k , k = 1, 2, associated with ρ k , λ k and μ k , are well‐known. We study A by considering it as a ‘compact perturbation’ of the pair ( A 1 , A 2 ). The difficulty is: if ψ ∈ C   ∞ 0 (ℝ 2 ) and u ∈ D(A) then ψ u does not necessarily belong to D(A) . It has already been encountered in other studies concerning elasticity (cf. [10,18]). Adapting the techniques used there to overcome this difficulty imposes restrictive conditions on λ k and μ k . The purpose of this paper is to propose a new method, which removes definitively this difficulty and enables us without restrictive conditions on λ k and μ k to prove a limiting absorption principle for A . Copyright © 1999 John Wiley & Sons, Ltd.