Scattering Theory for a Stratified Acoustic Strip with Short- or Long-Range Perturbations

Abstract

We consider the acoustic propagator H=−∇·ρ∇ acting in L²(Ω) with Ω:=Ω'×\mathbb{R} and Ω' a bounded open set in \mathbb{R}ⁿ⁻¹, n≥ 2. The real-valued function ρ belongs to L^∞(Ω), and is bounded from below by c>0. We assume there exist two strictly positive constants c₁ and c₂ and two perturbations, δ^S of short-range type and δ^L of long-range type, such that ρ=c_j+δ^S+δ^L on Ω_j:={(x', x_n)∈Ω | (−1)^jx_n>0}, j=1, 2. We build two modified free evolutions U_j(t), j=1, 2, such that the wave operators Ω_j^±:=s−lim_t→±∞e^itHU_j(t), j=1, 2, exist and are asymptotically complete.