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

Abstract

We consider the acoustic propagator H = −∇ · _ρ_∇ acting in _L_2 (Ω) with Ω := Ω' × ℝ and Ω' a bounded open set in 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 c_1 and c_2 and two perturbations, δS of short-range type and δL of long-range type, such that ρ = cj + δS + δL on Ω_j := {(x', xn) ∈ Ω|(−1)j xn > 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_→±∞ eitH U__j(t), j = 1, 2, exist and are asymptotically complete.