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

Abstract

We consider the acoustic propagator $H=-\nabla \cdot \rho \nabla$ acting in $L^2(\Omega )$ with $\Omega :={\Omega }^{\prime }\times R$ and ${\Omega }^{\prime }$ a bounded open set in $R^{n-1}$, $n\ge 2$. The real-valued function $\rho$ belongs to $L^{\infty }(\Omega )$, and is bounded from below by $c>0$. We assume there exist two strictly positive constants $c_1$ and $c_2$ and two perturbations, ${\delta }^S$ of short-range type and ${\delta }^L$ of long-range type, such that $\rho =c_j+{\delta }^S+{\delta }^L$ on ${\Omega }_j:=\{(x_n^{\prime },xn)\in \Omega |(-1{)}_n^j>0\}$, $j=1,2$. We build two modified free evolutions $U_j(t)$, $j=1,2$, such that the wave operators ${\Omega }_j^{\pm }:=s-{\lim }_{t\to \pm \infty }{e}^{itH}{U}_j(t)$, $j=1,2$, exist and are asymptotically complete.