Inverse scattering in an asymptotically flat multilayer domain

Abstract

We consider a scattering problem for a wave equation ${\partial }_t^{2}u=\frac{1}{\sqrt{g}}{\partial }_i(\sqrt{g}{g}^{ij}{\partial }_j)u$ in a multilayer domain $\hat{I}\text{c◯}\subset {\mathbf{R}}_x^{n+1}={\mathbf{R}}_y^n\times {\mathbf{R}}_{x^{n+1}}^1$ of the form $\hat{I}\text{c◯}=\mathcal{K}\cup \hat{I}{\text{c◯}}_1\cup \cdots \cup \hat{I}{\text{c◯}}_N$, where $\mathcal{K}$ is a bounded open set and $\hat{I}{\text{c◯}}_k$ is asymptotically equal to a slab domain ${\mathbf{R}}^n\times (c_k,c_k+d_k)$ as $|y|\to \infty$. Assuming that ${\partial }_x^{\hat{I}}\pm (g_{ij}(x)-\hat{I}{´}_{ij})=O(|x{|}^{-|\hat{I}\pm |-\hat{I}{´}_0}),\hat{I}{´}_0>1,\forall \hat{I}\pm$, we show that $\hat{I}\text{c◯}$ and $g^{ij}$ are determined by one diagonal component $S_{11}(\hat{I}»)$, for all energies, of the S-matrix associated with the slab $\hat{I}{\text{c◯}}_1$, provided $\hat{I}{\text{c◯}}_1$ is flat: $\hat{I}{\text{c◯}}_1\cap \{|y|>R\}=\{|y|>R\}\times (c_1,c_1+d_1)$ for some constants $c_1,d_1,R>0$, and the metric is Euclidean on $\hat{I}{\text{c◯}}_1\cap \{|y|>R\}$.