Inverse scattering in an asymptotically flat multilayer domain

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 $\Omega \subset {\bf R}^{n+1}_x = {\bf R}^n_y \times {\bf R}^1_{x^{n+1}}$ of the form $\Omega = \mathcal K \cup \Omega_1 \cup \cdots \cup \Omega_N$, where $\mathcal K$ is a bounded open set and $\Omega_k$ is asymptotically equal to a slab domain ${\bf R}^n \times (c_k,c_k + d_k)$ as $|y| \to \infty$.

Assuming that $\partial_x^{\alpha}\big(g_{ij}(x) - \delta_{ij}\big) = O(|x|^{-|\alpha| - \delta_0}), \ \delta_0 > 1, \forall \alpha$, we show that $\Omega$ and $g^{ij}$ are determined by one diagonal component $S_{11}(\lambda)$, for all energies, of the S-matrix associated with the slab $\Omega_1$, provided $\Omega_1$ is flat: $\Omega_1 \cap \{|y| > R\} = \{|y| > R\} \times (c_1, c_1+d_1)$ for some constants $c_1, d_1, R > 0$, and the metric is Euclidean on $\Omega_1\cap \{|y| > R\}$.