1 Introduction↩︎

Consider a domain \(\Omega \subset {\boldsymbol{R}}^{n+1}_x = {\boldsymbol{R}}^n_y \times {\boldsymbol{R}}^1_{x_{n+1}}\), \(x = (y,x_{n+1})\), as in Figure 1 and \(\Delta_y + (\partial_{n+1})^2\) in \(\Omega\). Assume that there exists a constant \(C > 0\) such that \(\Omega \subset \{|x_{n+1}| < C\}\) and there exist \(f_{\pm}(y) \in C^{\infty}({\boldsymbol{R}}^n)\) and constants \(C_- < C_+\) such that the boundary \(\partial \Omega = \Gamma_+ \cup \Gamma_-\) is written as \(x_{n+1} = f_{\pm}(y)\) with \(\partial_y^{\alpha}(f_{\pm}(y) - C_{\pm}) = O( |y| ^{-|\alpha| - \delta_0})\) as \(|y| \to \infty\) for a constant \(\delta_0 > 1\), \(\forall \alpha\). Then, without loss of generality, we can assume that there exists \(R_0 > 0\) such that \(\Omega \cap \{|y| > R_0\}\) is diffeomorphic to \(\{|y| > R_0\} \times (0,d)\), \(d= C_+ - C_-\), equipped with the Laplacian \[- H := \Delta_G = \frac{1}{\sqrt{g}}\partial_i\big(\sqrt{g} g^{ij}\partial_j\big), \nonumber\] where \(G = ( g_{ij}) = (g^{ij})^{-1}\) is a Riemannian metric on \(\Omega\), \(\partial_i = \partial/\partial y_i\), \(i = 1, \dots, n\), \(\partial_{n+1} = \partial_{x_{n+1}}\), and \[\partial_x^{\alpha}\big(g_{ij}(x) - \delta_{ij}\big) = O(|x|^{-|\alpha| - \delta_0}), \quad \delta_0 > 1, \quad \forall \alpha. \label{Condgij40x41-edltaij}\tag{1}\] We call such \(\Omega\) an asymptotically flat slab, and \(\textcolor{blue}{d}\) the thickness of the slab. In the case that \(\Omega_0 = {\boldsymbol{R}}^n\times (0,1)\) equipped with the Euclidean metric and the Neumann boundary condition on \(x_{n+1} = 0,1\), we call it a model slab.

More generally, we consider a connected smooth open subset \(\Omega \subset {\boldsymbol{R}}^{n+1}\) such that \[\label{eq-defM:1} \overline{\Omega} = \overline{\mathcal{K}} \cup \overline{\Omega_1} \cup \cdots \cup \overline{\Omega_N},\tag{2}\] where \(\mathcal{K}\) is a bounded open set, \(\Omega_m\), \(m = 1, \dots, N\) are asymptotically flat slabs (see Figure 1 (B)) satisfying (1 ), where the metric on \(\Omega_m\), denoted by \((g_{m,ij})\) may be different for \(m = 1, 2, \dots, N\). We assume that \(\Omega_i \cap \Omega_j = \emptyset\) if \(i \neq j\), which means that \(\Omega_i\)’s are almost parallel near infinity. We are dealing with an idealized modelisation of concrete situations as can be found in geophysics with aquifer-systems. See e.g. [1] p. 9, [2] p. 34.

The problem we address is the inverse scattering on such \(\Omega\) for \(H\), the Laplacian \(-\Delta_G\) with Neumann boundary condition¹. Namely, we are interested in the following problem:

• Recovery of the topology and the metric on \(\Omega\) from the knowledge of the S-matrix.

Let us briefly recall the idea of the \(S-\)matrix, which is a representation of the scattering operator. Given a Hamiltonian \(H\) and the wave equation \(\partial_t^2 u = - Hu\), we transform it into a first order system \(i \partial_tv = \mathcal{H}v\) and consider the associated evolution operator \(\mathcal{U}(t) : f \to e^{-it\mathcal{H}}f\). The scattering operator links the behaviour of \(\mathcal{U}(t)f\) as \(t\to -\infty\) to that as \(t\to +\infty.\) Its representation by \(S\)-matrix is believed to contain knowledge equivalent to the system in question. For more details, see for example §3.1 of [3].

A similar problem for the case of asymptotically cylindrical manifolds was studied in [4]. In this case, each end \(\Omega_i\) \((i = 1, \dots, N)\) is diffeomorphic to \([R_i,\infty)\times M_i\), where \(M_i\) is a compact manifold of dim. \(n\) (with or without boundary). In our following argument, we assume that \(n \geq 2\) to avoid the case of cylindrical end.

We study the limiting absorption principle (LAP), and the asymptotic behavior of solutions of Helmholtz equation at infinity, from which the S-matrix is derived. The inverse scattering procedure will then be as follows. We are given two such asymptotically flat slabs \(\Omega^{(1)}\) and \(\Omega^{(2)}\) as in (2 ). The associated S-matrix is an operator-valued matrix \(\mathcal{S}(\lambda) = \big(\mathcal{S}_{ij}(\lambda)\big)\), \(1 \leq i, j \leq N\), where each entry \(\mathcal{S}_{ij}(\lambda)\) is a bounded operator from \(L^2(S^{n-1}\times (c_i, c_i + d_i))\) to \(L^2(S^{n-1}\times (c_j, c_j + d_j))\), where \(d_i\) is the thickness of \(\Omega_i\). Our aim is to show the following fact: The whole domain \(\Omega\) is determined by one diagonal entry of the S-matrix, i.e.

For the meaning of the notation \(\mathcal{E}(H^{(i)})\cup\mathcal{E}(H_{\emptyset}^{(i)})\), see (6 ) and §3.²

The paper is organized as follows. In §2 we describe the forward problem for the model space. More precisely, we construct the free and pertubed spectral representations and we define the S-matrix. Then we derive the analytic continuation of the scattering amplitude. In §3, we begin to reconstruct the domain \(\Omega\) by studying the interior problem on the basis of information obtained from the S matrix. We focus on the method involving the source to solution map. Finally in §4 we complete the proof.

2 Forward problem↩︎

2.1 Spectral theory for the model space↩︎

Let us make the definition of \(\Omega\) in (2 ) more precise. In \({\boldsymbol{R}}^{n+1}\), we are given a domain \(\Omega = \mathcal{K} \cup \Omega_1 \cup \cdots \cup \Omega_N\) satisfying the following conditions: Each \(\Omega_j\) is an open set of the form \(\{(y,x_{n+1})\, ; \, |y| > R_0\), \(c_j < x_{n+1} < c_j + d_j\}\) which is diffeomorphic to \(\{|y| > R_0\} \times (0,d_j)\) where \(R_0=2^{\ell_0}\) for some \(\ell_0 > 0\), which we simply denote as \(\Omega_j = \{|y| > R_0\} \times (0,d_j)\). We assume that \[\overline{\Omega_i} \cap \overline{\Omega_j} = \emptyset, \quad {\it if} \quad i \neq j. \nonumber\] Moreover, \(\mathcal{K} = \Omega \setminus \big(\overline{\Omega_1} \cup \cdots \cup \overline{\Omega_N}\big)\) is a bounded open set, and \(\overline{\Omega}\) is a \(C^{\infty}\) manifold with boundary. Assume that \(\Omega\) is equipped with a Riemannian metric satisfying (1 ) on each \(\Omega_j\).

In the following, \(X\) denotes a point in \(\Omega\), while \(x = (y,x_{n+1})\) denotes a corresponding point in the model slab or in \(\Omega_j\) ³. Letting \({\rm{dist}}\, (X,X')\) be the Riemannian distance of \(X, X' \in\Omega\), and fixing \(X_0 \in \Omega\) arbitrarily, we define for \(s \in {\boldsymbol{R}}\) \[L^{2,s} \ni f \Longleftrightarrow \|f\|_s^2 = \int_{\Omega} \big(1 + d(X,X_0)\big)^{2s}|f(X)|^2 d\Omega_X < \infty, \nonumber\] where \(d\Omega_X\) is the volume element of \(\Omega\). We also use the Agmon-Hörmander space \(\mathcal{B}, \mathcal{B}^{\ast}\), which are the Besov type spaces associated with \(\Omega\) defined as follows: For \(f \in L^2_{loc}(\Omega_j)\), we put \[\|f\|_{\mathcal{B}(\Omega_j)} = \sum_{\ell=\ell_0}^{\infty} 2^{\ell/2}\Big(\int_{2^{\ell} < |y| < 2^{\ell+1}}\|f(y,\cdot)\|^2_{L^2((0,d_j))}dy\Big)^{1/2}, \nonumber\] \[\|f\|_{\mathcal{B}(\Omega)} = \|f\|_{L^2(\mathcal{K})} + \sum_{j=1}^N\|f\|_{\mathcal{B}(\Omega_j)}. \nonumber\] with \(R_0=2^{\ell_0}\). The norm of \({\mathcal{B}}^{\ast}(\Omega)\) is defined as follows: \[\|u\|^2_{{\mathcal{B}}^{\ast}} = \|u\|^2_{L^2(\mathcal{K})} + \sum_{j=1}^N \sup_{R>R_0} \frac{1}{R}\int_{\Omega_j \cap \{|y| < R\}} |u(X)|^2 d\Omega_X < \infty. \nonumber\] We define also the following relation of equivalence on \(\mathcal{B}^{\ast}\): \[u \simeq v \Longleftrightarrow \lim_{R\to\infty} \sum_{j=1}^N \frac{1}{R} \int_{\Omega_j \cap \{|y| < R\}} |u(X)-v(X)|^2 d\Omega_X = 0. \label{ExpansioninBast40Omega41}\tag{3}\] The spaces \(\mathcal{B}(\Omega_j)\), \(\mathcal{B}^{\ast}(\Omega_j)\) and the relation \(u \simeq v\) on \(\Omega_j\) are defined similarly. We often omit \(\Omega\) or \(\Omega_j\) in the notation of \(\mathcal{B}, \mathcal{B}^{\ast}\) spaces. The following inclusion relations hold: For \(s > 1/2\), \[L^{2,s} \subset \mathcal{B} \subset L^{2,1/2} \subset L^2 \subset L^{2,-1/2} \subset \mathcal{B}^{\ast} \subset L^{2,-s}. \nonumber\]

First let us consider the model slab \({\boldsymbol{R}}^n\times (0,1)\). We know that for any \(\lambda > 0\), the limit \((- \Delta_y - \lambda \mp i0)^{-1}\) exists as a bounded operator from \(L^{2,s}({\boldsymbol{R}}^n)\) to \(L^{2,-s}({\boldsymbol{R}}^n)\) with \(s>1/2\), and from \(\mathcal{B}({\boldsymbol{R}}^n)\) to \(\mathcal{B}^{\ast}({\boldsymbol{R}}^n)\). Let \(- (\partial_{n+1})^2\) be \(- (\partial_{x_{n+1}})^2\) on \((0,1)\) with Neumann boundary condition. To prove the existence of these limits, i.e. the limiting absorption principle, abbreviated as LAP, for \(- \Delta_y - (\partial_{n+1})^2\) on \({\boldsymbol{R}}^n\times (0,1)\), we choose an orthnormal basis of \(- (\partial_{n+1})^2\) which enables us to prove these limits for each eigenspace associated to these basis. Expanding by the orthonormal basis of \(-(\partial_{n+1})^2\) one can prove the limiting absorption principle, abbreviated as LAP, for \(-\Delta_y - (\partial_{n+1})^2\) on \({\boldsymbol{R}}^n\times (0,1)\).

Consider the slab \(\Omega_j\), and let \(\chi_j \in C^{\infty}(\Omega;[0,d_j])\) be such that \(\chi_j = 1\) on \(\Omega_j\cap\{|y| > R_0 + 2\}\), \(\chi_j = 0\) on \(\Omega_j\cap\{|y| < R_0 + 1\}\), and also \(\chi_j = 0\) on \(\Omega\setminus\Omega_j\). Define \(\chi_0 = 1 - \sum_{j=1}^N\chi_j\). Then, \(\{\chi_j\}_{j=0}^N\) is a partition of unity on \(\Omega\). The conormal differentiation at the boundary with respect to the metric \(G\) is denoted by \(\partial_{\nu}\), and that with respect to the Euclidean metric is denoted by \(\partial_{\nu^{(0)}}\). We set, for \(R\ge R_0\), \[\partial \Omega_j(R) = \partial\Omega_j \cap \{|y| > R\} = \{(y,0), (y,d_j) \in \partial\Omega_j\, ; \, |y| > R\}. \nonumber\] \(H^{m,s}(\Omega_j)\) is the weighted Sobolev space defined by \(u \in H^{m,s}(\Omega_j) \Longleftrightarrow (1 + |y|)^su \in H^m(\Omega_j)\), and \(H^{m,s}(\partial\Omega_j)\) is defined similarly with weight \((1 + |y|)^s\). When \(s = 0\), \(H^{m,0}\) is denoted by \(H^m\). The following lemma can be proven in the same way as Lemmas 3.1 and 3.2 in [4].

We then have for \(u \in H^2(\Omega_j)\) satisfying \(\partial_{\nu_j^{(0)}}u = 0\) on \(\partial\Omega_j(R)\), \[\partial_{\nu}(\chi_j u) = w(x)^{-\delta_0}B_ju, \quad {\rm on} \quad \partial\Omega_j(R), \nonumber\] \[B_j = w(x)^{\delta_0}\big(\chi_j(\partial_{\nu} - \partial_{\nu^{(0)}}) + (\partial_{\nu}\chi_j)\big). \label{Bjdefine}\tag{4}\] We put \[\mathcal{E}_j = w(x)^{-\delta_0}\widetilde{\mathcal{E}}_j. \nonumber\] Letting \(G_j^{(0)}\) be the Euclidean metric on \(\Omega_j\), we also put \[\mathcal{V}_j(z) = [ - \Delta_G, \chi_j] + \chi_j(\Delta_{G_j^{(0)}} - \Delta_G) + (\Delta_G + z)\mathcal{E}_jB_j. \label{DefineVj40z41}\tag{5}\] Let \(H_j^{(0)} = - \Delta_y - (\partial_{n+1})^2\) in \(\Omega_j\) with Neumann boundary condition on the boundary and \(R_j^{(0)}(z) = (H_j^{(0)} - z)^{-1}\). Finally, let \(H\) be the Laplacian \(-\Delta_G\) on \(\Omega\) with Neumann boundary condition on \(\partial\Omega\) and \(R(z) = (H - z)^{-1}\). Then, as in Lemma 3.3 in [4], we have

Let \(\Delta_G\) be the Laplacian on \(\Omega\), and \(H = - \Delta_G\) with Neumann boundary condition on \(\partial\Omega\). By the method of singular sequence, we can prove

Using the resolvent equations (?? ), (?? ) in Lemma 2, one can prove LAP for \(H\). In fact, letting \(- (\partial_{j,n+1})^2\) be the Neumann Laplacian on the interval \((0, d_j)\), we first prove LAP for \(H_j^{(0)}\) on \(\Omega_j\) using LAP for \(- \Delta_y - (\partial_{j,n+1})^2\), and then prove LAP for \(H\) by perturbation arguments.

We define the set of thresholds for \(H\) by \[\mathcal{T}(H) = \cup_{j=1}^N\sigma_p(- (\partial_{j,n+1})^2), \nonumber\] and the set of exceptional points by \[\mathcal{E}(H) = \mathcal{T}(H) \cup \sigma_p(H). \label{DefinemathcalEH}\tag{6}\]

The above theorem is proven in the same way as Theorem 3.10 in [4]. Here, we need to mention the radiation condition. It suffices to consider the case of model band. Consider a solution \(u\in \mathcal{B}^{\ast}\) to the equation \((- \Delta_y - (\partial_{n+1})^2 - \lambda)u = f \in \mathcal{B}\), \(\lambda > 0\). Let \(P_m\) be the eigenprojection for the \(m\)-th eigenvalue \(\lambda_m\) of \(- (\partial_{n+1})^2\). Then, we have \((- \Delta_y - (\lambda - \lambda_m))P_mu= P_mf\). If \(\lambda < \lambda_m\), \(P_mu\) will belong to \(L^2\). Therefore, the radiation condition is required only when \(\lambda > \lambda_m\). We say that \(u\) satisfies the outgoing radiation condition if, for any \(1 \leq j \leq N\) and \(\lambda_{j,m} < \lambda\), \(u_{j,m} = P_{j,m}u\) satisfies \[\big(\frac{\partial}{\partial |y|} - i\sqrt{\lambda - \lambda_{j,m}}\big)u_{j,m} \simeq 0, \label{RadCond1}\tag{7}\] in the sense of (3 ). If \(i\sqrt{\lambda - \lambda_{j,m}}\) is replaced by \(-i\sqrt{\lambda - \lambda_{j,m}}\), \(u\) is said to satisfy the incoming radiation condition.

The condition (7 ) is rephrased as follows.

The following lemma is then proven in the same way as Lemma 3.8 of [4].

In the course of the proof of Theorem 2, it is shown that the resolvent \(R(\lambda \pm i0)\) satisfies the radiation condition. We then have the following corollary.

2.2 Free spectral representation↩︎

To derive the eigenfunction expansion, we first consider \(H^{(0)}_0 = -\Delta_y -(\partial_{n+1})^2\) in the euclidean cylinder \({\boldsymbol{R}}^n\times (0,1)\) with Neumann condition on \(x_{n+1} = 0, 1\). Let \(G_0(z) = (-\Delta_y -z)^{-1}\) be the Green operator for \(- \Delta_y\) on \({\boldsymbol{R}}^n\). Let \(\lambda_{\ell}\) and \(P_{\ell}\), \(\ell = 1, 2, \dots\) be the eigenvalues and eigenprojections of \(-(\partial_{n+1})^{2}\) on \(\lbrack 0, 1\rbrack\). Then for \(z \in {\boldsymbol{C}}\setminus [0,\infty)\), \(R^{(0)}_0(z) = (H^{(0)}_0 - z)^{-1}\) is written as \[R^{(0)}_0(z) = \sum _{\ell=1}^{\infty} G_0(z -\lambda_{\ell})\otimes P_{\ell}. \label{R0061G0Pell}\tag{8}\] We put \[c_0(\lambda) = \frac{\lambda^{(n-2)/4}}{{\sqrt 2}}, \quad \lambda>0, \label{DefineC0laambda}\tag{9}\] and define the Fourier transform \(\mathcal{F}_0(\lambda)\) on \({\boldsymbol{R}}^n\) by \[\mathcal{F}_0(\lambda) f = c_0(\lambda)\hat{f}(\sqrt{\lambda}\omega) = c_0(\lambda)(2\pi)^{-n/2}\int_{{\boldsymbol{R}}^n}e^{- i\sqrt{\lambda}\omega\cdot y}f(y) dy, \quad \omega \in S^{n-1}, \label{DefineF0lambdaf}\tag{10}\] \[\hat{f}(\sqrt{\lambda}\omega) = (2\pi)^{-n/2}\int_{{\boldsymbol{R}}^n} e^{-i\sqrt{\lambda}\omega\cdot y}f(y)dy. \nonumber\] We also define \[\mathcal{F}_0^{(\pm)}(\lambda)f(\omega) = \big(\mathcal{F}_0(\lambda)f\big)(\pm \omega). \nonumber\] It is well-known that for any \(f \in \mathcal{B}({\boldsymbol{R}}^n)\) and \(\lambda > 0\), \(G_0(\lambda \pm i0)f\) has the expansion \[G_0(\lambda \pm i0)f(\omega) \simeq c_1^{(\pm)}(\lambda)r^{-(n-1)/2} e^{\pm i\sqrt{\lambda}r}\hat{f}(\pm \sqrt{\lambda}\omega), \nonumber\] \[c_1^{(\pm)}(\lambda) = \sqrt{\frac{\pi}{2}}e^{\pm (3-n)\pi i/4}\lambda^{(n-3)/4}, \nonumber\] in \(\mathcal{B}^{\ast}({\boldsymbol{R}}^n)\) as \(r= |y| \to \infty\), \(\omega = y/r\). This together with (8 ) implies the following lemma.

Given a slab domain having \(N\) bands, let \(\lambda_{j,\ell}\) be the Neumann eigenvalue and \(\varphi_{j,\ell}\) the normalized Neumann eigenfunction for \(- (\partial_{n+1})^2\) of the \(j\)-th model band \(\Omega_j = {\boldsymbol{R}}^n\times (0,d_j)\). In the following, for the sake of simplicity of description, we assume that \(d_j = 1\). We put \[\Psi_{j,\ell}^{(0)}(y,x_{n+1};\lambda,\omega) = c_0(\lambda - \lambda_{j,\ell})(2\pi)^{-n/2} e^{i\sqrt{\lambda - \lambda_{j,\ell}}\omega\cdot y} \varphi_{j,\ell}(x_{n+1}), \quad \lambda > \lambda_{j,\ell}. \label{DefinePsijell0ycn431}\tag{11}\] In view of (10 ) and (?? ), we define the free Fourier transformation \[\mathcal{F}_{j,\ell}^{(0)}(\lambda)f(\omega,x_{n+1}) = \left(\int_{{\boldsymbol{R}}^n\times (0,1)} \overline{\Psi^{(0)}_{j,\ell}(y,x_{n+1}';\lambda,\omega)}f(y,x_{n+1}')dydx_{n+1}'\right) \varphi_{j,\ell}(x_{n+1}), \nonumber\] \[\mathcal{F}^{(0)}_j f(\lambda) = \sum_{\ell=1}^{\infty} \mathcal{F}^{(0)}_{j,\ell}(\lambda)f \nonumber\] for each model band \(\Omega_j\), which is first defined on \(C_0^{\infty}(\Omega_j)\) and then extended to whole \(L^2(\Omega_j)\). The total free Fourier transformation is defined by \[\mathcal{F}^{(0)}= (\mathcal{F}^{(0)}_1, \mathcal{F}^{(0)}_2, \ldots, \mathcal{F}^{(0)}_N), \label{totalFouriertransformF0}\tag{12}\] which is unitary from \(L^2(\cup_{j=1}^N\Omega_j)\) to \(\oplus_{j=1}^NL^2((0,\infty);L^2(S^{n-1}\times (0,d_j));d\lambda)\) ⁵.

2.3 Perturbed spectral representation↩︎

When \(\delta_0 > (n+1)/2\), the generalized eigenfunctions for \(H\) are defined by \[\Psi_{j,\ell,\pm}(\lambda) = (\chi_j -\mathcal{E}_jB_j)\Psi^{(0)}_{j,\ell}(\lambda) - R(\lambda \textcolor{blue}{\pm} i 0)\mathcal{V}_j(\lambda) \Psi^{(0)}_{j,\ell}(\lambda). \label{Psijlpm61464646-R40lambda-i041}\tag{13}\] It satisfies \[\left\lbrace \begin{align} &(H-\lambda)\Psi_{j,\ell,\pm}(\lambda) = 0\quad in\quad \Omega,\\ &\partial _\nu \Psi_{j,\ell,\pm}(\lambda) = 0 \quad on \quad \partial\Omega. \end{align} \right. \nonumber\]

Let \({\boldsymbol{h}}_j(\lambda) = L^2(S^{n-1})\times {\rm span}\,\{\varphi_{j,\ell} \, ; \, \lambda_{j,\ell} < \lambda\}\), where \({\rm span}\, A\) means the linear hull of the set \(A\), and put \[{\boldsymbol{h}}(\lambda) = \oplus_{j=1}^N{\boldsymbol{h}}_j(\lambda). \label{hlambdadefine}\tag{14}\]

To deal with the general case, we consider the operator (cf. (?? )) \[\mathcal{F}_{j,\ell,\pm}(\lambda) = \mathcal{F}^{(0)}_{j,\ell}\big(\lambda) J_j \Big(\chi_j -(\mathcal{E}_jB_j)^{\ast} -\mathcal{V}_j(\lambda)^{\ast}R(\lambda \mp i 0)\Big), \label{mathcalFjellpmlambdadefine}\tag{15}\] which is well-defined on \(\mathcal{B}\) when \(\delta_0 > 1\), i.e. \[\mathcal{F}_{j,\ell,\pm}(\lambda) \in {\boldsymbol{B}}(\mathcal{B}; {\boldsymbol{h}}_j(\lambda)), \quad \lambda \in (0,\infty)\setminus \mathcal{E}(H). \nonumber\] If \(\delta_0 > (n+1)/2\), by (13 ), \(\mathcal{F}_{j,\ell,\pm}(\lambda)^{\ast}\) is the operator with integral kernel \(\Psi_{j,\ell,\pm}(y, x_{n+1};\lambda,\omega)\). Letting \(\chi_{\lambda_{j,\ell}}(\lambda)\) be the characteristic function of the interval \((\lambda_{j,\ell},\infty)\), we define \[\mathcal{F}_{j,\pm}(\lambda) = \sum_{\ell=1}^{\infty} \chi_{\lambda_j,\ell}(\lambda) \mathcal{F}_{j,\ell,\pm}(\lambda) = \sum _{\lambda_{j,\ell}<\lambda}\mathcal{F}_{j,\ell,\pm}(\lambda), \nonumber\] \[\mathcal{F}_\pm (\lambda) = (\mathcal{F}_{1,\pm}(\lambda), \ldots, \mathcal{F}_{N,\pm}(\lambda)). \nonumber\] By the resolvent equations (?? ), (?? ), the expansion in Lemma 5 is extended to the perturbed operator.

The following Parseval’s formula is proven in the same way as [4], Lemma 4.5.

We define \[\widehat {\mathbb{H}}_j = \left\{\sum_{\ell=1}^{\infty} f_{\ell}(\xi)\varphi_{j,\ell}(x_{n+1})\, ; \, f_{\ell} \in L^2(|\xi|^2 > \lambda_{j,\ell}),\; \sum_{\ell=1}^{\infty} \|f_{\ell}\|^2_{L^2(|\xi|^2 > \lambda_{j,\ell})}<\infty \right\}, \nonumber\] where \(L^2(|\xi|^2 > \lambda_{j,\ell})\) is the space of functions \(f \in L^2({\boldsymbol{R}}^n)\) with support in \(|\xi|^2 > \lambda_{j,\ell}\). We put \[\widehat{\mathbb{H}} = \oplus_{j=1}^N\widehat{\mathbb{H}}_j.\] These preparations are sufficient to prove the following theorem. Let \({\mathcal{H}}_{ac}(H)\) be the absolutely continuous subspace for \(H\).

2.4 S-matrix↩︎

The time-dependent scattering theory can be developed in the same way as in Theorem 4.7 of [4]. Let \[H_j^{(0)} = - \Delta_y - (\partial_{j,n+1})^2 \nonumber\] be the Laplacian on the slab \(\Omega_j\). The wave operator \(W_{\pm} : \oplus_{j=1}^NL^2(\Omega_j) \to L^2(\Omega)\) is defined by \[W_{\pm} = {\mathop{\rm s-lim}_{t\to \pm\infty}}\sum_{j=1}^N e^{it\sqrt{H}}\chi_j e^{-it\sqrt{H_j^{(0)}}}. \nonumber\] Then, \(W_{\pm}\) exists and is complete, i.e. \({\rm Ran}\, W_{\pm} = \mathcal{H}_{ac}(H)\). Moreover, \(W_{\pm} = (\mathcal{F}_{\pm})^{\ast}\mathcal{F}^{(0)}\) (see (12 )). The scattering operator is defined by \[S = (W_+)^{\ast}W_-. \nonumber\] We consider its Fourier transform: \[\widehat S = \mathcal{F}^{(0)}S(\mathcal{F}^{(0)})^{\ast}. \nonumber\] It has the following representation. For each \(\lambda \in (0,\infty)\setminus{\mathcal{E}}(H)\), there exists a unitary operator \(\widehat S(\lambda)\) on \({\boldsymbol{h}}(\lambda)\) such that \[(\widehat Sf)(\lambda) = \widehat S(\lambda)f(\lambda), \quad \forall f \in \widehat H, \quad \lambda \in (0,\infty) \setminus \mathcal{E}(H). \nonumber\] Here \(\widehat S(\lambda)\) is an \(N\times N\) matrix operator whose entry has the form ([4], Lemma 4.9) \[\widehat S_{jk}(\lambda) = \delta_{jk}I_k - 2\pi i \mathcal{F}_{j,+}(\lambda) \mathcal{V}_k(\lambda)\big(\mathcal{F}_k^{(0)}(\lambda)\big)^{\ast}, \quad 1 \leq j, k \leq N. \nonumber\] \(I_k\) being the identity on \({\boldsymbol{h}}_k(\lambda)\). The scattering amplitude is defined by \[A_{jk}(\lambda) = \mathcal{F}_{j,+}(\lambda)\mathcal{V}_k(\lambda)\big(\mathcal{F}_k^{(0)}(\lambda)\big)^{\ast} \in {\boldsymbol{B}}({\boldsymbol{h}}_k(\lambda) \, ; \,{\boldsymbol{h}}_j(\lambda)). \label{DefineAjklambda}\tag{16}\] Projecting to the \(m\)-th and the \(n\)-th eigenspaces for \(- (\partial_{j,n+1})^2\) and \(- (\partial_{k,n+1})^2\), we put \[A_{jm,kn}(\lambda) = \mathcal{F}_{j,m,+}(\lambda)\mathcal{V}_k(\lambda)\big(\mathcal{F}_{k,n}^{(0)}(\lambda)\big)^{\ast}. \nonumber\] Then, we have \[\label{rel46SjkAjmkn} \widehat S_{jk}(\lambda) - \delta_{jk}I_k = - 2\pi i \sum_{\lambda_{j,m}<\lambda, \: \lambda_{k,n}<\lambda} A_{jm,kn}(\lambda).\tag{17}\]

We now fix \(j\) and \(k\), and assume that \(\delta_0 > (n+1)/2\) on the slabs \(\Omega_j\) and \(\Omega_k\). Then, the operator \(A_{jm,kn}(\lambda)\) has an integral kernel written by \(\Psi_{j,m,+}(\lambda)\) and \(\Psi^{(0)}_{k,n}(\lambda)\)⁶: \[A_{jm,kn}(\lambda;\omega,x_{n+1},\omega',x_{n+1}') = \int_{\Omega}\overline{\Psi_{j,m,+}(\lambda; X,\omega,x_{n+1})}\Big(\mathcal{V}_{k}(\lambda)\Psi^{(0)}_{k,n}(\lambda;X,\omega',x_{n+1}')\Big)d\Omega_X. \nonumber\] We contract the variables \(x_{n+1}, x'_{n+1}\), and define \(\mathcal{A}_{jm,kn}(\lambda;\omega,\omega')\) by \[\mathcal{A}_{jm,kn}(\lambda;\omega,\omega') = \iint_{(0,1)\times (0,1)}\varphi_{j,m}(x_{n+1})A_{jm,kn}(\lambda;\omega,x_{n+1},\omega',x_{n+1}')\varphi_{k,n}(x_{n+1}')dx_{n+1}dx'_{n+1}. \label{DefinemathcaAjmkncontraction}\tag{18}\] Let \(\mathcal{A}_{jm,kn}(\lambda)\) be the integral operator with kernel \(\mathcal{A}_{jm,kn}(\lambda;\omega,\omega')\). Then letting \[\langle h,\varphi_{k,n}\rangle = \int_0^1h(\omega,x_{n+1})\varphi(x_{n+1})dx_{n+1}, \nonumber\] we have for \(h \in {\boldsymbol{h}}_{k}(\lambda)\) \[A_{jm,kn}(\lambda)h = \varphi_{j,m}\mathcal{A}_{jm,kn}(\lambda)\langle h,\varphi_{k,n}\rangle. \nonumber\] By virtue of (13 ) and Lemma 6, we have \[\Psi_{j,\ell,\pm}(\lambda) - \chi_j\Psi_{j,\ell}^{(0)}(\lambda) \simeq - \sum_{\lambda_{j,m}<\lambda}c_{\mp}(\lambda - \lambda_{j,m}) \frac{e^{\mp i\sqrt{\lambda - \lambda_{j,m}}|y|}}{|y|^{(n-1)/2}} \mathcal{F}_{j,m,\mp}(\lambda)\mathcal{V}_j(\lambda)\Psi^{(0)}_{j,\ell}(\lambda). \nonumber\] This formula implies the following lemma.

2.5 Analytic continuation of the scattering amplitude↩︎

We now observe the slab \(\Omega_1\). When \(j = k = 1\), the scattering amplitude is written as (see (10 ), (11 ), (15 ) and (16 ))⁷ \[\begin{align} A_{1m,1n}(\lambda) = \mathcal{F}_0(\lambda - \lambda_{1,m})P_{1,m}J_1\Big(\chi_1 - (\mathcal{E}_1B_1)^{\ast} - \mathcal{V}_1(\lambda)^{\ast}R(\lambda + i0) \Big)\mathcal{V}_1(\lambda)\mathcal{F}_0(\lambda - \lambda_{1,n})^{\ast} P_{1,n}. \end{align} \nonumber\] Assuming that \(\Omega_1\) is flat for \(|y| > R\), we take \(\chi_1(y)\) so that \(\chi_1(y) = 0\) for \(|y| < R +1\), and \(\chi_1(y) = 1\) for \(|y| > R+2\). Then, in view of (4 ), we have \(B_1 = 0\) and that \(\mathcal{V}_1(z)\) in (5 ) is independent of \(z\) and supported in \(|y| < R\). Then \(A_{1m,1n}(\lambda)\) is written as \[\begin{align} A_{1m,1n}(\lambda) = \mathcal{F}_0(\lambda - \lambda_{1,m})P_{1,m} \Big(\chi_1 - \mathcal{V}_1^{\ast}R(\lambda + i0) \Big)\mathcal{V}_1\mathcal{F}_0(\lambda - \lambda_{1,n})^{\ast} P_{1,n}. \end{align} \label{NewA1m1n}\tag{19}\] Then, \(A_{1m,1n}(\lambda)\) defined for \(\lambda > \max\{\lambda_{1,m},\lambda_{1,n}\}\) is analytically continued to the upper-half plane \({\boldsymbol{C}}_+ = \{{\rm Im}\, \lambda > 0\}\), and is extended to a continuous function on \({\boldsymbol{C}}_+\cup ({\boldsymbol{R}}\setminus \mathcal{E}(H))\). We denote the obtained function for \(\lambda < \max\{\lambda_{1,m},\lambda_{1,n}\}\) by \(A_{1m,1n}^{(nph)}(\lambda)\) and call it the non-physical scattering amplitude. It then follows from this definition that the non-physical scattering amplitude \(A_{1m,1n}^{(nph)}(\lambda)\) coincides with the physical scattering amplitude \(A_{1m,1n}(\lambda)\) for \(\lambda > \max\{\lambda_{1,m},\lambda_{1,n}\}\). Since the function \(\sqrt{\lambda - \lambda_{\ell}}\) defined for \(\lambda > \lambda_{\ell}\) is analytically extended to \(i\sqrt{\lambda_{\ell} - \lambda}\) defined for \(\lambda < \lambda_{\ell}\), the following Lemma 9 is obvious from (19 ). We put for \(\mu < 0\) \[\mathcal{G}_{0}(\mu)f = c_0( - \mu)(2\pi)^{-n/2}\int_{{\boldsymbol{R}}^n}e^{\sqrt{-\mu}\omega\cdot y}f(y)dy, \label{DefinemathcalG0mu}\tag{20}\] \[\widetilde{\mathcal{G}_{0}(\mu)} \psi = c_0(-\mu)(2\pi)^{-n/2}\int_{S^{n-1}}e^{- \sqrt{-\mu}\omega\cdot y}\psi(\omega)d\omega. \nonumber\]

For \(\lambda < \lambda_{1,n}\), we put \[\Phi^{(0)}_{1,n}(x,\lambda,\omega) = c_0(\lambda_{1,n} - \lambda)) e^{-y\cdot\omega\sqrt{\lambda_{1,n}-\lambda}}\varphi_{1,n}(x_{n+1}), \label{FormulaPhi01n61e-yomegavarphi1}\tag{21}\] which is an exponentially growing solution to the equation \((- \Delta_y - (\partial_{1,n+1})^2)u = \lambda u\) and \[\widetilde{\mathcal{G}}_0(\lambda - \lambda_{1,n})\psi \otimes \varphi_{1,n} = (2\pi)^{-n/2}\int_{S^{n-1}}\Phi^{(0)}_{1,n}(x,\lambda,\omega)\psi(\omega)d\omega, \quad \psi \in L^2(S^{n-1}). \nonumber\] Define the non-physical eigenfunction by \[\Phi_{1,n,-}(\lambda) = \chi_1 \Phi_{1,n}^{(0)}(\lambda) - R(\lambda + i0)\mathcal{V}_1\Phi^{(0)}_{1,n}(\lambda), \label{FormulaPhi1n-61byPhi01n}\tag{22}\] which is an exponentially growing solution to the equation \((H - \lambda)u = 0\). Compared with the physical eigenfunction defined by (13 ) for \(\lambda > \lambda_{1,n}\) \[\Psi_{1,\ell,-}(\lambda) = \chi_1\Psi^{(0)}_{1,\ell}(\lambda) - R(\lambda + i0)\mathcal{V}_1\Psi^{(0)}_{1,\ell}(\lambda), \nonumber\] the non-physical eigenfunction is its analytic continuation to the interval \(\lambda < \lambda_{1,n}\). By Lemma 9, the non-physical scattering amplitude is computed from the asymptotic behavior of non-physical eigenfunction in the following way. First let us note the following lemma.

We put as in (18 ) \[\mathcal{A}^{(nph)}_{1m,1n}(\lambda) = (A^{(nph)}_{1m,1n}(\lambda)\varphi_{1,n},\varphi_{1,m}).\]

We put \[\mathcal{F}^{(phy)}_{1,-}(\lambda)f = \sum_{\lambda > \lambda_{\ell}}\int_{\Omega} \overline{\Psi_{1,\ell,-}(\lambda;X,\omega, x_{n+1})}f(X)d\Omega_X, \nonumber\] \[\mathcal{F}^{(nph)}_{1,-}(\lambda)f = \sum_{\lambda < \lambda_{\ell}}\int_{\Omega} \overline{\Phi_{1,\ell,-}(\lambda;X, \omega, x_{n+1})}f(X)d\Omega_X. \nonumber\]

Arguing in the same way as Lemma 6, we obtain the following lemma.

3 From the scattering matrix to the interior problem↩︎

In this section, we consider the reconstruction of the domain \(\Omega\). Suppose we are given two slab domains \(\Omega^{(i)} = \mathcal{K}^{(i)} \cup \Omega_1^{(i)} \cup \cdots \cup \Omega_{N^{(i)}}^{(i)}\), \(i = 1, 2\), satisfying the assumptions in §1. Note that we do not assume \(N^{(1)} = N^{(2)}\). It will be proven in our inverse procedure given below.

The first step is to reduce the issue to the interior boundary value problem. For this purpose, there are two ways. The first one is used in [4], and the second one is used in [6].

3.1 Boundary spectral projections↩︎

The first method uses the boundary spectral projection. Take a constant \(R_0 > 0\) so that \(\Omega_1 \cap \{|y| > R_0\}\) is flat. We put \[\Omega_{ext} = \Omega_1 \cap \{|y| > R_0\}, \quad \Omega_{int} = \Omega \setminus \overline{\Omega_{int}}.\] Take an open set \(\mathcal{O} \subset\subset \Omega_{int}\) such that it has a smooth boundary not intersecting \(\partial\Omega_{int}\) and that \(\Omega_{int}\setminus \mathcal{O}\) is connected. Let \(H_{\mathcal{O}}\) be \(- \Delta_G\) on \(\Omega_{int} \setminus \overline{\mathcal{O}}\) with Neumann boundary condition. We can construct the generalized Fourier transformation \(\mathcal{F}(\lambda)\) for \(H_{\mathcal{O}}\) as in Subsection 2.3. Let \((\lambda_i, P_i)\) be the set of eigenvalues and eigenprojections for \(H_{\mathcal{O}}\). We put \(\Gamma_{\mathcal{O}} = \Omega_1 \cap \{|y| = R_0\}\) if \(\mathcal{O} = \emptyset\), and \(\Gamma_{\mathcal{O}} = \partial \mathcal{O}\) if \(\mathcal{O} \neq \emptyset\). Let \(r_{\mathcal{O}} \in {\boldsymbol{B}}(H^1(\Omega_{\mathcal{O}});H^{1/2}(\Gamma_{\mathcal{O}}))\) be the trace operator to \(\Gamma_{\mathcal{O}}\), \[r_{\mathcal{O}} : H^1(\Omega_{\mathcal{O}}) \ni f \to f\big|_{\partial{\mathcal{O}}} \in H^{1/2}(\Gamma_{\mathcal{O}}). \nonumber\]

We call the set \[\left\{(\lambda,r_{\mathcal{O}}\mathcal{F}(\lambda)^{\ast}\mathcal{F}(\lambda) r_{\mathcal{O}}^\ast);\lambda \in (0,\infty)\setminus \mathcal{E}(H_{\mathcal{O}})\right\} \cup \left\{(\lambda_i,r_{\mathcal{O}} P_i r_{\mathcal{O}}^{\ast}) \right\}_{i=1}^{\textcolor{blue}{d_p}} \nonumber\] the boundary spectral projection (BSP) for \(H_{\mathcal{O}}\) on \(\Gamma_{\mathcal{O}}\), where \(d_p\) is the dimension of the point spectral subspace for \(H_{\mathcal{O}}\). Then, arguing in the same way as in [4] Subsections 5.2 and 5.3, one can prove that \(S_{11}(\lambda)\), the \((1,1)\)-component of the S-matrix, determines BSP. See Lemma 5.7 of [4].

3.2 Source-to-solution map↩︎

The second method uses the source-to-solution map. Using the above notations \(\Omega_{ext}\) and \(\Omega_{int}\), assume that \(\Omega_{ext}\) is flat and \(H = H^{(0)}_0\) there. Take a bounded open set \(\mathcal{O} \subset\subset \Omega_{ext}\), and consider the following boundary value problem \[\left\{ \begin{align} & (H - \lambda)u = F \quad {\it in}\quad \Omega, \quad {\rm supp}\, F \subset \mathcal{O},\\ & \partial_{\nu}u = 0 \quad {\it on} \quad \partial\Omega, \\ & u \;{\it satisfies}\; {\it the} \;{\it outgoing} \;{\it or} \;{\it incoming} \;{\it radiation} \;{\it condition} \end{align} \right. \nonumber\] for \(\lambda \in (0,\infty) \setminus \mathcal{E}(H)\). Extending \(F \in L^2(\mathcal{O})\) to be 0 outside \(\mathcal{O}\), the solution \(u\) exists uniquely by Theorem 3. The source-to-solution map is defined by \[\mathbb{U}_{\mathcal{O}, \pm }(\lambda) : F \to u_{\pm}\big|_{\mathcal{O}} = R(\lambda \pm i0)F\big|_{\mathcal{O}}. \nonumber\] Let us assume that for \(i = 1, 2\), \(\Omega^{(i)}_{ext}\) is flat and \[\Omega_{ext}^{(1)} = \Omega_{ext}^{(2)}, \label{Omega4014161Omega40241Y62R}\tag{24}\] more precisely isometric in the Euclidean sense. We observe the incoming and outgoing waves in the infinity of \(\Omega^{(i)}_1\). We put the super-script \(\, ^{(i)}\) for all notations relevant to \(\Omega^{(i)}\). However, due to the assumption (24 ), we sometimes omit the super-script \(\, ^{(i)}\) for operators subordinate to \(\Omega^{(i)}_1\). We denote the set \(\Omega_{ext}^{(1)} = \Omega_{ext}^{(2)}\) as \(\Omega_{ext}\). Let \(S_{11}^{(i)}(\lambda)\) be the (1,1) entry of the S-matrix \(S^{(i)}(\lambda)\).

4 The remaining proof↩︎

If we pass to the BSP, we are in the same situation as in [4], §6, where the problem is reduced to \((\Omega_1 \cap \{|x| < R_1\})\cup \cdots \cup \Omega_N \cup \mathcal{K}\), each \(\Omega_i\) being a cylindrical domain, and the boundary is \(\overline{\Omega_1} \cap \{x= R_1\}\), \(x \in {\boldsymbol{R}}\). In the present paper, the cylindrical domain \(\Omega_i\) is replaced by a slab and the boundary is a band \(\overline{\Omega_1} \cap \{|y| = R_1\}\), \(y \in {\boldsymbol{R}}^n, n \geq 2\). We can then mimick the BC method in [4], §6 word by word to get the same conclusion. We have thus completed the proof of Theorem 1.

If we use the source-to-solution map, the arguments in [6] work in the same way with a slight change of notation. We do not repeat the details.

References↩︎