The limiting absorption principle for the acoustic wave operators in two unbounded media (Q1327639)

The author proves that the limiting absorption principle is valid and that eigenvalues are absent for the elliptic operator \(L = - a(x)^2 \Delta\) in \(\mathbb{R}^n\) with \(n \geq 2\). The space \(\mathbb{R}^n\) is divided into two disjoint regions \(\Omega_+\) and \(\Omega_-\) with common boundary \(S = \partial \Omega_+ = \partial \Omega_-\) given by \[ S = \bigl\{ (y,z) \in \mathbb{R}^{n - 1} \times \mathbb{R} \mid z = \varphi (y) \bigr\} \] with a suitable function \(\varphi \in C^1 (\mathbb{R}^{n - 1} \backslash \{0\})\). This function and the first derivatives must satisfy several conditions for \(y \mapsto 0\) and \(|y |\to \infty\), which guarantee that the surface \(S\) behaves asymptotically like the conic surface \(z = b |y |\) with \(b \geq 0\). The propagation speed can jump across \(S\) and must also satisfy several asymptotic conditions. To justify the limiting absorption principle, \textit{E. Mourre}'s method [Commun. Math. Phys. 78, 391-408 (1981; Zbl 0489.47010)] is used, which is based on some commutator estimates.