## Resonances near the real axis for transparent obstacles.(English)Zbl 0951.35036

Let $$O\subset \mathbb{R}^n$$, $$n\geq 2$$, be a bounded strictly convex domain with $$C^\infty$$ boundary $$\Gamma$$ and $$\Omega= \mathbb{R}^n/\overline O$$, $$\alpha> 0$$. This paper deals with the transmission problem $\begin{cases} (c^2\Delta+ \lambda^2)u_1= 0\quad &\text{in }O,\\ (\Delta+ \lambda^2) u_2= 0\quad &\text{in }\Omega,\\ u_1- u_2= 0\quad &\text{on }\Gamma,\\ \partial_{-n}u_1+ \partial_n u_2= 0\quad &\text{on }\Gamma,\\ u_2\text{-}\lambda\text{-outgoing},\end{cases}\tag{1}$ $$\lambda\in \mathbb{C}$$ will be said to be a resonance for the transmission problem associated to $$O$$, if the problem (1) has a nontrivial solution. On this basis the authors show: There exists an infinite sequence $$\{\lambda_j\}$$ of different resonances of (1) such that $0< \text{Im }\lambda_j\leq C_N|\lambda_j|^{-N},\quad \forall N\geq 1.$ The basic step concerns the construction of a quasi mode of the frequency support, which is concentrated at the glancing manifold $$\kappa$$ of the interior problem.
To do this, they provide a global symplectic normal form for pairs of glancing hypersurfaces in a neighbourhood of $$\kappa$$ and then separate the variables microlocally near the whole glancing manifold. The authors conclude their investigation by looking at the geometry of the billiard flow.

### MSC:

 35J25 Boundary value problems for second-order elliptic equations 35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions) 58J05 Elliptic equations on manifolds, general theory
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