Abstract
We examine the cut-off resolvent R χ (λ) = χ (–Δ D – λ2)–1χ, where Δ D is the Laplacian with Dirichlet boundary condition and \(\chi \in C_0^{\infty}(\mathbb{R}^n)\) equal to 1 in a neighborhood of the obstacle K. We show that if R χ (λ) has no poles for \(\Im \lambda \geq -\delta,\: \delta > 0\), then \(\|R_{\chi}(\lambda)\|_{L^2 \to L^2} \leq C|\lambda|^{n-2},\: \lambda \in \mathbb{R},\: |\lambda| \geq C_0.\) This estimate implies a local energy decay. We study the spectrum of the Lax-Phillips semigroup Z(t) for trapping obstacles having at least one trapped ray.
Keywords: Trapping obstacles, Resonances, Local energy decay, Cut-off resolvent
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Bony, JF., Petkov, V. Resolvent estimates and local energy decay for hyperbolic equations . Ann. Univ. Ferrara 52, 233–246 (2006). https://doi.org/10.1007/s11565-006-0018-1
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DOI: https://doi.org/10.1007/s11565-006-0018-1