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Eigenvalues and limiting absorption principle in an acoustic strip. (Valeurs propres et principe d’absorption limite dans une bande acoustique.) (French. Abridged English version) Zbl 0878.76069

Summary: One considers the acoustic propagator \(A=-\nabla\cdot c^2\nabla\) in the strip \(\Omega=\{(x,z)\in\mathbb{R}^2\), \(0< z<H\}\). The velocity \(c\), which describes the multistratification of the strip \(\Omega\), depends for large \(|x|\) only on the variable \(z\); it is assumed to be a function in \(L^\infty(\Omega)\), bounded from below by \(c_{\min}>0\), and there exists a real number \(M\geq 0\) such that \(c(x,z)= c_1(z)\) if \(x<-M\) and \(c(x,z)= c_2(z)\) if \(x>M\). Thanks to the known results on spectral analysis, limiting absorption principle and division theorems for the free propagators \(A_j\), \(j=1,2\), associated to the velocities \(c_j\), the spectrum of \(A\) is described and a limiting absorption principle is obtained outside of a countable set \(\Gamma\). The points of \(\Gamma\) can only accumulate on the left of the thresholds of the free propagators.

MSC:

76Q05 Hydro- and aero-acoustics
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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