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Bound on the counting function for the eigenvalues of an infinite multistratified acoustic strip

Abstract : Let N (µ) be the counting function of the eigenvalues associated with the self– adjoint operator −−(ρ(x, z)·) in the domain Ω = R×]0, h[, h > 0, with Neuman or Dirichlet conditions at z = 0, z = h. If ρ = 1 in the exterior of a bounded rectangular region O, that is, for |x| large, then N (µ) is known to be sublinear: the proof consists in the spectral analysis of a quadratic form obtained from a Green formula for −−(ρ(x, z)·) on O. In our case, the medium is multistratified: the function ρ(x, z) satisfies ρ(x, z) = ρ(z) for |x| large. Since the direct use of the previous proof fails, we modify the quadratic form and obtain the estimate N (µ) ≤ Cµ 3/2 .
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Olivier Poisson. Bound on the counting function for the eigenvalues of an infinite multistratified acoustic strip. 1998. ⟨hal-01429917⟩

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