Bound on the counting function for the eigenvalues of an infinite multistratified acoustic strip
Olivier Poisson · Mathematical Methods in the Applied Sciences · 1999
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, 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) ⩽ Cμ3/2. Copyright © 1999 John Wiley & Sons, Ltd.