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Body resonances for classical waves

Andrea Mantile, Andrea Posilicano

Abstract

We provide a detailed study of the spectral properties of the linear operator $H(\varepsilon)=-(\varepsilon^{2}χ_{Ω_{\varepsilon}}+χ_{Ω^{c}_{\varepsilon}})Δ$ modeling, through the wave equation $(\partial_{tt}+H(\varepsilon))u=0$, the dynamics of acoustic waves in the presence of a small inhomogeneity of size $\varepsilon$ having high contrast $\varepsilon^{-2}$. In particular, we give precise results on the localization of the resonances of $H(\varepsilon)$ and their first-order $\varepsilon$-expansions; the latter are explicitly expressed in terms of the eigenvalues and eigenvectors of the Newton potential operator of the set $Ω$ whose rescaling of size $\varepsilon$ defines $Ω_{\varepsilon}$.

Body resonances for classical waves

Abstract

We provide a detailed study of the spectral properties of the linear operator modeling, through the wave equation , the dynamics of acoustic waves in the presence of a small inhomogeneity of size having high contrast . In particular, we give precise results on the localization of the resonances of and their first-order -expansions; the latter are explicitly expressed in terms of the eigenvalues and eigenvectors of the Newton potential operator of the set whose rescaling of size defines .

Paper Structure

This paper contains 7 sections, 11 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Results
  4. The model operator and its spectrum
  5. The limiting absorption principle
  6. Localization and $\varepsilon$-expansions of the resonances
  7. Proof of Theorem \ref{['Theorem_main']}

Key Result

Theorem \oldthetheorem

Let $\lambda_{\circ}\in\sigma_{disc}(N_{0})$ with $\dim(\ker(\lambda_{\circ}-N_{0}))=m$; let $e_{j}^{\circ}$, $1\leq j\leq m$, be the corresponding orthonormal eigenvectors. Then, whenever $\varepsilon$ is sufficiently small, close to $\lambda_{\circ}^{-1}$ there are $m$ (not necessarily distinct) r

Theorems & Definitions (22)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • ...and 12 more