High-Contrast Transmission Resonances for the Lam\'e System

TL;DR

This paper analyzes elastic scattering resonances in high-contrast isotropic inclusions, providing asymptotic descriptions near real axis and classifying long-lived resonances, with explicit expansions for different regimes.

Contribution

It offers a sharp asymptotic characterization of elastic resonances in high-contrast inclusions, including near zero and wavelength-scale regimes, with explicit resolvent expansions and resonance lifetime classification.

Findings

Resonances cluster near nonzero Neumann eigenvalues with imaginary parts of order τ.

Near zero, resonances have real parts of order √τ, with a lifetime dichotomy.

Explicit resolvent expansions are derived for both fixed-size and microresonators.

Abstract

We consider the Lam\'e transmission problem in $\mathbb{R}^3$ with a bounded isotropic elastic inclusion in a high-contrast setting, where the interior-to-exterior Lam\'e moduli and densities scale like $1/\tau$ as $\tau\to0$. We study the scattering resonances of the associated self-adjoint Hamiltonian, defined as the poles of the meromorphic continuation of its resolvent. We obtain a sharp asymptotic description of resonances near the real axis as $\tau\to0$. Near each nonzero Neumann eigenvalue of the interior Lam\'e operator there is a cluster of resonances lying just below it in the complex plane; in this wavelength-scale regime the imaginary parts are of order $\tau$ with non-vanishing leading coefficients. In addition, near zero (a subwavelength regime), we identify resonances with real parts of order $\sqrt{\tau}$ and prove a lifetime dichotomy: their imaginary parts are of order $\tau$ generically, but of order $\tau^2$ for an explicit admissible set $\mathcal E$. This yields a classification of long-lived elastic resonances in the high-contrast limit. We also establish resolvent asymptotics for both fixed-size resonators and microresonators. We derive explicit expansions with a finite-rank leading term and quantitative remainder bounds, valid near both wavelength-scale and subwavelength resonances. For microresonators, at the wavelength scale the dominant contribution is an anisotropic elastic point scatterer. Near the zero eigenvalue, the leading-order behaviour is of monopole or dipole type, and we give a rigorous criterion distinguishing the two cases.