dorsal/arxiv
View SchemaHigh-Contrast Transmission Resonances for the Lam\'e System
| Authors | Long Li, Mourad Sini |
|---|---|
| Categories | |
| ArXiv ID | 2601.10290vv1 |
| URL | https://arxiv.org/abs/2601.10290 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
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.
{
"annotation_id": "b4ad4dfe-9a0f-4719-a535-d5f9d00ede58",
"date_created": "2026-02-17T05:53:24.331000Z",
"date_modified": "2026-02-17T05:53:24.331000Z",
"file_hash": "40582deb79d8b069f5460433dcc6036e86dff04389a10cc202d28ef86dc29252",
"private": false,
"record": {
"abstract": "We consider the Lam\\\u0027e transmission problem in $\\mathbb{R}^3$ with a bounded isotropic elastic inclusion in a high-contrast setting, where the interior-to-exterior Lam\\\u0027e 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.\n We obtain a sharp asymptotic description of resonances near the real axis as $\\tau\\to0$. Near each nonzero Neumann eigenvalue of the interior Lam\\\u0027e 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.\n 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.",
"arxiv_id": "2601.10290",
"authors": [
"Long Li",
"Mourad Sini"
],
"categories": [
"math.AP"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "High-Contrast Transmission Resonances for the Lam\\\u0027e System",
"url": "https://arxiv.org/abs/2601.10290",
"version": "v1"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "cbb0814c-603e-4667-b766-60141a05ce73",
"id": "arXiv Dataset",
"type": "Model",
"variant": "snapshot-2026-01-17",
"version": "0.1.0"
},
"user_id": 1000002
}