A Uniform Pole-Subtracted Limiting Absorption Principle for High-Contrast Elastic Resonator Clusters
Abstract
We establish a uniform pole-subtracted limiting absorption principle for a fixed cluster of N disjoint three-dimensional high-contrast elastic resonators when the contrast tends to infinity and ω=δ1/2τ approaches the zero threshold. The exterior Dirichlet-to-Neumann map and a variational Grushin--Feshbach reduction give an exact decomposition of the cutoff resolvent into a uniformly bounded regular part and a finite-rank term governed by cMδ±(ω) = δK−ω2Im ∓iiδωΓ0 +𝒪(δ2+δω2). The cutoff-resolvent norm is uniformly equivalent to 1+|(cMδ±(ω))−1|; hence the finite-dimensional channel carries every loss of uniformity. An elastic optical identity factors the leading radiation matrix through one total-force map into C3. Thus rankΓ0=3 and dimKerΓ0=6N−3. Compression to a static eigenspace of dimension r leaves at most three leading radiative channels. Simple bright poles have width O(δ), whereas force-dark poles with a positive second radiation form have width O(δ2); the corresponding real-axis peaks have orders δ−3/2 and δ−5/2. We also treat multiple static eigenvalues, compute the spherical coefficients, and derive a conditional two-parameter crossover for a symmetry-broken dimer.
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BibTeX
@misc{gao2026uniform,
title = {{A Uniform Pole-Subtracted Limiting Absorption Principle for High-Contrast Elastic Resonator Clusters}},
author = {Yixian Gao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.09367},
eprint = {2608.09367},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09367},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Uniform Pole-Subtracted Limiting Absorption Principle for High-Contrast Elastic Resonator Clusters
AU - Yixian Gao
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.09367
UR - https://arxiv.org/abs/2608.09367
AB - We establish a uniform pole-subtracted limiting absorption principle for a fixed cluster of N disjoint three-dimensional high-contrast elastic resonators when the contrast tends to infinity and ω=δ^1/2τ approaches the zero threshold. The exterior Dirichlet-to-Neumann map and a variational Grushin--Feshbach reduction give an exact decomposition of the cutoff resolvent into a uniformly bounded regular part and a finite-rank term governed by cM_δ^±(ω) = δK−ω^2I_m ∓iiδωΓ_0 +𝒪(δ^2+δω^2). The cutoff-resolvent norm is uniformly equivalent to 1+|(cM_δ^±(ω))^−1|; hence the finite-dimensional channel carries every loss of uniformity. An elastic optical identity factors the leading radiation matrix through one total-force map into C^3. Thus rankΓ_0=3 and dimKerΓ_0=6N−3. Compression to a static eigenspace of dimension r leaves at most three leading radiative channels. Simple bright poles have width O(δ), whereas force-dark poles with a positive second radiation form have width O(δ^2); the corresponding real-axis peaks have orders δ^−3/2 and δ^−5/2. We also treat multiple static eigenvalues, compute the spherical coefficients, and derive a conditional two-parameter crossover for a symmetry-broken dimer.
ER -