A Pole-Subtracted Limiting Absorption Principle for Clusters of High-Contrast Elastic Subwavelength Resonators

Abstract

We establish a pole-subtracted limiting absorption principle for a fixed cluster of \(N\) disjoint three-dimensional high-contrast elastic resonators in the joint regime in which the material contrast tends to infinity and the frequency approaches the zero threshold on the subwavelength scale \(ω=δ^{1/2}τ\). Eliminating the homogeneous exterior by the elastic Dirichlet-to-Neumann map and applying a variational Grushin--Feshbach reduction on the inclusion domain yield an exact decomposition of the cutoff resolvent into a uniformly bounded regular part and a finite-rank resonant part governed by \[ \cM_δ^\pm(ω) = δK-ω^2I_m \mp\iiδωΓ_0 +\mathcal O(δ^2+δω^2). \] Moreover, the cutoff resolvent norm is uniformly equivalent to \(1+\|(\cM_δ^\pm(ω))^{-1}\|\), so every loss of uniformity is carried by the finite-dimensional channel. Although the rigid space has dimension \(6N\), an elastic optical identity shows that the leading radiation matrix is generated by a single total-force map into \(\C^3\); consequently, \(\rankΓ_0=3\) for every \(N\) and \(\dim\kerΓ_0=6N-3\). On force-dark resonant branches, a second nonnegative radiation form yields a threshold Fermi golden rule. Bright poles have width \(O(δ)\) and real-axis peaks \(O(δ^{-3/2})\), whereas second-order-bright dark poles have width \(O(δ^2)\) and peaks \(O(δ^{-5/2})\). The degenerate theory captures dark--bright mixing inside multiple static eigenspaces. A spherical example is fully explicit, and a symmetric dimer exhibits the corresponding symmetry-breaking crossover.

Disclosure

“plication by 12 I + K∗ (ζ) yields the DtN expansion. All factors are holomorphic, so the differentiated remainder obeys the same uniform bound. The optical identity identifies the Hermitian coefficient Λ1 and proves its positivity. Use of generative AI and AI-assisted technologies During the preparation of this work, AI-assisted tools were used to support language editing, exposition refinement, and preliminary consistency checks of notation and references. All mathematical statements, p”

PDF page 37

Classification

Rewriting existing author-written text

Multiplier

4

Verified

2026-08-23T05:07:20.957812Z

Structural counts