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 \(ω=δ^{1/2}τ\) 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 \[ \cM_δ^\pm(ω) = δK-ω^2I_m \mp\iiδωΓ_0 +\mathcal O(δ^2+δω^2). \] The cutoff-resolvent norm is uniformly equivalent to \(1+\|(\cM_δ^\pm(ω))^{-1}\|\); 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 \(\dim\KerΓ_0=6N-3\). 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.

51pages