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 \(\omega=\delta^{1/2}\tau\). 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 \[ \cM_\delta^\pm(\omega) = \delta K-\omega^2I_m \mp\ii\delta\omega\Gamma_0 +\mathcal O(\delta^2+\delta\omega^2). \] Moreover, the cutoff resolvent norm is uniformly equivalent to \(1+\|(\cM_\delta^\pm(\omega))^{-1}\|\), 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, \(\rank\Gamma_0=3\) for every \(N\) and \(\dim\ker\Gamma_0=6N-3\). On force-dark resonant branches, a second nonnegative radiation form yields a threshold Fermi golden rule. Bright poles have width \(O(\delta)\) and real-axis peaks \(O(\delta^{-3/2})\), whereas second-order-bright dark poles have width \(O(\delta^2)\) and peaks \(O(\delta^{-5/2})\). 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.