On subwavelength resonances in three-dimensional elastic scattering by multi-layered high-contrast nested resonators of arbitrary shape

This paper delves into subwavelength resonances within three dimensional elastic scattering systems. These systems are made up of multi-layered, high-contrast nested resonator structures with arbitrary shapes. Initially, we use the Dirichlet-to-Neumann map and a variational method to describe these resonances. We show that they arise when the determinant of a particular matrix becomes zero. Secondly, by applying the Gohberg–Sigal theory and Puiseux expansions, we characterize the subwavelength characteristic values near the origin and derive their asymptotic expansions. For an N -layer structure, the number of subwavelength characteristic values is 12 N , counted with algebraic multiplicity, including 6 N branches with positive real parts and 6 N branches with negative real parts. Finally, we offer a modal decomposition formula for the total field of the elastic scattering problem and obtain the transverse and longitudinal far-field patterns related to the elastic far-field modes scattered by the N -layer nested resonators.