Abstract

We survey functional analytic methods for studying subwavelength resonator systems. In particular, rigorous discrete approximations of Helmholtz scattering problems are derived in an asymptotic subwavelength regime. This is achieved by reframing the Helmholtz equation as a nonlinear eigenvalue problem in terms of integral operators. In the subwavelength limit, resonant states are described by the eigenstates of the generalised capacitance matrix, which appears by perturbing the elements of the kernel of the limiting operator. Using this formulation, we are able to describe subwavelength resonances and related phenomena. In particular, we demonstrate large-scale effective parameters with exotic values. We also show that these systems can exhibit localised and guided waves on very small length scales. Using the concept of topologically protected edge modes, such localisation can be made robust against structural imperfections.

Keywords

Mathematical Subject Classification

Milestones

Authors