  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 re-framing the Helmholtz equation as a non-linear 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
resonance 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.
