We show that defect modes in infinite systems of resonators have
corresponding modes in finite systems which converge as the size of the system
increases. We study the generalized capacitance matrix as a model for
three-dimensional coupled resonators with long-range interactions and consider
defect modes that are induced by compact perturbations. If such a mode exists,
then there are elements of the discrete spectrum of the corresponding
truncated, finite system converging algebraically to each element of the pure
point spectrum. This result, which concerns periodic lattices of arbitrary
dimension in a three-dimensional differential system, is in contrast with the
exponential convergence observed in one-dimensional problems. This is due to
the presence of long-range interactions in the system, which gives a dense
matrix model and shows that exponential convergence cannot be expected in
physical systems.
