Convergence rates for defect modes in large finite resonator arrays

TL;DR

This paper demonstrates how defect modes in infinite resonator systems have finite-system counterparts that converge as the system size grows, with convergence rates depending on lattice dimension and interaction range.

Contribution

It introduces a model using the generalized capacitance matrix to analyze convergence of defect modes in finite resonator arrays with long-range interactions.

Findings

Convergence of defect modes depends on lattice dimension.

Exponential convergence occurs when lattice and physical space dimensions match.

Algebraic convergence occurs when lattice dimension is less than physical space dimension.

Abstract

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 that converge to each element of the pure point spectrum. The rate of convergence depends on the dimension of the lattice. When the dimension of the lattice is equal to that of the physical space, the convergence is exponential. Conversely, when the dimension of the lattice is less than that of the physical space, the convergence is only algebraic, because of long-range interactions arising due to coupling with the far field.