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Spectral convergence of defect modes in large finite resonator arrays

by H. Ammari and B. Davies and E.O. Hiltunen

(Report number 2023-05)

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 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.

Keywords: finite crystals, metamaterials, edge effects, capacitance coefficients, subwavelength resonance

@Techreport{ADH23_1042,
  author = {H. Ammari and B. Davies and E.O. Hiltunen},
  title = {Spectral convergence of defect modes in large finite resonator arrays},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-05},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-05.pdf },
  year = {2023}
}

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