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Universal estimates for the density of states for aperiodic block subwavelength resonator systems

by H. Ammari and S. Barandun and B. Davies and E.O. Hiltunen and A. Uhlmann

(Report number 2025-15)

Abstract
We consider the spectral properties of aperiodic block subwavelength resonator systems in one dimension, with a primary focus on the density of states. We prove that for random block configurations, as the number of blocks goes to infinity, the integrated density of states converges to a non-random, continuous function. We show both analytically and numerically that the density of states exhibits a tripartite decomposition: it vanishes identically within bandgaps; it forms smooth, band-like distributions in shared pass bands (a consequence of constructive eigenmode interactions); and, most notably, it exhibits a distinct fractal-like character in hybridisation regions. We demonstrate that this fractal-like behaviour stems from the limited interaction between eigenmodes within these hybridisation regions. Capitalising on this insight, we introduce an efficient meta-atom approach that enables rapid and accurate prediction of the density of states in these hybridisation regions. This approach is shown to extend to systems with quasiperiodic and hyperuniform arrangements of blocks.

Keywords: Disordered system, random block system, subwavelength regime, density of states, hybridisation, fractal behaviour, quasiperiodic sampling, hyperuniform sampling, Jacobi matrices and operators, metric transitivity, ergodicity.

@Techreport{ABDHU25_1136,
  author = {H. Ammari and S. Barandun and B. Davies and E.O. Hiltunen and A. Uhlmann},
  title = {Universal estimates for the density of states for aperiodic block subwavelength resonator systems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-15},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-15.pdf },
  year = {2025}
}

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