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Uniform Hyperbolicity, Bandgaps and Edge Modes in Aperiodic Systems of Subwavelength Resonators

by H. Ammari and C. Thalhammer and A. Uhlmann

(Report number 2025-27)

Abstract
We aim to characterise the spectral distributions of bi-infinite, semi-infinite, and finite aperiodic one-dimensional arrays of subwavelength resonators, constructed by sampling from a finite library of building blocks. By adopting the modern formalism of uniform hyperbolicity, we are able to strengthen and rigorously prove a Saxon-Hutner-type result, fully characterising the spectral gaps of the composite bi-infinite aperiodic system in terms of its constituent blocks. Crucial to this approach is a change of basis from transfer matrices to propagation matrices, allowing for a block-level characterisation. This approach also enables an explicit characterisation of edge-induced eigenmodes in the semi-infinite setting. Finally, we leverage finite section methods for Jacobi operators to extend our results to finite systems-providing strict bounds for their spectra in terms of their constituent blocks.

Keywords: Saxon-Hutner theorem, propagation matrices, pseudo-ergodicity, dominated splitting, uniform hyperbolicity, invariant cone criterion, Jacobi operators, block disordered systems, Johnson's theorem, Coburn's lemma, semi-infinite systems.

@Techreport{ATU25_1148,
  author = {H. Ammari and C. Thalhammer and A. Uhlmann},
  title = {Uniform Hyperbolicity, Bandgaps and Edge Modes in Aperiodic Systems of Subwavelength Resonators},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-27},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-27.pdf },
  year = {2025}
}

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