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Long-range interactions and Anderson localisation for one-dimensional high-contrast resonator chain

by H. Ammari and S. Barandun and J. Qiu and A. Uhlmann

(Report number 2026-27)

Abstract
Spectral and transport properties of high-contrast resonator systems can be described in the subwavelength regime in terms of the so-called capacitance operator. In this paper, we consider an infinitely periodic chain of high-contrast resonators in three dimensions. The first result is a precise estimate of the off-diagonal decay rate of the capacitance operator \(\mathcal{C}\). Importantly, we demonstrate that the decay rate is long-range and critical: as \(|n-m|\to\infty\), \begin{equation*} \mathcal{C}(n,m)\sim \frac{1}{|n-m|\log^2|n-m|}, \end{equation*} which is \(\ell^1\) summable but slower than quadratic. This borderline decay of the off-diagonal entries makes the present proof of Anderson localisation with arbitrary disorder, which is observed numerically in this paper, out of reach; we hope that this physical example of classical wave systems with critical long-range interactions provides new insight in the field of Anderson localisation. As the second main result, based on the off-diagonal decay estimate, we prove a strong convergence of the finite capacitance operator, which corresponds to a truncated chain, to the capacitance operator as the size of the truncated chain grows to infinity. Using this strong convergence, we improve the results of [Ammari et al., SIAM J. Math. Anal., 2023 and Bull. London Math Soc., 2025] by presenting a rigorous estimate of the convergence rate of the spectrum.

Keywords: Anderson localisation, high-contrast resonator system, capacitance operator, algebraic borderline decay of off-diagonal entries, critical long-range interactions

@Techreport{ABQU26_1182,
  author = {H. Ammari and S. Barandun and J. Qiu and A. Uhlmann},
  title = {Long-range interactions and Anderson localisation for one-dimensional high-contrast resonator chain},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-27},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-27.pdf },
  year = {2026}
}

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