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

Summary · qwen2.5:32b

The paper examines spectral and transport properties in high-contrast resonator chains, revealing long-range interactions characterized by an off-diagonal decay rate $C(n,m)\sim \frac{1}{|n-m|\log^2|n-m|}$, crucial for understanding Anderson localisation with arbitrary disorder. Additionally, it establishes the strong convergence of finite capacitance operators to the full operator as chain size increases, enhancing spectral convergence rate estimates from previous studies.

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Excerpt

arXiv:2607.04971v1 Announce Type: new 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 $C$. Importantly, we demonstrate that the decay rate is long-range and critical: as $|n-m|\to\infty$, \begin{equation*} 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.