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arXiv · 2605.27572

Frequency-dependent capacitance matrix formulation for Fabry-Perot resonances in two and three dimensional systems

Habib Ammari·Bowen Li·Ping Liu·Yingjie Shao·Alexander Uhlmann
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Abstract

We study scattering resonances of finite and infinite periodic two- and three-dimensional systems of high-contrast resonators beyond the subwavelength regime. At each fixed admissible nonzero reference frequency, we introduce frequency-dependent capacitance matrices and derive quantitative asymptotic expansions of hybridized Fabry--Pérot resonant frequencies and their corresponding eigenmodes in terms of the material contrast parameter. We provide a partial differential equation (PDE) formulation of the frequency-dependent capacitance matrix analogous to the one for the capacitance matrix in the subwavelength regime. Based on this PDE formulation, we establish key properties of the frequency-dependent capacitance matrix and estimate its norm at high frequencies for a single smooth resonator with nontrapping exterior in three dimensions, identifying a regime in which the reduced residual remains perturbative. For infinite periodic resonator arrays, we prove bandgap opening under a uniform exterior non-resonance condition and generic Dirac cones for a honeycomb scaling family. Our results extend the use of discrete approximations as a powerful tool for characterizing the resonant properties of finite and infinite periodic systems of high-contrast resonators at arbitrarily high frequencies and understanding their anomalous localization and transport properties arising from strong coupling to a discrete set of eigenfrequencies.

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Habib Ammari, Bowen Li, Ping Liu, Yingjie Shao, Alexander Uhlmann. 2026-08-07. Frequency-dependent capacitance matrix formulation for Fabry-Perot resonances in two and three dimensional systems. https://arxiv.org/abs/2605.27572

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