Frequency-dependent capacitance matrix formulation for Fabry-P\'erot resonances. Part I: One-dimensional finite systems

We study scattering resonances of finite one-dimensional systems of high-contrast resonators beyond the subwavelength regime. Introducing a novel tridiagonal frequency-dependent capacitance matrix, we derive quantitative asymptotic expansions of the hybridized Fabry-P\'erot resonant frequencies in terms of the material contrast parameter. The leading-order shifts are governed by the eigenvalues of this matrix, while the corresponding eigenmodes are approximated, to leading order, by trigonometric functions on selected spacings between resonators. Our results extend the use of discrete approximations as a powerful tool for characterizing the resonant properties of a system of high-contrast resonators at arbitrarily high frequencies.