Line-defect waveguiding at an exterior Dirichlet spectral frequency

Extend the frequency-dependent capacitance framework to high-contrast waveguiding problems with a line defect when the reference exterior wavenumber squared belongs to the spectrum of the exterior Dirichlet Laplacian, where the exterior Dirichlet-to-Neumann map is singular and the leading spectral splitting is expected to be of order \(\delta^{1/2}\).

Background

The paper develops its effective capacitance-operator reduction only in the regular exterior regime, where the reference exterior wavenumber squared lies in the resolvent set of the exterior Dirichlet Laplacian. This condition guarantees that the exterior Dirichlet-to-Neumann map is analytic and bounded near the reference frequency.

The authors identify the complementary singular regime in which the reference exterior wavenumber squared belongs to the exterior Dirichlet spectrum. In that case, the Dirichlet-to-Neumann map becomes singular, and the usual first-order $\mathcal O(\delta)$ reduction is no longer expected to apply; instead, the leading splitting is anticipated to scale like $\delta^{1/2}$. The unresolved task is to establish an analogous theory specifically for line-defect waveguides.