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arXiv · 2610.18483

Sign-indefinite transmission conditions and near-zero eigenvalues and resonances

Andrea Mantile·Andrea Posilicano
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Abstract

We consider the linear operator for metamaterial modeling $$L_{c,d }=(χ_{\RE^{3}\backslash\overline Ω}+c)\nabla\!\cdot (χ_{\RE^{3}\backslash\overline Ω}+d)\nabla\,,$$ where the material parameters $c>0$ and $d$ are constant on each connected component of the bounded domain $Ω$, $d$ is negative on at least one component of $Ω$ and transmission boundary conditions hold at the interfaces. It turns out that $L_{c,d }$ has, as in the sign-definite case, an absolutely continuous spectrum equal to $[0,+\infty)$; however, contrary to the latter, it has a non-empty point spectrum; in particular, we find conditions on the parameters so that zero is a threshold eigenvalue. In the case $Ω$ is connected, we study how such a threshold eigenvalue behaves under $\varepsilon$-small analytic perturbations of the parameters. We show that, depending on the position of the number $\frac12\,\frac{1+d(\ve)}{1-d(\ve)}$ relative to the nearest eigenvalue of the Neumann-Poincaré operator of the interface, the zero eigenvalue splits into two branches (analytic in $\varepsilon^{1/2}$), consisting either of two branches of resonances, or of one branch of negative eigenvalues and one of resonances.

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Andrea Mantile, Andrea Posilicano. 2026-10-13. Sign-indefinite transmission conditions and near-zero eigenvalues and resonances. https://arxiv.org/abs/2610.18483

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