Sign-indefinite transmission conditions and near-zero eigenvalues and resonances
Abstract
We consider the linear operator for metamaterial modeling Lc,d =(χRE3backslashΩ̅+c)∇ · (χRE3backslashΩ̅+d)∇ , 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 Lc,d has, as in the sign-definite case, an absolutely continuous spectrum equal to [0,+∞); 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 ε-small analytic perturbations of the parameters. We show that, depending on the position of the number 1/2 (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 ε1/2), consisting either of two branches of resonances, or of one branch of negative eigenvalues and one of resonances.
Related
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BibTeX
@misc{mantile2026signindefinite,
title = {{Sign-indefinite transmission conditions and near-zero eigenvalues and resonances}},
author = {Andrea Mantile and Andrea Posilicano},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18483},
eprint = {2610.18483},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18483},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sign-indefinite transmission conditions and near-zero eigenvalues and resonances
AU - Andrea Mantile
AU - Andrea Posilicano
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18483
UR - https://arxiv.org/abs/2610.18483
AB - We consider the linear operator for metamaterial modeling L_c,d =(χ_RE^3backslashΩ̅+c)∇ · (χ_RE^3backslashΩ̅+d)∇ , 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,+∞); 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 ε-small analytic perturbations of the parameters. We show that, depending on the position of the number 1/2 (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 ε^1/2), consisting either of two branches of resonances, or of one branch of negative eigenvalues and one of resonances.
ER -