Essential spectral geometry of the Maxwell system in unbounded domains
Abstract
We analyse the essential spectrum of ℳ = curl curl acting on divergence-free vector fields in unbounded domains of ℝ3. We show that σe(ℳ) = [0,+∞) in quasi-conical domains and σe(ℳ) ≠ ∅ in quasi-cylindrical domains. For horn-shaped domains with circular cross-section and eventually mean-convex boundary, we establish that σe(ℳ) = ∅, independently of their volume. For horns with annular cross-section, the transverse normal harmonic field determines an effective one-dimensional Schrödinger operator HV with σe(ℳ) ⊇ σe(HV). Finally, for a concrete family of perforated exponential horns with a double-exponential hole, we show that depending on the rate of shrinking of the hole at infinity, either σe(ℳ) = ∅, or σe(ℳ) = [γ2, +∞), or σe(ℳ) = [0,+∞).
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BibTeX
@misc{ferraresso2026essential,
title = {{Essential spectral geometry of the Maxwell system in unbounded domains}},
author = {Francesco Ferraresso and Marco Marletta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20339},
eprint = {2609.20339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20339},
}
RIS
TY - UNPB
TI - Essential spectral geometry of the Maxwell system in unbounded domains
AU - Francesco Ferraresso
AU - Marco Marletta
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20339
UR - https://arxiv.org/abs/2609.20339
AB - We analyse the essential spectrum of ℳ = curl curl acting on divergence-free vector fields in unbounded domains of ℝ^3. We show that σ_e(ℳ) = [0,+∞) in quasi-conical domains and σ_e(ℳ) ≠ ∅ in quasi-cylindrical domains. For horn-shaped domains with circular cross-section and eventually mean-convex boundary, we establish that σ_e(ℳ) = ∅, independently of their volume. For horns with annular cross-section, the transverse normal harmonic field determines an effective one-dimensional Schrödinger operator H_V with σ_e(ℳ) ⊇ σ_e(H_V). Finally, for a concrete family of perforated exponential horns with a double-exponential hole, we show that depending on the rate of shrinking of the hole at infinity, either σ_e(ℳ) = ∅, or σ_e(ℳ) = [γ^2, +∞), or σ_e(ℳ) = [0,+∞).
ER -