Essential spectral geometry of the Maxwell system in unbounded domains

Abstract

We analyse the essential spectrum of ${\mathcal M} = {\operatorname{curl}} {\operatorname{curl}}$ acting on divergence-free vector fields in unbounded domains of $\mathbb{R}^3$. We show that $σ_e({\mathcal M}) = [0,+\infty)$ in quasi-conical domains and $σ_e({\mathcal M}) \neq \emptyset$ in quasi-cylindrical domains. For horn-shaped domains with circular cross-section and eventually mean-convex boundary, we establish that $σ_e({\mathcal M}) = \emptyset$, 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 $σ_e({\mathcal M}) \supseteq σ_e(H_V)$. 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 $σ_e({\mathcal M}) = \emptyset$, or $σ_e({\mathcal M}) = [γ^2, +\infty)$, or $σ_e({\mathcal M}) = [0,+\infty)$.