无界域中Maxwell系统的本质谱几何
Essential spectral geometry of the Maxwell system in unbounded domains
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中文总结 AI 辅助
本文研究无界域中Maxwell算子curl curl的本质谱，揭示了其依赖于域几何（锥形、柱形、喇叭形及穿孔喇叭），并给出不同几何下的谱刻画。
中文摘要 AI 辅助
我们分析了在$\mathbb{R}^3$的无界域中作用于无散度向量场的算子${\mathcal M} = {\operatorname{curl}} {\operatorname{curl}}$的本质谱。我们证明了在拟锥形域中$\sigma_e({\mathcal M}) = [0,+\infty)$，在拟柱形域中$\sigma_e({\mathcal M}) \neq \emptyset$。对于具有圆形截面且边界最终为平均凸的喇叭形域，我们建立了$\sigma_e({\mathcal M}) = \emptyset$，这与它们的体积无关。对于环形截面的喇叭，横向法向调和场确定了一个有效的一维薛定谔算子$H_V$，且$\sigma_e({\mathcal M}) \supseteq \sigma_e(H_V)$。最后，对于一类具有双指数孔的穿孔指数喇叭的具体族，我们展示了根据孔在无穷远处收缩的速率，要么$\sigma_e({\mathcal M}) = \emptyset$，要么$\sigma_e({\mathcal M}) = [\gamma^2, +\infty)$，要么$\sigma_e({\mathcal M}) = [0,+\infty)$。
英文摘要
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 $σ_e({\mathcal M}) = [0,+\infty)$.
发表机构
- University of Verona（维罗纳大学）
- Cardiff University（卡迪夫大学）
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