标题： 全矢量麦克斯韦方程组与连续角指数 显示英文标题

标题： Full Vectorial Maxwell Equations with Continuous Angular Indices

摘要： 本文提出了一种数学框架，用于在具有连续角指数的圆柱和球形几何中求解麦克斯韦方程组。 我们超越了标准的离散谐波分解，采用广义谱积分进行连续谱表示，捕捉表现出奇异行为但又在几何中心产生有限能量场的电磁解。 对于连续角指数$\ell, m \in \mathbb{R}$，我们在加权Sobolev空间$H^s_{\alpha(\ell,m)}(\Omega)$中研究解的存在性和唯一性，遵循~\cite{adams2003, reed1975}中建立的框架，证明$\ell > -\frac{1}{2}$的有限能量，并通过双正交函数系统构造显式谱核。 该框架涵盖了具有连续方位指数$\nu \in (0,1)$的可分离圆柱模式以及通过矢量旋度运算耦合场分量的不可分离球形模式。 我们给出了奇异场行为的渐近分析，研究了谱近似的收敛速率，并通过Galerkin投影方法和数值谱积分验证了理论框架。 显示英文摘要

摘要： This article presents a mathematical framework for solving Maxwell's equations in cylindrical and spherical geometries with continuous angular indices. We extend beyond standard discrete harmonic decomposition to a continuous spectral representation using generalized spectral integrals, capturing electromagnetic solutions that exhibit singular behavoiur yet yield finite-energy fields at the geometric center. For continuous angular indices $\ell, m \in \mathbb{R}$, we study existence and uniqueness of solutions in weighted Sobolev spaces $H^s_{\alpha(\ell,m)}(\Omega)$ following the framework established in ~\cite{adams2003, reed1975}, prove finite energy for $\ell > -\frac{1}{2}$, and construct explicit spectral kernels via biorthogonal function systems. The framework encompasses both separable cylindrical modes with continuous azimuthal index $\nu \in (0,1)$ and non-separable spherical modes where field components couple through vectorial curl operations. We present asymptotic analysis of singular field behavior, investigate convergence rates for spectral approximations, and validate the theoretical framework through Galerkin projection methods and numerical spectral integration.