Full Vectorial Maxwell Equations with Continuous Angular Indices

Abstract

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.