Full vectorial Maxwell equations with continuous angular indices—part I: analytic framework

Abstract

This article presents a mathematical framework for solving Maxwell's equations in 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 behaviour yet yield finite-energy fields at the geometric centre. For continuous angular indices ℓ,m∈R, we study the existence and uniqueness of solutions in weighted Sobolev spaces Hα(ℓ,m)s(Ω) following the framework established in Adams & Fournier (Adams, Fournier. 2003 Sobolev spaces, 2nd edn) and Reed & Simon (Reed, Simon. 1975 Methods of modern mathematical physics II: Fourier analysis, self-adjointness), prove finite energy for ℓ>−12, and construct explicit spectral kernels via biorthogonal function systems. The framework encompasses non-separable spherical modes where field components couple through vectorial curl operations.