Electromagnetic Modes in Spherical Cavities: Complete Theory of Angular Spectra, Dispersion Relations, and Self-Adjoint Extensions

Abstract: We present a complete theory of electromagnetic modes in spherical cavities, resolving fundamental questions about the nature of angular quantization. The standard result that angular indices $(\ell,m)$ must be integers is shown to be a consequence of domain constraints -- regularity at both poles and single-valuedness in the azimuthal coordinate -- rather than a requirement imposed by Maxwell's equations themselves. We prove that, for the sectoral case $ν=m$, the function $\sin^{m}θ$ exactly solves the angular eigenvalue equation for any real $m>0$, giving rise to a continuous dispersion curve. We demonstrate why non-sectoral modes (tesseral and zonal) appear only at isolated integer points on the full sphere, and show how boundary modifications such as cones and wedges convert these isolated points into continuous families of modes. Complete field solutions, wave impedances, and energy integrability conditions are derived. At the limiting point $(ν, m) = (0, 0)$, the electromagnetic field vanishes identically while the underlying Debye potential remains non-trivial -- a distinction with implications for mode counting that connects to longstanding questions in gauge theory and cavity quantization. Full-wave simulations validate the theoretical predictions with sub-percent accuracy. These results raise the possibility of structural analogues in wave equations on curved spacetimes, where conical deficits or horizon excisions similarly modify the angular domain.