The Zero-Frequency Limit of Spherical Cavity Modes: On the Formal Endpoint at v=1

Abstract

The transverse magnetic (TM) modes of a spherical cavity satisfy a dispersion relation connecting the angular eigenvalue $\nu$ to the resonant frequency through zeros of the spherical Bessel function derivative. Analytic continuation of this dispersion relation to $\nu = -1$ yields a formal zero-frequency endpoint where $j_{-1}(x) = \cos x / x$ admits the root $x = 0$. We examine this limit in detail, showing that while the mathematics is well-defined, the endpoint does not correspond to a physical electromagnetic mode. The positivity of the angular Sturm-Liouville operator restricts physical eigenvalues to $\nu \geq 0$, placing $\nu = -1$ outside the admissible spectrum. We demonstrate that all electromagnetic field components vanish in this limit, even though the underlying Debye potential $\Pi = \cos(kr)/kr$ remains non-trivial and exhibits a monopole-type singularity at the origin. This distinction between potential and field reflects the kernel structure of the curl-curl operator for spherically symmetric configurations. The analysis clarifies the boundary between propagating electromagnetic modes and static field configurations in spherical geometry, connecting the formal endpoint to longstanding questions about mode counting in cavity quantization.