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

TL;DR

This paper investigates the mathematical and physical implications of the zero-frequency limit of spherical cavity modes, clarifying that the formal endpoint at ν=-1 does not correspond to a physical electromagnetic mode and exploring the nature of the potential versus the field.

Contribution

It provides a detailed analysis of the ν=-1 limit in spherical cavity modes, showing it is a mathematical boundary without physical electromagnetic modes, and clarifies the distinction between potential and field in this context.

Findings

The ν=-1 point is outside the physical spectrum due to positivity constraints.

All electromagnetic fields vanish at ν=-1, despite a non-trivial potential.

The potential exhibits a monopole-like singularity at the origin.

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.