Spectral Gap and Anomalous Nonlinear Stability of the Electrostatic Breathing Mode in a Plasma Sphere

The electrostatic eigenmodes of a plasma sphere include the radial breathing mode (= 0) at the plasma frequency ωₚ and the Mie surface modes (≥ 1) at frequencies ω_ = ωₚ√/(2+1), all within the band [ωₚ/√3,\,ωₚ/√2]. We prove that no resonant three-wave parametric decay exists between any modes in the surface-mode sector or from the breathing mode into any pair of modes---surface or volume---at any electron temperature. The result is a spectral gap: the minimum pairwise frequency sum of any two eigenmodes accessible as daughters exceeds the breathing mode frequency ωₚ, with the closest approach 2ωₚ/√3 ≈ 1.155\,ωₚ occurring for the = 1 dipole pair. The gap arises from the bounded, compressed Mie spectrum, which is itself a consequence of the long-range Coulomb restoring force, and is insensitive to electron temperature because the Mie frequencies are determined by boundary conditions at the sphere surface. The three-wave parametric threshold for the breathing mode, computed explicitly via the Mathieu equation for the density-modulated dipole mode, is εcrit ≈ 17% fractional displacement---a large amplitude rarely achieved in laboratory plasmas. In cold plasma, four-wave processes are also globally gapped; at finite temperature, the Tonks-Dattner volume mode spectrum opens potential four-wave channels that require separate analysis. Independently of the four-wave question, we identify the first resonant decay channel: a five-wave process 2ωₚ = 3ω₄ in which two breathing mode quanta decay into three = 4 surface mode quanta, at fifth order in perturbation theory with quality factor Q ~ 1/ε⁸. The breathing mode is radiatively trapped (no monopole radiation) and protected against resonant three-wave decay into radiating multipoles at any temperature. This double protection is anomalous among wave systems and specific to Coulomb-restored oscillations in spherical geometry: acoustic modes in the same geometry (stellar p-modes) admit resonant three-wave decay at all amplitudes.