Classify the complete higher-multipole spectrum

Classify the complete finite-frequency higher-multipole spectrum of the Lorentz-violating Kalb–Ramond field on a fixed Schwarzschild background, including the more involved higher-frequency structure for multipoles with ell=2 and above.

Background

The paper derives an exact tower of static zero-frequency modes and numerically follows two low-frequency branches for ell=1. It also examines ell=2, finding distinct static zero modes and evidence that the two-branch structure persists beyond the lowest multipole.

Only the low-frequency portion of the ell=2 spectrum is analyzed. The authors explicitly leave the complete classification of its higher-frequency structure unresolved, motivating further spectral analysis of the coupled two-form perturbation equations.

References

The higher-frequency structure of the $\ell=2$ spectrum is more involved, and we leave its complete classification for future work.

— Zero modes and oscillatory instabilities of a Lorentz-violating Kalb-Ramond field on a Schwarzschild background  (2609.39936 - Lyu et al., 30 Sep 2026) in Section “Higher-Multipole Perturbations,” paragraph following the discussion of the ell=2 spectrum