Sensitivity of black hole spectral instability to ultraviolet perturbations

Abstract

Black hole quasinormal modes are known to exhibit spectral instability under ultraviolet perturbations of the effective potential. In the present work, we investigate the sensitivity of the fundamental mode to different types of localized perturbations through a combination of analytic and numerical analyzes. We show that the instability is governed primarily by the effective size of the perturbation rather than by its specific shape. In particular, the instability may persist even in the limit where the width of the perturbation vanishes, provided that the integrated strength of the perturbation is not zero. While a delta-function perturbation destabilizes the fundamental mode through an outward spiral, its interplay with a jump-discontinuity-type perturbation gives rise to competing inward and outward spiral motions. We further show that the stability of the fundamental mode depends sensitively on how the magnitude of the perturbation decreases as it moves away from the compact object, leading to qualitatively distinct outward spirals, inward spirals, and rotational trajectories. Finally, we investigate the motion of the fundamental mode in perturbed Regge-Wheeler potentials containing a jump discontinuity associated with a thin matter shell surrounding the black hole. The resulting behavior qualitatively resembles the spiral structure observed in double-sided P\"oschl-Teller potentials, suggesting that the mechanisms identified in analytically tractable models persist in more realistic black hole effective potentials. The present results indicate that the spectral instability of low-lying black hole modes is considerably richer than previously anticipated and may have important implications for black hole spectroscopy in realistic astrophysical environments.