Stability of perturbations around black holes under spatially extended potential deformations

It has been shown that the spectrum of black hole quasi-normal modes is extremely sensitive to localized deformations of the geometry away from the potential peak. Deformations associated with a real astronomical environment may extend over a broad spatial region rather than be localized in a finite region. Whether spatially extended fluctuations distributed throughout spacetime can produce similar effects remains unclear. In this work, we construct a toy model to investigate spatially extended effective-potential deformations motivated by geometric or environmental fluctuations. We study how such deformations affect the corresponding perturbation system in the time domain. Using both the P"oschl-Teller and scalar Regge-Wheeler potentials as representative backgrounds, we show that sufficiently wide oscillatory perturbations of zero mean can drive the corresponding perturbation systems from decay to exponential growth in the time domain. Through numerical analysis, we uncover a scaling relation for the instability threshold and further provide an analytical interpretation of its physical origin.