The pseudospectrum and spectrum (in)stability of quantum corrected Schwarzschild black hole

Abstract

In this study, we investigate the pseudospectrum and spectrum (in)stability of quantum corrected Schwarzschild black hole. Methodologically, we use the hyperboloidal framework to cast the quasinormal mode (QNM) problem into an eigenvalue problem associated with a non-selfadjoint operator, and then the spectrum and pseudospectrum are depicted. Besides, the invariant subspace method is exploited to improve the computational efficiency for pseudospectrum. The investigation into the spectrum (in)stability entails two main aspects. On the one hand, we calculate the spectra of the quantum corrected black hole, then by the means of the migration ratio, the impact of the quantum correction effect on the Schwarzschild black hole has been studied. The results indicate that the so-called “migration ratio instability” will occur for small black holes with small angular momentum number l. In the eikonal limit, the migration ratios remain the same for each overtone. On the other hand, we study the spectrum (in)stability of the quantum corrected black hole by directly adding some particular perturbations into the effective potential, where perturbations are located at the event horizon and null infinity, respectively. There are two interesting observations under the same perturbation energy norm. First, perturbations at infinity are more capable of generating spectrum instability than those at the event horizon. Second, we find that the peak distribution can lead to the instability of QNM spectrum more efficiently than the average distribution.