## Title: Investigating Spectrum Instability in Quantum Corrected Black Holes: Pseudospectrum Analysis

In this study, we investigate the pseudospectrum and spectrum (in)stability

of a quantum corrected black hole. Methodologically, we use the hyperboloidal

framework to cast the QNM problem into an eigenvalue problem associated with a

non-selfadjoint operator, and exploit the invariant subspace method to improve

the computational efficiency for pseudospectrum. The investigation of the

spectrum (in)stability have two aspects. On the one hand, we calculate the

spectra of the quantum corrected black hole, then the impact of the quantum

correction effect on the Schwarzschild black hole has been studied through

migration ratios. 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.

In this study, the authors investigate the pseudospectrum and spectrum instability of a quantum corrected black hole. They use the hyperboloidal framework and the invariant subspace method to analyze the problem. The investigation of the spectrum instability has two aspects.

## First Aspect: Quantum Correction Effect

The authors calculate the spectra of the quantum corrected black hole and study the impact of the quantum correction effect on the Schwarzschild black hole through migration ratios. The results show that “migration ratio instability” occurs for small black holes with small angular momentum. In the eikonal limit, the migration ratios remain the same for each overtone.

## Second Aspect: Perturbations

The authors study the spectrum instability of the quantum corrected black hole by adding particular perturbations into the effective potential. These perturbations are located at the event horizon and null infinity. Two interesting observations are made:

- Perturbations at infinity are more capable of generating spectrum instability compared to those at the event horizon.
- The peak distribution of perturbations can lead to the instability of QNM spectrum more efficiently than the average distribution.

## Roadmap for Readers

Based on the conclusions of the study, readers interested in this topic can explore several potential directions:

- Further investigation into the migration ratio instability in small black holes with small angular momentum numbers.
- Exploration of the eikonal limit and its implications for migration ratios in black hole spectra.
- Further research on the effects of perturbations located at the event horizon and null infinity, and their role in spectrum instability.
- Investigation into why perturbations at infinity are more effective in generating spectrum instability compared to those at the event horizon.
- Study of the impact of different distribution patterns of perturbations on QNM spectrum instability.

## Challenges and Opportunities

Researchers delving into this field will encounter several challenges and opportunities:

**Complex Mathematics:**The study involves the use of the hyperboloidal framework, non-selfadjoint operators, and eigenvalue problems, which require a solid understanding of advanced mathematical concepts.**Data Efficiency:**The investigation of pseudospectrum and spectrum instability requires computationally efficient methods to handle large datasets and calculate migration ratios.**Experimental Validation:**Theoretical findings should be tested through experiments or numerical simulations to validate their accuracy and practical applications.**Potential Applications:**Understanding spectrum instability in quantum corrected black holes can have implications for various fields, such as astrophysics, cosmology, and quantum gravity research.

Disclaimer:The information presented in this study is based on theoretical analysis. Further research and empirical evidence are needed to fully understand the implications and applications of the findings.