# Pseudospectra of Holographic Quasinormal Modes

**Authors:** Daniel Arean, David Garcia-Fariña, Karl Landsteiner  
**arXiv:** [2307.08751](https://arxiv.org/abs/2307.08751)  
**Date:** 2023-07-17  
**Summary page:** https://www.summarizepaper.com/en/paper/2307.08751v1/

## Key points

- Quasinormal modes and frequencies are crucial for understanding black holes in asymptotically Anti-de Sitter geometries
- Non-Hermitian operators exhibit instability under small perturbations, unlike normal Sturm-Liouville operators
- Study focuses on stability analysis of quasinormal frequencies related to AdS black holes using pseudospectrum analysis
- Specific differential operators describing fluctuations of scalar and transverse gauge fields in AdS black hole background highlight non-Hermiticity at the horizon
- Numerical methods based on pseudospectral methods are employed to test stability under random local potential perturbations
- Pseudospectrum calculations reveal open contour lines indicating instability for both scalar field and transverse vector pseudospectra
- Researchers' findings, conclusions, and outlook on potential future studies are summarized in section 7

## Summary

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The paper delves into the analysis of quasinormal modes and frequencies, which are crucial in understanding the physics of black holes, particularly in asymptotically Anti-de Sitter geometries. These modes play a significant role in studying strongly coupled quantum many-body systems through gauge/gravity duality. Non-Hermitian operators exhibit instability under small perturbations, making their eigenvalues inherently unstable unlike normal Sturm-Liouville operators. The study focuses on the stability analysis of quasinormal frequencies related to asymptotically planar AdS black holes using pseudospectrum analysis. Specific differential operators that describe fluctuations of scalar and transverse gauge fields in an AdS black hole background are constructed in section 4, highlighting the concentration of non-Hermiticity at the horizon. The local origin of non-Hermiticity and non-normality is discussed in detail. Section 5 outlines the numerical methods employed, based on pseudospectral methods, introducing selective pseudospectra to test stability under random local potential perturbations. Relevant regimes are explored further through construction of specific potentials. In section 6, core findings from pseudospectrum calculations for energy norms are presented. The analysis reveals open contour lines indicating instability for both scalar field and transverse vector pseudospectra. Finally, section 7 summarizes the researchers' findings, draws conclusions from the study, and provides an outlook on potential future studies. Hydrodynamic modes are hinted at as a highly interesting case for further investigation. Technical details on calculations are also provided for those interested in a deeper understanding of the methodology employed in this research.