Exceptional line and pseudospectrum in black hole spectroscopy

TL;DR

This paper explores the presence of exceptional points and their pseudospectra in black hole perturbation theory, revealing a continuous line of EPs and their impact on spectral stability and topological properties.

Contribution

It introduces a model with a Gaussian bump in the Regge-Wheeler potential, identifying a continuous exceptional line and analyzing its topological and spectral properties using matrix perturbation theory.

Findings

A continuous line of exceptional points is found in the parameter space.

The Berry phase and vorticity characterize loops around the exceptional line.

Spectral instability is enhanced at exceptional points, with pseudospectrum scaling as epsilon^{1/q}.

Abstract

We investigate the exceptional points (EPs) and their pseudospectra in black hole perturbation theory. By considering a Gaussian bump modification to the Regge-Wheeler potential with variable amplitude, position, and width parameters, $(\varepsilon,d,\sigma_0)$, a continuous line of EPs (exceptional line, EL) in this three-dimensional parameter space is revealed. We find that the vorticity $\nu=\pm1/2$ and the Berry phase $\gamma=\pi$ for loops encircling the EL, while $\nu=0$ and $\gamma=0$ for those do not encircle the EL. Through matrix perturbation theory, we prove that the $\epsilon$-pseudospectrum contour size scales as $\epsilon^{1/q}$ at an EP, where $q$ is the order of the largest Jordan block of the Hamiltonian-like operator, contrasting with the linear $\epsilon$ scaling at non-EPs. Numerical implements confirm this observation, demonstrating enhanced spectral instability at EPs for non-Hermitian systems including black holes.