Exceptional line and pseudospectrum in black hole spectroscopy

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.