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, $(\epsilon ,d,{\sigma }_0)$, a continuous line of EPs (exceptional line, EL) in this three-dimensional parameter space is revealed. Notably, the EL exhibits an anisotropic spectral response: parameters migrating along the EL direction leaves the coalesced QNM spectra nearly unchanged, while moving parameters away from the EL induces the characteristic ${\epsilon }^{1\mathrm{/}2}$ scaling, highlighting the directional nature of spectral instability in exceptional structures. We find that the vorticity $\nu =\pm 1\mathrm{/}2$ and the Berry phase $\gamma =\pi$ for loops encircling the EL, while $\nu =0$ and $\gamma =0$ for those do not encircle the EL. In the neighborhood of an eigenvalue, through matrix perturbation theory, we prove that the $\epsilon$-pseudospectrum contour size scales as ${\epsilon }^{1\mathrm{/}q}$ at an EP , where $q$ is the order of the largest Jordan block of the Hamiltonian-like operator associated with that eigenvalue, 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.