Black hole Quasi-Normal Mode instabilities: Pseudospectrum, asymptotics and resonant expansions

Can we measure the 'effective regularity' of spacetime from the instabilities of black hole quasi-normal mode (QNM) overtones? Assessing the structural stability of QNM frequencies of self-gravitating compact objects is an important open problem in gravitational physics, given the fundamental role that QNMs play in diverse areas of gravity theory. A particular avenue to this problem is provided by Kato’s perturbation theory of linear operators. However, such an approach requires first to cast QNMs in terms of a proper eigenvalue problem, in particular with proper (finite norm) QNM eigenfunctions in a Hilbert space. The hyperboloidal approach to black hole perturbations provides a geometric framework to such a characterization of QNMs in terms of the spectrum of a non-selfadjoint operator. In this talk we discuss some of the concepts and tools imported from the spectral theory of non-selfadjoint operators. Focus will be placed on the notion of pseudospectrum, paying special attention to QNM resonant expansions of a scattered field and, time allowing, possible connections to spacetime asymptotic symmetries. Black hole QNM instability is a classical general relativity low-regularity phenomenon, agnostic to possible detailed descriptions of gravity at higher-energies and potentially observationally accessible.