Energy scales and black hole pseudospectra: the structural role of the scalar product

A pseudospectrum analysis has recently provided evidence of a potential generic instability black hole (BH) quasinormal mode (QNM) overtones under high-frequency perturbations. Such depends on the assessment size This is encoded in scalar product and its choice not unique. We address impact choice, founding it physical energy scales problem. The article organized three parts: basics, applications heuristic proposals. In first part, we revisit used hyperboloidal approach to QNMs, extending previous effective analyses placing them solid spacetime basis. second part focuses QNM problem: i) demonstrate that an artifact spectral numerical schemes, by implementing finite elements calculation from weak formulation; ii) using Keldysh's theorem, provide resonant expansions for gravitational waveform, with explicit expressions expansion coefficients; iii) propose notion "epsilon-dual expansions" exploit BH spectroscopy, exploiting both non-perturbed perturbed former informing large latter probing small scales. third enlarges conceptual scope proposing: a) spiked perturbations are more efficient triggering instabilities than smooth ones, b) general picture problem given, supporting conjecture (built Burnett's conjecture) Nollert-Price branches converge universally logarithmic Regge limit c) aiming at fully geometric description BMS states hinted as possible asymptotic/boundary degrees freedom inverse scattering