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