In this talk we explore the issue of black hole quasi-normal mode (QNM) instability.
Specifically, by adopting a hyperboloidal approach to QNMs
we cast the QNM problem as an eigenvalue problem for a non-selfadjoint
operator. Such operators suffer potentially of instabilities in their
spectrum and this point is addressed by specific tools in
non-selfadjoint spectral theory, among them the notion of pseudospectrum.
At an exploratory stage prior to the full analytical study,
we adopt here a numerical methodology based on pseudospectral methods.
After studying the Pöschl-Teller potential as a toy model already containing the main features of the discussion, we address the black hole Schwarzschild case. As a result of our analysis, we find strong support to claim: i) the stability of the slowest decaying mode, and ii) the instability of all QNM
overtones. We will conclude with a discussion of the potential implications in astrophysics and fundamental physics.