  In this work, we wish to address the question -- whether the quasi-normal
modes, the characteristic frequencies associated with perturbed black hole
spacetimes, central to the stability of these black holes, are themselves
stable. Though the differential operator governing the perturbation of black
hole spacetimes is self-adjoint, the boundary conditions are dissipative in
nature, so that the spectral theorem becomes inapplicable, and there is no
guarantee regarding the stability of the quasi-normal modes. We have provided a
general method of transforming to the hyperboloidal coordinate system, for both
asymptotically flat and asymptotically de Sitter spacetimes which neatly
captures the dissipative boundary conditions, and the differential operator
becomes non-self-adjoint. Employing the pseudospectrum analysis and numerically
implementing the same through Chebyshev's spectral method, we present how the
quasi-normal modes will drift away from their unperturbed values under external
perturbation of the scattering potential. Intriguingly, for strong enough
perturbation, even the fundamental quasi-normal mode, associated with
gravitational perturbations, drifts away from its unperturbed position for
asymptotically de Sitter black holes, in stark contrast to the case of
asymptotically flat black holes. Besides presenting several other interesting
results, specifically for asymptotically de Sitter black holes, we also discuss
the implications of the instability of the fundamental quasi-normal mode on the
strong cosmic censorship conjecture.
