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In this work, we wish to address the question -- whether the quasi-normalmodes, the characteristic frequencies associated with perturbed black holespacetimes, central to the stability of these black holes, are themselvesstable. Though the differential operator governing the perturbation of blackhole spacetimes is self-adjoint, the boundary conditions are dissipative innature, so that the spectral theorem becomes inapplicable, and there is noguarantee regarding the stability of the quasi-normal modes. We have provided ageneral method of transforming to the hyperboloidal coordinate system, for bothasymptotically flat and asymptotically de Sitter spacetimes which neatlycaptures the dissipative boundary conditions, and the differential operatorbecomes non-self-adjoint. Employing the pseudospectrum analysis and numericallyimplementing the same through Chebyshev's spectral method, we present how thequasi-normal modes will drift away from their unperturbed values under externalperturbation of the scattering potential. Intriguingly, for strong enoughperturbation, even the fundamental quasi-normal mode, associated withgravitational perturbations, drifts away from its unperturbed position forasymptotically de Sitter black holes, in stark contrast to the case ofasymptotically flat black holes. Besides presenting several other interestingresults, specifically for asymptotically de Sitter black holes, we also discussthe implications of the instability of the fundamental quasi-normal mode on thestrong cosmic censorship conjecture.

Perturbing the perturbed: Stability of quasinormal modes in presence of a positive cosmological constant