  It has been suggested that the spectrum of quasinormal modes of rotating
black holes is unstable against additional potential terms in the perturbation
equation, as the operator associated with the equation is non-self-adjoint. We
point out that a bi-linear form has been constructed previously to allow a
perturbation analysis of the spectrum, which was applied to study the
quasinormal modes of weakly charged Kerr-Newman black holes [Phys. Rev. D 91,
044025 (2015)]. The proposed spectral instability should be restated as
instability against potential terms that have infinitesimal ``energy" norm that
is specifically defined by the type of inner products introduced by Jaramillo
et al. and preserving the physical meaning of energy. We argue that it is
necessary to address the stability of all previous mode analysis results to
reveal their susceptibility to energetically infinitesimal perturbations. In
particular, for near extremal Kerr spacetime, we show that the spectrum of
zero-damping modes, which have slow decay rates, is unstable (with order unity
fractional change in decay rates) with fine-tuned modification of the
potential. The decay rates are, however, always positive with energetically
infinitesimal perturbations. If finite potential modifications are allowed near
the black hole, it is possible to find superradiantly unstable modes, i.e., a
``black hole bomb" without an explicit outer shell. For the zero-damping modes
in near-extremal Reissner-Nordstr\"{o}m-de Sitter black holes, which are
relevant for the breakdown of strong cosmic censorship, we find that the
corresponding spectrum is stable under energetically infinitesimal
perturbations.
