  We study the scattering of gravitational waves by a Schwarzschild black hole
and its perturbed siblings to investigate influences of proposed spectral
instability of quasinormal modes on the ringdown signal. Our results indicate
that information of dominant ringdown signals, which are ascribed to the
fundamental (i.e., least damping) quasinormal mode of unperturbed Schwarzschild
black holes, is imprinted in the phase shift defined from the transmission
amplitude (1/A_{in} in our notation). This approximately parallels the fact
that the resonance of quantum systems is imprinted in the phase shift of the
S-matrix. The phase shift around the oscillation frequency of the fundamental
mode is modified only perturbatively even if the quasinormal-mode spectrum is
destabilized by a perturbative bump at a distant location, signifying the
stability of the ringdown signal. At the same time, the phase shift at low
frequencies is modulated substantially reflecting the late-time excitation of
echo signals associated with the quasinormal-mode spectrum after
destabilization.
