Quasinormal modes of Schwarzschild black holes on the real axis

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.