On the mapping between bound states and black hole quasinormal modes via analytic continuation: a spectral instability perspective

Abstract

In this work, we investigate the relation between bound states and quasinormal modes within black hole perturbation theory in the context of spectral instability. Our analysis indicates that the reliability of such spectral mapping stretches beyond the domain of validity of the analytic continuation employed to connect the perturbative bound-state problem to the corresponding open-system dynamics. However, for the numerical scheme proposed by Völkel to work, the transformations of the metric parameters must be carried out in a region where the underlying Taylor expansion is convergent. As analytically accessible explicit examples, we explore the perturbed delta-function and Pöschl-Teller potential barriers. For the latter, we construct two distinct perturbative setups for which the convergence of the series expansion involved in the perturbation theory can be rigorously controlled. When the deformation is placed near the potential's extremum, the resulting corrections to the bound-state energies can be analytically continued to yield perturbed quasinormal frequencies, in agreement with known semi-analytic results. In contrast, when the perturbation is localized asymptotically far from the compact object, the bound states are only mildly modified and are accurately described by a perturbative expansion to the first order. However, the associated analytic continuation yields a strongly deformed spectrum that shows no clear connection to the quasinormal modes. These findings contribute to the effort to scrutinize the conditions under which bound states faithfully encode quasinormal spectra and to shed light on the underlying physics of black hole spectral instability.

Figures (6)

• Figure 1: The QNMs for the modified Pöschl-Teller effective potential Eq. \ref{['MPT']} evaluated using different approaches. The discontinuity is placed at the origin $y_c=-1$. The numerical calculations are carried out using the parameters $V_{0}=1$, $V_{0+}=0.99$, and $b=1$. The QNMs of the original Pöschl-Teller effective potential, with $\mathrm{Re}(\omega_n)=0.8660$, are showen in orange crosses. The results obtained by Eqs. \ref{['Del_omey_full-1']} and \ref{['Del_omey_full+1']} using the maping from bound states are represented by empty red triangles, and those evaluated using the matrix method and the semi-analytic ones are shown in filled blue dots and empty green squares. The insets amplify part of the QNM spectrum to highlight the comparison among the various approaches.
• Figure 2: The same as Fig. \ref{['fig1']}, but the discontinuity is placed near the origin at $y_c=-0.98$.
• Figure 3: The same as Fig. \ref{['fig1']}, but the discontinuity is placed at $y_c=-0.5$. The inset in the upper right corner shows the mode $\omega_{13}=10.45-9.22i$, which is one order higher than the highest-order mode shown in the main figure, as it lies outside the frame.
• Figure 4: The same as Fig. \ref{['fig1']}, but the discontinuity is placed at $y_c=0.62$. The inset in the upper right corner shows the mode $\omega_{6}=44.11-103.22i$.
• Figure 5: The same as Fig. \ref{['fig1']}, but the discontinuity is placed near spatial infinity at $y_c=0.96$. The inset in the upper right corner shows the mode $\omega_{4}=480.49-99.62i$.