Validity of analytic continuation from bound states to quasinormal modes

Determine rigorous conditions under which the analytic continuation that maps bound-state energies of inverted black hole perturbation potentials to quasinormal mode frequencies is valid, specifying its domain of applicability for cases such as the inverted Regge–Wheeler and modified Pöschl–Teller potentials and in regimes exhibiting spectral instability.

Background

The paper investigates mapping quasinormal modes (QNMs) from bound-state spectra by analytic continuation, originally proposed for exactly solvable potentials and recently extended numerically by Völkel’s Taylor-expansion-based method. While this mapping works in certain controlled cases, its broader validity—especially under spectral instability where small far-field deformations induce large changes in QNMs—has been questioned.

The authors note that prior numerical attempts on inverted Regge–Wheeler potentials faced difficulties obtaining QNMs, and that the validity of the analytic continuation itself has been debated. Clarifying when and why the analytic continuation faithfully reproduces QNMs is therefore an explicit unresolved issue highlighted in the introduction.

References

The inability to find the modes, together with open questions about the validity of the analytic continuation itself, has led to some ambiguity and subsequent criticism.

— On the mapping between bound states and black hole quasinormal modes via analytic continuation: a spectral instability perspective  (2603.29704 - Li et al., 31 Mar 2026) in Section 1 (Introduction)

We were unable to obtain stable results with this approach. If such a procedure exists, it appears to be more complicated than calculating the quasi-bound states directly by numerical shooting and matching solutions of the Riccati equation.

— WKB approximation for quasi-bound states and trapped modes  (2609.10880 - Konoplya et al., 9 Sep 2026) in Section 7.2, “Higher-order corrections to the action integrals” (also summarized in Section 8, “Conclusions”)