Extend the near-extremal expansion beyond the established order

Systematically extend the exact-WKB computation of the near-extremal zero-damped-mode spectrum of massless scalar perturbations in Reissner–Nordström black holes to orders beyond $\mathcal{O}(\varepsilon^{4j})$ in the near-extremality parameter $\varepsilon$.

Background

The paper computes the zero-damped-mode spectrum as a near-extremality expansion in ε\varepsilon, using exact-WKB Voros symbols. For fixed angular momentum j, the nonresonant Riccati construction becomes insufficient at higher order because resonant poles and associated logarithmic contributions enter the contour integrals; at the same time, previously neglected contributions from the parametrically separated Voros symbol become relevant.

The authors identify systematic extension beyond O(ε4j)\mathcal{O}(\varepsilon^{4j}) as an unresolved problem. Higher-order control is also motivated by the possibility of determining resurgent connections between the zero-damped and ordinary quasinormal-mode branches.

References

We note that our treatment still leaves open two unanswered questions, namely how to address the $j=0$ $s$-wave perturbations and how to systematically extend the result to higher orders beyond $\mathcal{O}\big(\varepsilon{4j}\big)$.

— Zero-damped modes of near-extremal Reissner--Nordström black holes from exact WKB  (2609.02816 - Chiatto et al., 2 Sep 2026) in Section 6, Conclusions (Section \ref{sec:Outlook})