Treat the s-wave zero-damped-mode sector

Analyze the zero-damped-mode spectrum of massless scalar perturbations in the near-extremal Reissner–Nordström black hole for the angular-momentum sector j=0, which is not covered by the exact-WKB derivation for j\geq1.

Background

The paper derives an exact-WKB quantization condition and the near-extremal zero-damped-mode spectrum for massless, neutral scalar perturbations of Reissner–Nordström black holes, but the analysis explicitly excludes the s-wave sector j=0 because its turning-point structure differs qualitatively from the j\ne0 case. Although substituting j=0 into the leading-order formula reproduces the standard result, the authors state that this agreement is accidental because the derivation is not valid in that sector.

A systematic treatment of j=0 is therefore needed to determine its spectrum and establish whether the higher-order exact-WKB framework can be extended consistently to the s-wave channel.

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})