This paper shows that the very rapidly damped vibrations of a black hole — called asymptotic quasinormal modes (QNMs) — carry direct information about the geometry inside the horizon for a class of exact quantum-corrected black holes in two spatial dimensions plus time. The authors compute those high‑overtone frequencies and find they follow a simple pattern: each frequency looks like an offset plus an integer times a gap. They confirm this pattern with numerical checks and then use it to read off properties of the singularity inside the hole.

The black holes they study are “quantum” in the semi-classical sense: the geometry solves Einstein’s equations with the average effect of quantum matter included. Exact non-perturbative solutions of that semi-classical system are hard to find. Here they work in a braneworld holography setting, where a lower‑dimensional semi-classical black hole appears as a classical solution in a higher‑dimensional anti‑de Sitter (AdS) bulk. This gives access to explicit, fully resummed geometries for neutral and charged black holes in (2+1) dimensions, and also neutral flat‑space examples.

To extract the asymptotic QNMs the authors reduce the wave problem to an effective one‑dimensional wave equation and then analytically continue the radial coordinate into the complex plane. They use complex‑analysis tools — Stokes line matching and the monodromy theorem — to match solutions near three regions: the curvature singularity inside, the horizon, and infinity. That matching yields quantization conditions for the frequencies. At leading order the frequencies indeed take the generic form “offset plus n times gap,” with the offset and gap set by properties such as the horizon surface gravity, the AdS boundary position in tortoise coordinates, and the length scale that controls quantum backreaction. The results are expected to be reliable when the overtone number n is large.