Exact WKB and Quantum Periods for Extremal Black Hole Quasinormal Modes

TL;DR

This paper uses exact WKB analysis to derive precise quantization conditions for black hole quasinormal modes, computing quantum periods for extremal black holes and confirming frequencies with high accuracy.

Contribution

It introduces a high-order computation of quantum periods for extremal black holes and applies Borel-Padé resummation to accurately determine quasinormal mode frequencies.

Findings

Resummed quantization conditions match known quasinormal mode frequencies.

High-order quantum periods improve the precision of spectral predictions.

Method applies to extremal Reissner–Nordström and Kerr black holes.

Abstract

We apply exact WKB analysis to the spectral problem arising in black hole perturbation theory. The boundary conditions for quasinormal modes lead to exact quantization conditions for the complex frequencies. To solve these conditions, one needs to evaluate the so-called quantum periods, or Voros symbols. For scalar perturbations of extremal Reissner--Nordstr\"om and Kerr black holes, we compute these quantities up to very high orders in the WKB expansion and perform Borel--Pad\'e resummation. The resulting resummed quantization conditions successfully reproduce the correct quasinormal mode frequencies with high precision.