Exact WKB and Quantum Periods for Extremal Black Hole Quasinormal Modes

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.