Review of Analytic Results on Quasinormal Modes of Black Holes

Abstract: We present a brief review of known analytic results for quasinormal modes (QNMs) of black holes and related space-times. We consider regimes in which the perturbation equations admit exact or perturbative solutions, providing insights complementary to numerical or semi-analytic approaches. We discuss solvable cases in lower-dimensional space-times, algebraically special modes, and exact results in higher-curvature gravity theories. Particular attention is given to the eikonal mode and its correspondence with null geodesics, as well as to beyond-eikonal approximations based on inverse multipole expansions in parametrized metrics. We review analytic solutions obtained in the near-extremal limit of Schwarzschild–de Sitter black holes, in the regime of large field mass, and in pure de Sitter and anti–de Sitter space-times, where boundary conditions play a crucial role. Being not exhaustive, this overview highlights the diversity of techniques and physical insights made possible by analytic treatments of quasinormal spectra. © Pleiades Publishing, Ltd. 2025.