Quasinormal modes are the gravitational wave analogue to the overtones heard after striking a bell; like many physical systems, black holes emit radiation as a response to perturbations. After a dynamical event, for example a black hole merger, the system is expected to relax to a stationary black hole solution. After sufficient time, the system can be treated as a perturbation to this stationary solution in what is called the ringdown phase. The observed gravitational wave signal is dominated by the ringing associated with these solutions to the linear perturbation equations in this period of the evolution. Each quasinormal mode is characterised by a complex frequency which encodes its behaviour in time: the imaginary part determines its oscillation and the real part its exponential decay.

In light of the observation of gravitational wave signals in the past few years, quasinormal modes are important from an astronomical perspective. By comparing the observed gravitational wave signal from some dynamical event with the predictions provided by computing quasinormal frequencies, one can compare the fit given by general relativity against some modified theory of gravity and test which is a better model for these phenomena. This black hole spectroscopy could also be used to deduce the parameters of an astrophysical object from the gravitational wave signal.

As horizons become extremal, various computations (from a range of authors including Detweiler, Hod and Zimmerman) have shown that in many cases, there exists a sequence of frequencies which become purely oscillatory in the limit and which cluster on a line in the complex plane. These zero-damped modes are typically the most slowly decaying resonances of the equation and hence are key to understanding stability. In the case of a positive cosmological constant, they are closely tied to the Strong Cosmic Censorship Conjecture: if the spectral gap is too large, the modes don't decay slowly enough to destabilise the Cauchy horizon.

From the large variety of examples in the literature of nearly extremal black holes with zero-damped modes, it is natural to conjecture that this phenomenon is generic. This thesis explores mathematically rigorous results that can be obtained toward resolving this question. In particular, we shall review the literature on quasinormal modes (focussing on zero-damped modes), discuss the mathematical definition of these objects and the idea of co-modes or dual resonant states: solutions to the adjoint problem which can make identifying the frequencies easier. Finally, we shall use this framework and Gohberg-Sigal theory to prove existence results for zero-damped modes: firstly in the case of wave equations with potentials which decay sufficiently rapidly, then for a large class of static, spherically symmetric black hole spacetimes. There are also partial results toward resolving the question for the Kerr-de Sitter spacetime.