Quasinormal modes in extremal Reissner-Nordstr\"om spacetimes

Understand

We present a new framework for characterizing quasinormal modes (QNMs) or resonant states for the wave equation on asymptotically flat spacetimes, applied to the setting of extremal Reissner-Nordstr\"om black holes.

• We show that QNMs can be interpreted as honest eigenfunctions of generators of time translations acting on Hilbert spaces of initial data, corresponding to a suitable time slicing.
• The main difficulty that is present in the asymptotically flat setting, but is absent in the previously studied cases of asymptotically de Sitter or anti de Sitter sub-extremal black hole spacetimes, is that $L^2$-based Sobolev spaces are not suitable Hilbert space choices.
• Instead, we consider Hilbert spaces of functions that are additionally Gevrey regular at infinity and at the event horizon.