  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. We introduce $L^2$-based Gevrey
estimates for the wave equation that are intimately connected to the existence
of conserved quantities along null infinity and the event horizon. We relate
this new framework to the traditional interpretation of quasinormal frequencies
as poles of the meromorphic continuation of a resolvent operator and obtain new
quantitative results in this setting.
