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–Nordström 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 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.

我们提出了一种新的框架，用于表征渐近平坦时空中波动方程的准正规模式 (QNM) 或共振态，应用于极值 Reissner-Nordström 黑洞的设置。我们表明 QNM 可以解释为时间平移生成器的真实特征函数，它们作用于初始数据的希尔伯特空间，对应于合适的时间切片。在渐近平坦设置中存在但在先前研究的渐近 de Sitter 或反 de Sitter 次极值黑洞时空中不存在的主要困难是基于\(L^2\)的 Sobolev 空间不适合 Hilbert空间选择。相反，我们考虑函数的希尔伯特空间，这些空间在无穷远和事件视界处也是 Gevrey 正则的。介绍\(L^2\)基于 Gevrey 对波动方程的估计，这些方程与沿零无穷大和事件视界的守恒量的存在密切相关。我们将这个新框架与作为求解算子亚纯连续极点的准正规频率的传统解释联系起来，并在此设置中获得新的定量结果。