We study the stability of quasinormal modes (QNM) in asymptotically flat black hole spacetimes by means of a pseudospectrum analysis. The construction of the Schwarzschild QNM pseudospectrum reveals the following: (i) the stability of the slowest-decaying QNM under perturbations respecting the asymptotic structure, reassessing the instability of the fundamental QNM discussed by Nollert [H. P. Nollert, About the Significance of Quasinormal Modes of Black Holes, Phys. Rev. D 53, 4397 (1996)] as an “infrared” effect; (ii) the instability of all overtones under small-scale (“ultraviolet”) perturbations of sufficiently high frequency, which migrate towards universal QNM branches along pseudospectra boundaries, shedding light on Nollert’s pioneer work and Nollert and Price’s analysis [H. P. Nollert and R. H. Price, Quantifying Excitations of Quasinormal Mode Systems, J. Math. Phys. (N.Y.) 40, 980 (1999)]. Methodologically, a compactified hyperboloidal approach to QNMs is adopted to cast QNMs in terms of the spectral problem of a non-self-adjoint operator. In this setting, spectral (in)stability is naturally addressed through the pseudospectrum notion that we construct numerically via Chebyshev spectral methods and foster in gravitational physics. After illustrating the approach with the Pöschl-Teller potential, we address the Schwarzschild black hole case, where QNM (in)stabilities are physically relevant in the context of black hole spectroscopy in gravitational-wave physics and, conceivably, as probes into fundamental high-frequency spacetime fluctuations at the Planck scale.

中文翻译：

---

伪谱和黑洞准正规模式不稳定性

我们通过伪谱分析研究了渐近平坦黑洞时空中的准正态模式 (QNM) 的稳定性。Schwarzschild QNM 伪谱的构建揭示了以下内容：(i) 在关于渐近结构的扰动下，衰减最慢的 QNM 的稳定性，重新评估 Nollert 讨论的基本 QNM 的不稳定性 [H. P. Nollert，关于黑洞准正规模式的重要性，物理学。修订版 D 53, 4397 (1996)] 作为“红外线”效应；(ii) 在足够高频率的小尺度（“紫外线”）扰动下所有泛音的不稳定性，沿着伪光谱边界向通用 QNM 分支迁移，揭示了 Nollert 的开创性工作以及 Nollert 和 Price 的分析 [H. P. Nollert 和 R. H. Price，量化准正规模式系统的激发，J. Math。物理。(纽约) 40, 980 (1999)]。在方法论上，针对非自伴随算子的谱问题，采用 QNM 的紧凑双曲面方法来投射 QNM。在这种情况下，光谱（不）稳定性通过我们通过切比雪夫光谱方法在数值上构建并在引力物理学中培育的伪光谱概念自然地得到解决。在说明了 Pöschl-Teller 势的方法之后，我们解决了 Schwarzschild 黑洞情况，其中 QNM（不）稳定性在引力波物理学中的黑洞光谱背景下是物理相关的，并且可以想象，作为对基本高普朗克尺度的频率时空波动。