AI 中文总结
本研究通过局域势微扰分析Schwarzschild黑洞的奇偶宇称准正规模，发现等谱性被破缺，例外点呈宇称选择性合并，并揭示了渐近Chandrasekhar-Detweiler因子对分离和临界振幅的调控作用。
AI 中文摘要
Schwarzschild黑洞的奇偶宇称引力微扰分别由不同的Regge-Wheeler势和Zerilli势控制，然而在真空中它们的准正规模（QNM）谱是相同的。在我们先前对奇宇称Regge-Wheeler谱的研究[M. Ould El Hadj and S. R. Dolan, arXiv:2608.16283]的基础上，我们研究了相应的偶宇称Zerilli问题，并在半径$r_0$处施加相同振幅$\epsilon$的微扰下比较了两个扇区。利用Chandrasekhar-Detweiler关系，我们推导了归一化Wronskian之间的精确关系，并表明在相同的非零局域缺陷下，Schwarzschild QNM等谱性不被保持。我们发现偶宇称和奇宇称的排斥点分支在大$r_0$处趋近于相同的Schwarzschild QNM频率，它们首次宇称依赖的分离出现在$1/\mathcal R^2$阶，其中$\mathcal R=r_{0*}/M$是缺陷的无量纲乌龟坐标位置。相应的例外点（EPs）形成两个平移的序列，在乌龟坐标中具有相同的渐近间距，而它们的相对位移由渐近Chandrasekhar-Detweiler因子的相位固定。临界微扰振幅在两个扇区中具有相同的指数衰减率，而它们的渐近前因子之比由同一因子的模固定。我们进一步发现偶宇称和奇宇称的EPs连接相同的相邻QNM泛音，对应的谱桥具有几乎相同的弧长。最后，例外简并是宇称选择性的：在相同的$(r_0,\epsilon)$值下，一对QNMs可能在一个宇称扇区中于EP处合并，而在另一个扇区中保持不同，导致铃宕响应的局域极点结构相应不同。
英文摘要
The odd- and even-parity gravitational perturbations of a Schwarzschild black hole are governed by the distinct Regge-Wheeler and Zerilli potentials, yet their quasinormal-mode (QNM) spectra are identical in vacuum. Building on our previous study of the odd-parity Regge-Wheeler spectrum [M. Ould El Hadj and S. R. Dolan, arXiv:2608.16283], we investigate the corresponding even-parity Zerilli problem and compare the two sectors under the same perturbation of amplitude $ε$ at radius $r_0$. Using the Chandrasekhar-Detweiler relation, we derive an exact relation between the normalized Wronskians and show that Schwarzschild QNM isospectrality is not preserved under an identical nonzero localized defect. We find that the even- and odd-parity repelling-point branches approach the same Schwarzschild QNM frequencies at large $r_0$, with their first parity-dependent separation appearing at order $1/\mathcal R^2$, where $\mathcal R=r_{0*}/M$ is the dimensionless tortoise-coordinate location of the defect. The corresponding exceptional points (EPs) form two shifted sequences with the same asymptotic spacing in the tortoise coordinate, while their relative displacement is fixed by the phase of the asymptotic Chandrasekhar-Detweiler factor. The critical perturbation amplitudes share the same exponential decay rate in the two sectors, whereas the ratio of their asymptotic prefactors is fixed by the modulus of the same factor. We further find that the even- and odd-parity EPs connect the same neighboring QNM overtones, with nearly identical arclengths for the corresponding spectral bridges. Finally, the exceptional degeneracy is parity selective: at identical values of $(r_0,ε)$, a pair of QNMs may coalesce at an EP in one parity sector while remaining distinct in the other, leading to a corresponding difference in the local pole structure of the ringdown response.
Comments18 pages, 7 figures, 4 tables
