AI 中文总结
该研究针对黑洞例外点处准简正模式合并的病态问题，通过构造Riesz簇的围道矩推导散射与源响应的Riesz–Laurent表示，得到无需模式标记的精确物理响应系数。
AI 中文摘要
在黑洞例外点（EP）处，两个准简正模式发生合并，其各自的留数变得病态。我们未采用近简正模式拟合的假设，而是从复标度Regge–Wheeler–Zerilli预解式出发，通过构造方法推导响应，将这些模式视为一个孤立的二阶Riesz簇。其第零阶和第一阶围道矩确定了EP两侧及EP处的精确配对预解式，无需标记单个模式或构造归一化Jordan链。在二阶EP处，这些矩确定了单极点和双极点Laurent算子。尽管模式分解是奇异的，但只要该簇保持孤立、互补预解式正则且无极点到达物理轴，固定实频率的透射系数和灰体因子仍在EP处保持实解析性。Laurent算子的源-观测者矩阵元定义了有限的、与归一化无关的振幅，并确定了因果衰减中的常数项和线性时间项。它们与Jost双零展开得到的系数相等，表明这些系数是指定物理响应的算子定义系数，而非拟合参数。因此，两个簇矩提供了无需模式标记的数据，由此可同时得到散射响应和驱动响应。
英文摘要
At a black-hole exceptional point (EP), two quasinormal modes coalesce and their separate residues become ill-conditioned. Rather than postulating a near-degenerate modal fit, we derive the response constructively from the complex-scaled Regge--Wheeler--Zerilli resolvent, treating the modes as one isolated rank-two Riesz cluster. Its zeroth and first contour moments determine an exact pair resolvent on both sides of, and at, the EP, without labeling the individual modes or constructing a normalized Jordan chain. At a second-order EP, these moments determine the simple- and double-pole Laurent operators. Although the modal decomposition is singular, fixed-real-frequency transmission and the greybody factor remain real-analytic through the EP, provided that the cluster remains isolated, the complementary resolvent is regular, and no pole reaches the physical axis. Source--observer matrix elements of the Laurent operators define finite, normalization-independent amplitudes and fix both the constant and linear-in-time terms in the causal ringdown. Their equality with the coefficients from the Jost double-zero expansion shows that they are operator-defined coefficients of the specified physical response, rather than fitting parameters. Thus two cluster moments provide mode-label-free data from which both scattering and driven responses follow.
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