负虚轴上的史瓦西谱梯子：端点非选择、割线相位与Jost分类
Schwarzschild spectral ladders on the negative imaginary axis: Endpoint nonselection, branch-cut phase, and Jost classification
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中文总结 AI 辅助
该研究分析负虚轴上的史瓦西谱梯子，发现有限矩阵收敛无法确定准正规极点，精确有限pencil本征值可能不满足Jost/Evans极点准则，相关收敛性问题仍待探究。
中文摘要 AI 辅助
紧致化谱离散化可产生稳定的负虚轴（NIA）本征值梯子，但有限矩阵收敛性并不能确定准正规（QNM）极点。对于轴向史瓦西扰动，入-出因式分解会留下一个在Ω=-iα处表现为t^{4α}的多余视界解，以及一个C^∞平坦的多余无穷远解。因此，当4α>k时，端点C^k正则性不具选择性，故C^2连续问题在α≈15-32区间上不可能存在离散NIA谱。精确切比雪夫网格公式给出根网格最大半径O(n^{2p})（对应x~C(1-y)^{-p}），解释了C1（x=2/(1-y)）与C2（x=4/(1-y)^2）的差异。任意精度 pencil 却揭示出可重复的68点C1梯子，而C2会对其进行重组。两个物理侧向Jost行列式在每个C1频率下均保持稳定非零。间隔0.05的割线扫描未发现零点，经细化的辐角原理计算显示两个延拓带内均无零环绕数。这排除了全部68个候选解，尽管网格与带证据未获区间认证。同阻尼控制可恢复史瓦西QNM n=60、100、130，其|D̂|=10^{-48}至10^{-56}。该梯子具有表面重力四分之一间距，遵循无参数Casals-Ottewill割线强度相位，并与QNM n=62,…,129的阻尼投影配对。在其第一个成员处，有限频率计算发现α_q=15.07832396512359处存在割线强度零点，介于渐近预测与C1根之间，支持（但未证明）序列范围的割线相位锁定。因此，精确有限pencil本征值可能不满足不变Jost/Evans极点准则。带有Keldysh权重的依赖表示节点是否会集体收敛至史瓦西割线响应与Price尾，仍待研究。
英文摘要
Compactified spectral discretisations may yield stable negative-imaginary-axis (NIA) eigenvalue ladders, but finite-matrix convergence does not establish quasinormal poles. For axial Schwarzschild perturbations, ingoing--outgoing factorisation leaves an unwanted horizon solution behaving as $t^{4α}$ at $Ω=-iα$, and a $C^\infty$-flat unwanted infinity solution. Thus endpoint $C^k$ regularity is nonselective for $4α>k$, so the $C^2$ continuum problem cannot have a discrete NIA spectrum over $α\simeq15$--32. Exact Chebyshev-grid formulas give roots-grid maximum radius $O(n^{2p})$ for $x\sim C(1-y)^{-p}$, explaining C1, $x=2/(1-y)$, versus C2, $x=4/(1-y)^2$. Arbitrary-precision pencils nevertheless reveal a reproducible 68-point C1 ladder, while C2 reorganises it. Both physical lateral Jost determinants remain stably nonzero at every C1 frequency. A 0.05-spaced on-cut scan finds no zero, and refined argument-principle calculations give zero winding in both continuation strips. This rejects all 68 candidates, although the mesh and strip evidence is not interval-certified. Same-damping controls recover Schwarzschild QNMs $n=60,100,130$, with $|\widehat{\mathcal D}|=10^{-48}$--$10^{-56}$. The ladder has surface-gravity quarter spacing, follows the parameter-free Casals--Ottewill branch-cut-strength phase, and pairs with damping projections of QNMs $n=62,\ldots,129$. At its first member, a finite-frequency calculation finds a branch-strength zero at $α_q=15.07832396512359$, between the asymptotic prediction and the C1 root, supporting, but not proving, sequence-wide cut-phase locking. Thus accurate finite-pencil eigenvalues can fail the invariant Jost/Evans pole criterion. Whether representation-dependent nodes with Keldysh weights converge collectively to the Schwarzschild cut response and Price tail remains open.
发表机构
- Khalifa University of Science and Technology（哈利法大学）
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