Schwarzschild spectral ladders on the negative imaginary axis: Endpoint nonselection, branch-cut phase, and Jost classification
Abstract
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 t4α at Ω=−iα, and a C∞-flat unwanted infinity solution. Thus endpoint Ck regularity is nonselective for 4α>k, so the C2 continuum problem cannot have a discrete NIA spectrum over α≃15--32. Exact Chebyshev-grid formulas give roots-grid maximum radius O(n2p) for x∼ 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 |𝒟̂|=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,…,129. At its first member, a finite-frequency calculation finds a branch-strength zero at αq=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.
Related
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BibTeX
@misc{batic2026schwarzschild,
title = {{Schwarzschild spectral ladders on the negative imaginary axis: Endpoint nonselection, branch-cut phase, and Jost classification}},
author = {Davide Batic and Denys Dutykh and Mark Essa Sukaiti},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29095},
eprint = {2608.29095},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29095},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Schwarzschild spectral ladders on the negative imaginary axis: Endpoint nonselection, branch-cut phase, and Jost classification
AU - Davide Batic
AU - Denys Dutykh
AU - Mark Essa Sukaiti
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29095
UR - https://arxiv.org/abs/2608.29095
AB - 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^∞-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≃15--32. Exact Chebyshev-grid formulas give roots-grid maximum radius O(n^2p) for x∼ 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 |𝒟̂|=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,…,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.
ER -