Darboux Isospectrality Constraints on Quasinormal Modes Deformed by Bumps
Abstract
A localized Gaussian or Pöschl-Teller bump is widely used to probe the sensitivity of Schwarzschild quasinormal modes, where it is typically added to both the Regge-Wheeler and Zerilli potentials. We show that such an additive prescription is generically incompatible with the Darboux transformation connecting the two parity sectors. The reason is that such a transformation restricts a parity-blind bump to a fixed family of bumps with an unavoidable r−2 tail, excluding any Gaussian or Pöschl-Teller profile with arbitrary amplitude, center, or width in the additive prescription. We give the complete classification of Darboux-admissible axial and polar pairs of bumps and propose two consistent scenarios: Darboux generator prescription in which a Gaussian or Pöschl-Teller profile is retained as a generator of the bump pair rather than as the bump itself, and Riccati completion prescription in which a Gaussian or Pöschl-Teller profile is regarded as one of the bump pair and its partner will be solved from the Riccati equation. Our frequency-domain calculations confirm that a finite and spurious axial-polar splitting of quasinormal modes will be produced if the Darboux-consistent scenarios are absent, while the centroid shift of the fundamental mode is nearly unchanged.
Related
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BibTeX
@misc{zhang2026darboux,
title = {{Darboux Isospectrality Constraints on Quasinormal Modes Deformed by Bumps}},
author = {Zhen-Xiao Zhang and Chen Lan and Yan-Gang Miao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07323},
eprint = {2609.07323},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07323},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Darboux Isospectrality Constraints on Quasinormal Modes Deformed by Bumps
AU - Zhen-Xiao Zhang
AU - Chen Lan
AU - Yan-Gang Miao
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07323
UR - https://arxiv.org/abs/2609.07323
AB - A localized Gaussian or Pöschl-Teller bump is widely used to probe the sensitivity of Schwarzschild quasinormal modes, where it is typically added to both the Regge-Wheeler and Zerilli potentials. We show that such an additive prescription is generically incompatible with the Darboux transformation connecting the two parity sectors. The reason is that such a transformation restricts a parity-blind bump to a fixed family of bumps with an unavoidable r^−2 tail, excluding any Gaussian or Pöschl-Teller profile with arbitrary amplitude, center, or width in the additive prescription. We give the complete classification of Darboux-admissible axial and polar pairs of bumps and propose two consistent scenarios: Darboux generator prescription in which a Gaussian or Pöschl-Teller profile is retained as a generator of the bump pair rather than as the bump itself, and Riccati completion prescription in which a Gaussian or Pöschl-Teller profile is regarded as one of the bump pair and its partner will be solved from the Riccati equation. Our frequency-domain calculations confirm that a finite and spurious axial-polar splitting of quasinormal modes will be produced if the Darboux-consistent scenarios are absent, while the centroid shift of the fundamental mode is nearly unchanged.
ER -