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arXiv · 2609.07323

Darboux Isospectrality Constraints on Quasinormal Modes Deformed by Bumps

Zhen-Xiao Zhang·Chen Lan·Yan-Gang Miao
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Abstract

A localized Gaussian or P\"oschl-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 P\"oschl-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 P\"oschl-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 P\"oschl-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.

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Zhen-Xiao Zhang, Chen Lan, Yan-Gang Miao. 2026-09-07. Darboux Isospectrality Constraints on Quasinormal Modes Deformed by Bumps. https://arxiv.org/abs/2609.07323

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