Pair-Dependent Drift of Kerr Neighboring-Overtone Gap Minima

TL;DR

This paper investigates the behavior of Kerr quasinormal-mode overtones, focusing on the minima of their complex-frequency separation and how these minima shift with spin, revealing insights into spectral pole proximity.

Contribution

It introduces a diagnostic based on the spin derivative of the squared separation to analyze minima shifts in Kerr quasinormal modes without requiring differentiation of the modulus.

Findings

Interior minima are identified in the frequency separation.

Minima locations shift between neighboring mode pairs with spin.

The minima align with dominant zeros of the diagnostic and radial turning points.

Abstract

We study adjacent Kerr quasinormal-mode overtones under a spin scan with overtone labels held fixed, using a public Leaver-type solver on a uniform grid. The observable is the modulus of the complex-frequency separation between neighbors; its minima are analyzed through the spin derivative of the squared separation, which supplies a smooth real diagnostic without differentiating the modulus itself. Clear interior minima appear, but their spin locations shift between neighboring pairs even within one \((s,\ell,m)\) sector and align with dominant zeros of the diagnostic and with radial turning of the separation vector in the complex-frequency plane. Representative extra sectors and smooth no-trigger cases support selectivity. Minimum drift is naturally read as drift of that dominant zero; the language connects to complex-spectral pole proximity for Kerr flows without identifying each minimum with an exceptional-point coalescence or claiming a universal rule over the full spectrum.