Establish the perturbative expansion of the radial Floquet exponent

Establish a consistent perturbative expansion for the radial Floquet exponent \(\sigma\) of the five-singular-point Fuchsian radial equation governing scalar perturbations of the seven-dimensional Kerr–AdS black hole, in order to identify the poles of the retarded Green’s function and determine the imaginary parts of the quasinormal-mode frequencies.

Background

The paper computes accessory-parameter expansions for the radial Fuchsian equation and uses them to determine the Floquet exponent ρ\rho, which enters the radial quantization condition for the real parts of the quasinormal-mode frequencies. However, when the Floquet exponents are re-expressed in terms of the black-hole parameters, formally higher-order terms in the conformal moduli contribute at lower perturbative orders. The authors state that this prevents one Floquet exponent from being determined consistently order by order.

The imaginary parts of the quasinormal-mode frequencies are associated with poles of the retarded Green’s function, obtained from the vanishing of the connection coefficient multiplying the non-normalizable solution at the AdS boundary. Determining these poles requires a reliable perturbative expansion of σ\sigma, so completing that expansion is necessary to extend the paper’s results from the real parts to the full quasinormal-mode spectrum.

References

As a result, one of the Floquet exponents cannot be determined consistently order by order. Interestingly, the same feature occurs for different Möbius transformations, corresponding to distinct parameter regimes, suggesting that it is intrinsic to the asymptotically AdS black hole rather than to a particular parametrization of the Fuchsian equation. Nevertheless, the Floquet exponent entering the radial quantization condition remains unaffected and can therefore be used to determine the real part of the QNM frequencies. We obtain a perturbative expansion including terms up to fourth order in u_{3}. Our results indicate that the first corrections due to the rotation parameters appear at sixth total order in the perturbative expansion.

The imaginary part of the QNM frequencies requires determining the poles of the retarded Green's function, which correspond to the vanishing of the connection coefficient associated with the non-normalizable solution. In particular, identifying these poles depends on the precise computation of the Floquet exponent \sigma, whose perturbative expansion remains to be fully established.

— Scalar quasinormal modes of Kerr--AdS$_{\bf 7}$ via accessory parameter expansions  (2609.03037 - Amado, 2 Sep 2026) in Section 4, Discussion