Characterization of overtone numbers near exceptional points

Characterize the overtone numbers of Kerr-de Sitter quasinormal-mode curves near the exceptional point where the n=5 and n=6 modes exchange Riemann sheets, including a rigorous criterion for assigning overtone labels when the corresponding radial eigenfunctions are nearly identical.

Background

The paper studies the exceptional point formed by the Kerr-de Sitter quasinormal modes with (ℓ,m)=(2,2) and nominal overtone numbers n=5 and n=6. Near this exceptional point, the quasinormal-mode frequencies behave as branches of a multivalued function and exchange Riemann sheets when parameter-space paths cross a branch cut.

Because the radial profiles of the two eigenfunctions become very similar near the exceptional point, the authors explain that the overtone labeling is not intrinsically unambiguous there. They therefore label modes by their proximity to the corresponding Kerr modes, but explicitly state that a rigorous characterization of the overtone numbers remains unresolved.