Quasinormal-mode encounters: Pseudospectra near Kerr--Newman exceptional points

Exceptional points (EPs), at which two quasinormal frequencies and their eigenvectors coalesce, can strongly enhance the spectral instability of black hole quasinormal modes. We investigate this behavior in Kerr--Newman spacetimes through pseudospectra and condition numbers for neutral massless scalar fields and coupled gravitoelectromagnetic (GE) perturbations. For the scalar field, we follow the $(\ell,m)=(1,1)$ spheroidal branch and derive the energy norm from the stress-energy tensor. For GE perturbations, we use a coupled four-field system equipped with a positive radiative energy. At the selected near-EP backgrounds, the condition numbers of both modes in the close pair exceed those of the two more-damped modes shown for comparison at the same background. Along a matched GE parameter path, the frequency splitting follows a square-root law and the condition numbers grow as the inverse square root of the distance to the EP. The pseudospectral contours resolve the transition from individual-mode sensitivity to the collective response of the pair. Within the discretized problems and energy norms considered here, these results quantify the amplification of spectral instability near EPs and provide a basis for studying their contribution to black-hole ringdown.