Scalar quasinormal and quasibound states of weakly magnetized Ernst--Schwarzschild spacetime: a confluent-Heun Wronskian approach

RESUMO

We study the quasinormal-mode and quasibound-state spectra of a massless scalar field in weakly magnetized Ernst--Schwarzschild spacetime. In the separated $O(B_0^2)$ regime, the Klein--Gordon equation reduces to a radial confluent-Heun equation. We construct an endpoint-selected spectral Wronskian directly from the canonical confluent-Heun recurrence and its regular-to-irregular connection limit. Quasinormal modes and quasibound states are then obtained as zeros of the same analytic spectral function on the radiative and bound sheets, respectively, providing a unified treatment of the two spectral sectors within the separated weak-field problem. The resulting QNM and QBS frequencies agree with independent weak-field Ernst and massive-Schwarzschild benchmarks available in the literature. We further show, for representative cases, that the frequencies selected by the local Heun-series termination relation examined here do not satisfy the endpoint-selected quasibound-state condition. The construction therefore provides a direct horizon-to-infinity spectral condition for both radiative and bound states within the weak-field Ernst problem.