Ces\`aro convergence of the high-order WKB method and its applications to black-hole overtones and long-lived modes

We develop a fully automatic Mathematica implementation of the black-hole WKB method at very high orders based on the Bender-Wu algorithm, which in principle is limited only by memory and computational time, and show that when pushed to sufficiently high order and improved by diagonal Pad\'e approximants the method becomes efficient for two regimes which are usually regarded as difficult for the standard low-order WKB treatment: the first several overtones with n>l and the very long-lived quasinormal modes of massive fields. At the same time, we show that this efficiency has a nontrivial limitation: for black-hole metrics belonging to the non-moderate class, especially when higher coefficients of the near-horizon parametrization become large, the WKB sequence may exhibit an apparent convergence to values which are nevertheless far from the accurate quasinormal frequencies. Thus, numerical stabilization of the WKB output alone is not always a sufficient criterion of correctness. However, we observe that although the WKB method with diagonal or near-diagonal Pad\'e approximants does not exhibit monotonic convergence order by order, the corresponding Ces\`aro means become monotonically convergent once a sufficiently high WKB order is reached. This behavior may serve as an internal WKB criterion for the convergence of the method.