The crossover of parametrized Schwarzschild quasinormal-mode overtones: a one-parameter local connection problem, tested numerically

Abstract

Miyachi, Namba, Omiya and Oshita (arXiv:2512.18631) showed with exact WKB that parametrized corrections δV = f Σj αj r−j to the Regge–Wheeler potential change the high-overtone asymptotics of Schwarzschild quasinormal modes. For j = 4, Re ωN grows as N1/5 instead of approaching ln3/4π, and the limits N → ∞ and α → 0 do not commute. We study the regime between the Schwarzschild and divergent behaviours. Near r = 0 and at large |ω|, the problem reduces to a local equation with the single parameter κp = αp ω(p−3)/2. A Motl–Neitzke monodromy argument gives the leading-order condition e4πω = C(κ), where C is a connection coefficient computed numerically. C(0) = −3 recovers Schwarzschild; at large |κ| the growth of C matches the published N1/5 law, but for α4 < 0 that law is only an envelope because |C| dips repeatedly. We compare with pre-registered Leaver continued-fraction spectra (204 grid modes up to N = 3136, plus fixed-α series to N = 8000; every root checked for genuineness). At fixed y = α4(N+½)1/2, residuals in Re ω and Im ω fall as N−1/2 for both signs of α4; for α4 < 0 they extrapolate to within 1.6% of zero, but at the largest N computed they remain 7–28%, so the pre-registered 1% criterion fails. Where the theory predicts |C| = 1, tracked modes reach the imaginary axis: for α4 > 0 the crossing extrapolates to y = 3.79–3.83 (fit-window dependent) against the predicted 3.7936; for α4 = −1 a mode reaches the axis at N ≈ 356 against the predicted 346, a dip the power law misses. A second predicted dip near N ≈ 1.06×104 is found out of sample about 4% earlier in N. For j = 5 the pre-registered exponent test is inconclusive (e = 0.32 ± 0.09 vs 1/3). By-products: Leaver roots that fail an inversion-index test once α ≠ 0, and confirmation of the Musiri–Siopsis N−1/2 coefficient (|γ| = 25.85 vs 25.859). Files: paper (PDF) and source bundle (code, data, pre-registration and amendment log). Computations and drafting were assisted by an AI system (Claude, Anthropic).