On bifurcation and spectral instability of asymptotic quasinormal modes in the modified P\"oschl-Teller effective potential

The P\"ochl-Teller effective potential mimics an asymptotically de Sitter black hole bounded by an event horizon and a cosmological one. Owing to the benefit of being analytically soluble, the asymptotic quasinormal modes in the modified P\"oschl-Teller potential have been extensively explored in the literature by various authors, and the results bear distinct features. Specifically, for small discontinuities placed at the potential's peak, Skakala and Visser showed that the resulting modes lie primarily along the imaginary frequency axis, in line with the numerical results encountered for most black hole metrics. However, it was also suggested that under ultraviolet perturbations, asymptotic modes are expected to lie parallel to the real axis, closely intervening with recent developments on spectral instability. In this work, by numerical and semi-analytical approaches, we aim to resolve the above apparent ambiguity. The numerical scheme is based on an improved version of the matrix method, which is implemented in compactified hyperboloidal coordinates on the Chebyshev grid. It is demonstrated that both asymptotic behaviors indeed agree with the numerical findings, which is somewhat to one's surprise. Specifically, we report the emergence of a novel branch of purely imaginary modes originating from a bifurcation in the asymptotic quasinormal mode spectrum. Moreover, we demonstrate how the bifurcation and asymptotic modes evolve as the discontinuity moves away from the potential's peak, furnishing a dynamic picture as the spectral instability unfolds. It is further argued that they can be partly attributed to the observed parity-dependent deviations occurring for the low-lying perturbed modes of the original P\"oschl-Teller effective potential.