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Description
To improve the analytical description of the gravitational-wave ringdown phase, this work investigates black hole spectroscopy via the pseudo-spectral method, which solves eigenvalue problems to compute the full frequency spectrum of black holes. Although highly promising, the complete spectrum remains elusive; while quasinormal modes (QNMs) are well understood, the physical nature of the remaining frequency components is yet to be identified. This work addresses a specific part of the frequency domain: the purely imaginary eigenvalues in the very low-frequency regime. We analyze the perturbed Klein-Gordon equation using Green's function methods, focusing on the branch cut solutions that dominate late-time tails. Unlike traditional approaches utilizing the Wronskian, we adopt a Sturm-Liouville-like formulation to construct the Green’s function from the pseudo-spectral eigenvalues and eigenvectors. Through a comparative analysis of these two Green's functions, we aim to definitively map the low-frequency, purely imaginary eigenvalues to the late-time tail solutions.
