Quasinormal-mode analysis of a massive scalar field in a Schwarzschild background via the spectral method

Abstract: We study the massive scalar quasinormal-mode spectral problem on a Schwarzschild background using an extended Chebyshev spectral method. The massive dispersion relation has a two-sheeted analytic structure, which we uniformise before discretisation. This converts the radial problem into a seventh-degree polynomial eigenvalue problem while incorporating the quasinormal-mode boundary conditions at the horizon and spatial infinity. The method yields stable spectra for low multipoles, high overtones, and frequencies on both sides of the nominal mass threshold, including regimes in which WKB and continued-fraction approaches become difficult to apply. For complex frequencies, the far-field behaviour cannot be classified from the real part of the frequency alone, but depends on the complex wavenumber and its Riemann sheet. We also identify stable families of purely imaginary roots with approximately uniform spacing in their damping rates, whose physical interpretation remains open. The Schwarzschild scalar problem is intended as a controlled benchmark for massive-field quasinormal-mode calculations rather than as a direct model of tensorial Kerr ringdown.