Physical interpretation of purely imaginary spectral-root families

Determine the physical interpretation of the numerically stable families of purely imaginary roots that arise in the massive scalar quasinormal-mode spectral problem for a Schwarzschild black hole, including whether they constitute a physically meaningful mode family.

Background

The paper develops an extended Chebyshev spectral method for the massive scalar-field quasinormal-mode problem on a Schwarzschild background. Uniformising the two-sheeted dispersion relation produces a seventh-degree polynomial eigenvalue problem and yields numerically stable spectra, including families of roots lying on the purely imaginary frequency axis with approximately uniform damping-rate spacing.

Although these roots pass the paper’s resolution and precision-stability tests, their physical status is unresolved. In particular, the archived frequency data do not retain sufficient information to assign each root unambiguously to the outgoing or reciprocal Riemann sheet or to determine its spatially decaying or growing asymptotics. The paper therefore does not identify the purely imaginary roots as physical quasinormal modes, and their interpretation remains an open problem.