Black–Hole Echo Resonance Spectra and Source Dependence in a Controlled Transfer–Function Model

Abstract

Echo models phenomenologically encode possible near–horizon structure by replacing the purely ingoing horizon–side condition with an effective reflecting inner boundary near the would–be horizon. We study this idea in a controlled transfer–function model consisting of a compactly supported one–dimensional barrier and a Robin wall at [Formula: see text], where [Formula: see text] is the cavity length measured in the tortoise coordinate. The aim is not to propose a new echo mechanism or to make an observational claim, but to analyze the standard cavity denominator in a controlled model with explicit normalizations. For this model, we prove a local one–zero–per–cell result with an [Formula: see text] localization error and derive a source–to–observer factorization which separates the homogeneous poles from the source–dependent residues. To test the physical robustness of the mechanism, we also perform direct numerical calculations for the untruncated axial Regge–Wheeler potential. The computed resonances form the same nearly equally spaced comb, their deviations from the first asymptotic centers are numerically consistent with [Formula: see text] scaling, and smooth source profiles modify the peak weights without changing the homogeneous pole locations. The rigorous [Formula: see text] theorem remains restricted to the compactly supported model.