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

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 a large negative 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 benchmark model with explicit normalizations. All rigorous $O(L^{-2})$ localization estimates are proved for this compactly supported model.