Stably Determining a generalised Impedance Obstacle from a Single Far-Field Pattern

Inverse scattering focuses on recovering unknown scatterers from wave measurements. A fundamental challenge is determining whether an inverse obstacle problem can be resolved from a single far-field measurement, a task particularly demanding for non-convex polytope obstacles under generalized impedance boundary conditions and closely linked to the long-standing Schiffer problem. In this paper, we develop a novel \emph{Artificial Test Domain} (ATD) framework for single-measurement inverse scattering of impenetrable polytope obstacles. Based on microlocal analysis near exterior-visible flat boundary patches, this approach transcends traditional methods reliant on observable corners. The ATD framework establishes two primary conceptual advancements: a unified \emph{generalized impedance hyperplane (GIH) exclusion mechanism}, which clarifies the structural role of uniqueness mechanisms, and a unified \emph{qualitative--quantitative principle} for the generalized impedance setting. Quantitatively, the method yields a \emph{far-field--geometry relation} where geometric discrepancy is controlled by far-field error, scaled by a leading ATD coefficient. Qualitatively, the non-vanishing of this coefficient reduces to the exclusion of exterior generalized impedance hyperplanes. Once uniqueness is established, this relation produces sharp stability estimates. Within this framework, the classical stability estimates for the sound-soft and sound-hard cases are recovered as special instances of a much more general stability theory. At the same time, we obtain several new sharp stability results that are of significant importance. These results unify currently available single-measurement uniqueness regimes for polytope geometry and provide new insights into the Schiffer problem across multiple generalized impedance settings.