Operator-Theoretic Dynamics: Spectral Admissibility and Decoherence at Physical Interfaces

Traditional formulations of quantum dynamics emphasize bulk evolution governed by unbounded differential operators. In this work, we introduce a unified operator-theoretic framework demonstrating that physical interaction, measurement, and macroscopic reality are fundamentally boundary-resolved phenomena. We define "boundary mechanics" as the spectral constraint structure governing admissible interactions across an interface, and "boundary dynamics" as the time evolution of interface-localized couplings. By embedding these concepts into bipartite open quantum systems (System-Bath architectures), we prove that classical macroscopic events emerge strictly as boundary-induced decoherence projections (einselection). Furthermore, we establish that dynamical coherence is preserved subject to a strict spectral admissibility condition, providing a necessary and physically robust sufficient criterion characterized by the non-vanishing of the regularized Birman-Schwinger determinant: Dboundary (z) = det2 (I + K (z) ) != 0. This work goes beyond standard decoherence theory by demonstrating that decoherence alone is insufficient as a criterion for classicality; it must be supplemented by a spectral stability condition at the interface. This yields a closed formalism in which macroscopic observable dynamics are governed entirely by spectrally admissible interface interactions, rigorously bridging Zurek's decoherence theory with the Kato-Rellich theorem and Birman-Schwinger perturbation theory.