Phase Derivative, Spectral Density\\and Windowed Readout:\\Unified Measurement Framework

Phase Derivative, Spectral Density and Windowed Readout: Unified Measurement Framework Auric Version 2.1 November 24, 2025 Abstract Establish unified framework connecting scattering phase derivative, spectral density, and windowed readout. Core formula holding a.e.: φ′(E) π=ρrel(E) = 1 2πtr Q(E) where φscattering phase, ρrel relative spectral density from Birman–Kre˘ın det S= e−2πiξ with ρrel =−ξ′,Q=−iS†∂ESWigner–Smith delay matrix. Windowed readout Rw=Rw(E)[h∗ρrel](E)dE with NPE three-term error decomposition. Applications: quantum metrology, scattering theory, condensed matter. 1 Core Definitions Definition 1.1 (Relative Spectral Density).For scattering pair (H, H0) with S(E) scattering matrix: ρrel(E) = −ξ′(E) = 1 2πi tr(S†∂ES) = 1 2πtr Q(E) where ξspectral shift function, QWigner–Smith delay. Definition 1.2 (Windowed Readout).For window w, kernel h: Rw[ρrel] = ZR w(E) [h∗ρrel](E)dE 2 Main Theorems Theorem 2.1 (Phase–Density Unification).On absolutely continuous spectrum a.e., singlechannel S=e2iφ gives φ′(E) = π ρrel(E) = 1 2tr Q(E). Multi-channel: 1 2πtr Q(E) = ρrel(E) = −ξ′(E). Proof. From Birman–Kre˘ın det S=e−2πiξ get ∂EArg det S=−2πξ′. Jacobi formula ∂Elog det S= tr(S−1∂ES) with unitarity S−1=S†gives tr(S†∂ES) = −2πiξ′. Definition Q=−iS†∂ES yields tr Q= 2πξ′, thus ρrel =−ξ′= (2π)−1tr Q. 1 Theorem 2.2 (Windowed Readout NPE Decomposition).For discrete approximation with step ∆, truncation N: |Rw−b Rw| ≤ |εalias|+|εEM|+|εtail| with εalias = 0 when bandlimited + Nyquist ∆≤π/Ω. Theorem 2.3 (Born Probability as I-Projection).Under alignment condition, Born probability pi=⟨ψ, Eiψ⟩equals I-projection minimizing DKL(p∥q)over constraint set. Theorem 2.4 (Pointer Basis as Ky Fan Minimum).Pointer basis {ek}minimizes Pk⟨ek, Wwek⟩ for window operator Ww=Rw(E)dEA(E)(Ky Fan minimum sum). 3 Applications 3.1 Quantum Metrology Phase derivative measurement via windowed readout provides optimal energy estimation within bandwidth constraints. 3.2 Scattering Theory Wigner–Smith delay directly measurable via phase–energy correlation, connection to Friedel sum rule. 3.3 Condensed Matter Local density of states (LDOS) in mesoscopic systems, quantum point contacts, resonant tunneling. 4 Discussion and Outlook Unified framework established connecting:  Phase derivative φ′(observable)  Spectral density ρrel (theoretical)  Delay trace tr Q(dynamical) via Birman–Kre˘ın–Wigner–Smith chain, with windowed readout providing experimental bridge. Key achievements: 1. Rigorous scale identity formula 2. NPE non-asymptotic error closure 3. Information-geometric Born probability 4. Ky Fan pointer basis characterization Future directions:  Extension to time-dependent scattering 2  Open quantum systems and decoherence  Experimental implementations in quantum optics  Connections to quantum field theory and gravity 3