Covariant Multi-Channel Windowed Scattering:\\ Phase--Density Unified Theorem

Covariant Multi-Channel Windowed Scattering: Phase–Density Unified Theorem Auric (S-series / EBOC) Version 1.2 November 24, 2025 Abstract For multi-channel scattering with N(E) channels at energy E, establish unified framework connecting scattering phase derivative, relative spectral density, and Wigner– Smith delay. Core formula holding a.e. on absolutely continuous spectrum: 1 N(E) d dE Arg det S(E) = 1 2π⟨tr Q(E)⟩N=ρrel(E) where Q(E) = −iS†(E)∂ES(E) is Wigner–Smith delay matrix, ⟨·⟩Ndenotes perchannel average, ρrel relative spectral density from Birman–Kre˘ın formula det S= e−2πiξ. Establish: (i) Channel covariance: formula invariant under unitary channel basis transformations; (ii) Threshold regularity: handle channel number jumps via proper boundary conditions; (iii) Windowed readout: finite-bandwidth measurement protocol with NPE error closure; (iv) Information geometry: Born probability as I-projection, pointer basis as Ky Fan minimum. 1 Setup and Notation Energy-dependent channel number N(E) with threshold set T={E:N(E+)=N(E−)}. Between thresholds, scattering matrix S(E)∈U(N(E)) unitary, differentiable a.e. Birman–Kre˘ın convention: det S(E) = e−2πiξ(E)where ξspectral shift function, ρrel(E) = −ξ′(E) relative spectral density. Wigner–Smith delay:Q(E) = −iS†(E)∂ES(E) self-adjoint, eigenvalues τj(E) individual channel delays. 2 Main Results Theorem 2.1 (Multi-Channel Phase–Density Unification).On threshold-regular intervals I (where I∩ T =∅), have a.e.: 1 2πtr Q(E) = d dE Arg det S(E) = −2π ξ′(E) = ρrel(E). Per-channel average delay ⟨τ⟩N(E) = 1 N(E)tr Q(E)satisfies ⟨τ⟩N(E) = 2πℏρrel(E) (restoring ℏ). 1 Proof. From det S=e−2πiξ get ∂Elog det S=−2πi ξ′. By Jacobi formula ∂Elog det S= tr(S−1∂ES) = tr(S†∂ES) (using unitarity). Thus tr(S†∂ES) = −2πi ξ′, giving tr Q=−itr(S†∂ES) = 2π ξ′. Birman–Kre˘ın ρrel =−ξ′completes chain. Theorem 2.2 (Channel Basis Covariance).Under unitary channel transformation U∈ U(N),S7→ ˜ S=USU†, have tr Q˜ S(E) = tr QS(E), ρrel[˜ S](E) = ρrel[S](E). Windowed readout Nw[S;E0] = Rw(E−E0)ρrel[S](E)dE invariant. Proof. Trace invariant under similarity. Spectral shift function gauge-invariant. Theorem 2.3 (Threshold Boundary Conditions).At threshold E∗∈ T where N(E− ∗) = N−, N(E+ ∗) = N+with N+> N−, impose: 1. Phase continuity:Arg det S(E)continuous at E∗after proper branch choice 2. Density regularization:ρrel(E)may have δ-function contribution at E∗from bound states entering/leaving continuous spectrum (Levinson theorem) 3. Windowed measurement: choose window wwith w(E∗−E0)sufficiently small or smooth to regularize threshold singularities 3 Windowed Readout and NPE Error Theorem 3.1 (NPE Three-Term Decomposition for Multi-Channel).For windowed readout with window w, kernel h, sampling step ∆E, truncation N: Nw[S;E0] = Zw(E−E0)[h∗ρrel](E)dE discrete approximation b N= ∆E N X n=−N w(En−E0)[h∗ρrel](En) satisfies error decomposition |Nw−b N| ≤ |εalias|+|εEM|+|εtail| with alias εalias = 0 when bandlimited + Nyquist. 4 Information Geometry and Pointer Basis For multi-channel measurement: Born probability: Measurement outcome probabilities pi=⟨ψ, Eiψ⟩for POVM {Ei}. I-projection: Minimal KL-divergence minp∈C DKL(p∥q) over constraint set Cyields exponential family. Pointer basis: Windowed operator Ww=Rw(E)dEA(E) minimal eigensubspace (Ky Fan) determines pointer basis. 2 5 Discussion Established for multi-channel scattering:  Phase–density–delay unification via Birman–Kre˘ın–Wigner–Smith  Channel basis covariance and threshold regularity  Windowed readout with NPE non-asymptotic error closure  Information-geometric Born probability and pointer basis Applications: quantum optics multi-mode scattering, mesoscopic transport, nuclear reactions, gravitational multi-polarization. 3