Negative Dwell Times as Resolvent Derivatives: A Structural Operator-Theoretic Explanation of 𝑻𝑻=𝒕𝒈

This dataset and manuscript present a structural operator‑theoretic explanation of negative group delays and post‑selected excitation times in resonant atomic media. The work demonstrates that the equality TT=tg arises naturally from the shared analytic structure of the internal resolvent R(ω)=(ω0−ω−iΓ/2)−1, which governs susceptibility, transmission amplitude, and excited‑state dynamics. A self‑contained conditional‑probability derivation (§5.7) establishes TT=Im[∂ωln⁡t(ω)]ω0 directly from the resolvent, closing the logical gap between quantum‑trajectory and structural proofs. Negative values of both quantities near resonance are shown to result from spectral reshaping, not superluminal propagation. Figures and appendices illustrate the Lorentzian resonance, group‑delay behavior, and causal envelope reshaping in a dilute two‑level medium. Keywords: group delay, weak value, resolvent, Lorentzian resonance, anomalous dispersion, causality, operator theory, quantum optics