Dawn and Twilight Time in Quantum Tunneling

Metastable decay exhibits a familiar exponential regime bracketed by early-time deviations and late-time power-law tails. We adopt the real-time, flux-based definition of the decay rate in the spirit of Andreassen et al.\ direct method and present a complete analysis of one-dimensional quantum-mechanical resonance models. We show that the kernel admits a universal pole--plus--branch decomposition and use it to define two computable time scales: a dawn time, when a single resonant contribution starts dominating and exponential decay sets in, and a twilight time, when the branch-cut tail overtakes exponential decay. The latter can be expressed in closed form via the Lambert W function, making its parametric dependence manifest without fitting. For square, modified square, and Pöschl--Teller barriers we obtain simple thick-barrier formulas, clarify the relation ΓT = T_trans between the decay rate Γ, oscillation period T, and transmission probability T_trans, and indicate how our spectral picture can be naturally extended to quantum field theoretic vacuum decay.