Application of the Master Formula to Nuclear Dynamics and Scattering: Amplitude–Phase Unification, Non-Hermitian Effects, and Scattering Observables

This work extends the Master Formula for amplitude–phase dynamics, originally derived from the time-dependent Schrödinger equation (TDSE) with dissipative and decoherence effects, to nuclear many-body systems and scattering theory. We map the Master Formula components—baseline amplitude (Φ), phase generator (Ω), and dissipative correction (Γ)—to effective Hamiltonian terms, complex optical potentials, and Lindblad-type environmental operators. This mapping yields compact expressions connecting coherence dynamics to standard scattering observables, including phase shifts, S-matrix elements, and cross sections. Analytic results for single-channel scattering are presented, along with practical recipes for numerical evaluation and generalizations to multichannel resonances. The unified framework clarifies how non-unitary and decoherence effects modify scattering amplitudes and provides a toolkit for applying the Master Formula in nuclear physics research.