Unified Evolution Equation

The Unified Evolution Equation (UEE) provides a common analytical framework that unifies reversible quantum dynamics (unitary evolution), dissipative dynamics of open systems (GKLS), and transport effects induced by boundaries and resonances (zero-area resonance kernels) as a single notion of time evolution of states. The purpose of this paper (UEE_01) is to define the UEE as a mathematically consistent analytical foundation and to establish its well-posedness, including existence, uniqueness, and invariance of states.We formulate the theory by taking the observable algebra as a von Neumann algebra and the state space as its predual, and by characterizing physically admissible time evolutions as preduals of normal, unital, completely positive maps. The UEE is formally expressed as a sum of reversible, dissipative, and resonance-transport generators. Rigorously, solutions are defined in the mild sense as trajectories generated by a strongly continuous completely positive and trace-preserving (CPTP) semigroup.Given the analytical data of the UEE, we construct the reversible, dissipative, and resonance-transport components separately as CPTP group or semigroup evolutions. Using a Chernoff/Trotter-type product formula, we prove that the composite limit evolution exists, forms a CPTP semigroup, and that its generator coincides with the closure of the sum of the individual generators. As a consequence, invariance of the set of normal states and the well-posedness of the UEE are rigorously established.This work provides a solid analytical foundation for the unified GKLS+$R$ representation employed in subsequent papers, ensuring consistency between physical modeling and operator-theoretic dynamics.