Transition representations of quantum evolution with application to scattering resonances

A Lyapunov operator is a self-adjoint quantum observable whose expectation value varies monotonically as time increases and may serve as a marker for the flow of time in a quantum system. In this paper it is shown that the existence of a certain type of Lyapunov operator leads to representations of the quantum dynamics, termed transition representations, in which an evolving quantum state ψ(t) is decomposed into a sum ψ(t) = ψ^b(t) + ψ^f(t) of a backward asymptotic component and a forward asymptotic component such that the evolution process is represented as a transition from ψ^b(t) to ψ^f(t). When applied to the evolution of scattering resonances, such transition representations separate the process of decay of a scattering resonance from the evolution of outgoing waves corresponding to the probability "released" by the resonance and carried away to spatial infinity. This separation property clearly exhibits the spatial probability distribution profile of a resonance. Moreover, it leads to the definition of exact resonance states as elements of the physical Hilbert space corresponding to the scattering problem. These resonance states evolve naturally according to a semigroup law of evolution.