Article

# Approximate resonance states in the semigroup decomposition of resonance evolution

Journal of Mathematical Physics (Impact Factor: 1.3). 01/2007; DOI: 10.1063/1.2383069

Source: arXiv

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**ABSTRACT:**The states (Schrödinger picture) and observables (Heisenberg picture) in the standard quantum theory evolve symmetrically in time, given by the unitary group with time extending over -∞ < t < +∞. This time evolution is a mathematical consequence of the Hilbert space boundary condition for the dynamical differential equations. However, this unitary group evolution violates causality. Moreover, it does not solve an old puzzle of Wigner: How does one describe excited states of atoms which decay exponentially, and how is their lifetime τ related to the Lorentzian width Γ? These question can be answered if one replaces the Hilbert space boundary condition by new, Hardy space boundary conditions. These Hardy space boundary conditions allow for a distinction between states (prepared by a preparation apparatus) and observables (detected by a registration apparatus). The new Hardy space quantum theory is time asymmetric, i.e, the time evolution is given by the semigroup with t0 <= t < +∞, which predicts a finite "beginning of time" t0, where t0 is the ensemble of time at which each individual system has been prepared. The Hardy space axiom also leads to the new prediction: the width Γ and the lifetime τ are exactly related by τ = hslash/Γ.Journal of Physics Conference Series 04/2013; 428(1):2016-. - [Show abstract] [Hide abstract]

**ABSTRACT:**We use the resonances of the spherical shell potential to present a thorough description of the Gamow (quasinormal) states within the rigged Hilbert space. It will be concluded that the natural setting for the Gamow states is a rigged Hilbert space whose test functions fall off at infinity faster than Gaussians.Journal of Mathematical Physics 10/2012; 53(10). · 1.30 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**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) = ψb(t) + ψf(t) of a backward asymptotic component and a forward asymptotic component such that the evolution process is represented as a transition from ψb(t) to ψf(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.Journal of Mathematical Physics 03/2011; 52(3):032106-032106-28. · 1.30 Impact Factor

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