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:**For selected classes of quantum mechanical scattering systems a canonical association of a decay semigroup is presented. The spectrum of the generator of this semigroup is a pure eigenvalue spectrum and it coincides with the set of all resonances. The essential condition for the results is the meromorphic continuability of the scattering matrix onto ∖(−∞,0] and the rims −±i0. Further finite multiplicity is assumed. The approach is based on an adaption of the Lax–Phillips scattering theory to semibounded Hamiltonians. It is applied to trace class perturbations with analyticity conditions. A further example is the potential scattering for central-symmetric potentials with compact support and angular momentum 0.Journal of Mathematical Physics 11/2010; 51(11):113508-113508-20. · 1.30 Impact Factor - [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:**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-.

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