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The resonant state of the open quantum system is studied from the viewpoint of the outgoing momentum flux. We show that the number of particles is conserved for a resonant state, if we use an expanding volume of integration in order to take account of the outgoing momentum flux; the number of particles would decay exponentially in a fixed volume of integration. Moreover, we introduce new numerical methods of treating the resonant state with the use of the effective potential. We first give a numerical method of finding a resonance pole in the complex energy plane. The method seeks an energy eigenvalue iteratively. We found that our method leads to a super-convergence, the convergence exponential with respect to the iteration step. The present method is completely independent of commonly used complex scaling. We also give a numerical trick for computing the time evolution of the resonant state in a limited spatial area. Since the wave function of the resonant state is diverging away from the scattering potential, it has been previously difficult to follow its time evolution numerically in a finite area.

We consider the problem of the meaning of quantum unstable states including their dressing. According to both Dirac and Heitler this problem has not been solved in the usual formulation of quantum mechanics. A precise definition of excited states is still needed to describe quantum transitions. We use our formulation given in terms of density matrices outside the Hilbert space. We obtain a dressed unstable state for the Friedrichs model, which is the simplest model that incorporates both bare and dressed quantum states. The excited unstable state is derived from the stable states through analytic continuation. It is given by an irreducible density matrix with broken time symmetry. It can be expressed by a superposition of Gamow density operators. The main difference from previous studies is that excited states are not factorizable into wave functions. The dressed unstable state satisfies all the criteria that we can expect: it has a real average energy and a nonvanishing trace. The average energy differs from Green's function energy by a small effect starting with fourth order in the coupling constant. Our state decays following a Markovian equation. There are no deviations from exponential decay neither for short nor for long times, as is the case for the bare state. The dressed state satisfies an uncertainty relation between energy and lifetime. We can also define dressed photon states and describe how the energy of the excited state is transmitted to the photons. There is another very important aspect: deviations from exponential decay would be in contradiction with indiscernibility as one could define, e.g., old mesons and young mesons according to their lifetime. This problem is solved by showing that quantum transitions are the result of two processes: a dressing process, discussed in a previous publication, and a decay process, which is much slower for electrodynamic systems. During the dressing process the unstable state is prepared. Then the dressed state decays in a purely exponential way. In the Hilbert space the two processes are not separated. Therefore it is not astonishing that we obtain for the unstable dressed state an irreducible density matrix outside the Liouville-Hilbert-space. This is a limit of Hilbert space states that are arbitrarily close to the decaying state. There are experiments that could verify our proposal. A typical one would be the study of the line shape, which is due to the superposition of the short-time process and the long-time process. The long-time process taken separately leads to a much sharper line shape, and avoids the divergence of the fluctuation predicted by the Lorentz line shape.

The charge transfer from an adatom to a semiconductor substrate of one-dimensional quantum dot array is evaluated theoretically. Due to the Van Hove singularity in the density of electron states at the band edges, the charge transfer decay rate is enhanced nonanalytically in terms of the coupling constant g as g 4/3 . The optical absorption spectrum for the ionization of a core level electron of the adatom to the conduction band is also calculated. The reversible non-Markovian process and irreversible Markovian process in the time evolution of the adatom localized state manifest themselves in the absorption spectrum through the branch point and pole contributions, respectively.

Key words non-Markovian decay, power law decay, bound stateIt is known that quantum systems yield non-exponential (power law) decay on long time scales, associated with continuum threshold effects contributing to the survival probability for a prepared initial state. For an open quantum system consisting of a discrete state coupled to continuum, we study the case in which a discrete bound state of the full Hamiltonian approaches the energy continuum as the system parameters are varied. We find in this case that at least two regions exist yielding qualitatively different power law decay behaviors; we term these the long time 'near zone' and long time 'far zone.' In the near zone the survival probability falls off according to a t −1 power law, and in the far zone it falls off as t −3 . We show that the timescale TQ separating these two regions is inversely related to the gap between the discrete bound state energy and the continuum threshold. In the case that the bound state is absorbed into the continuum and vanishes, then the time scale TQ diverges and the survival probability follows the t −1 power law even on asymptotic scales. Conversely, one could study the case of an anti-bound state approaching the threshold before being ejected from the continuum to form a bound state. Again the t −1 power law dominates precisely at the point of ejection.

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