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Long time properties of the evolution of an unstable state

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An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. In 1957 Khalfin proved that this amplitude tends to zero as t → ∞ more slowly than any exponential function of t. This can be described in terms of the time-dependent decay rate γ(t) which, when considered with the Khalfin result, means that this γ(t) is not a constant for large t but that it tends to zero as t → ∞. We find that a similar conclusion can be drawn for a large class of models of unstable states for a quantity, which can be interpreted as the “instantaneous energy” of the unstable state. This energy should be much smaller for suitably larger values of t than when t is of the order of the lifetime of the considered state. Within a given model we show that the energy corrections in the long (t → ∞) and relatively short (lifetime of the state) time regions, are different. This is a purely quantum mechanical effect. It is hypothesized that there is a possibility to detect this effect by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of astrophysical and cosmological parameters.
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