  We consider an effect generated by the nonexponential behavior of the
survival amplitude of an unstable state in the long time region: In 1957
Khalfin proved that this amplitude tends to zero as $t$ goes to the infinity
more slowly than any exponential function of $t$. This effect can be described
in terms of time-dependent decay rate $\gamma(t)$ and then the Khalfin result
means that this $\gamma(t)$ is not a constant for long times but that it tends
to zero as $t$ goes to the infinity. It appears that a similar conclusion can
be drawn for the energy of the unstable state for a large class of models of
unstable particles: This energy should be much smaller for suitably long times
$t$ than the energy of this state for $t$ of the order of the lifetime of the
considered state. Within a given model we show that the energy corrections in
the long ($t \to \infty$) and relatively short (lifetime of the state) time
regions, are different. It is shown that these corrections decrease to ${\cal
E} = {\cal E}_{min} < {\cal E}_{\phi}$ as $t \to \infty$, where ${\cal
E}_{\phi}$ is the energy of the system in the state $|\phi>$ measured at times
$t \sim \tau_{\phi}= \frac{\hbar}{\gamma}$. 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.
