Energy of unstable states at long times
  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 $t\to\infty$ more slowly than any
exponential function of $t$. For a time-dependent decay rate $\gamma(t)$
Khalfin's result means that this $\gamma(t)$ is not a constant for large $t$
but that it tends to zero as $t\to\infty$. We find that a similar conclusion
can be drawn for the instantaneous energy of the unstable state for a large
class of models of unstable states: This energy tends to the minimal energy of
the system ${\cal E}_{min}$ as $t\to\infty$ which is much smaller than the
energy of this state for $t$ of the order of the lifetime of the considered
state. Analyzing the transition time region between exponential and
non-exponential form of the survival amplitude we find that the instantaneous
energy of a considered unstable state can take large values, much larger than
the energy of this state for $t$ from the exponential time region. Taking into
account results obtained for a model considered, it is hypothesized that this
purely quantum mechanical effect may be responsible for the properties of broad
resonances such as $\sigma$ meson as well as having astrophysical and
cosmological consequences.
