Energy of unstable states at long times

Abstract

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 $γ(t)$ Khalfin's result means that this $γ(t)$ 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 $σ$ meson as well as having astrophysical and cosmological consequences.