Survival amplitude, instantaneous energy and decay rate of an unstable system: Analytical results. (arXiv:1802.01441v1 [quant-ph])

We consider a model of a unstable state defined by the truncated Breit-Wigner
energy density distribution function. An analytical form of the survival
amplitude $a(t)$ of the state considered is found. Our attention is focused on
the late time properties of $a(t)$ and on effects generated by the
non--exponential behavior of this amplitude in the late 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
using a time-dependent decay rate $\gamma(t)$ and then the Khalfin result means
that this $\gamma(t)$ is not a constant but at late times it tends to zero as
$t$ goes to the infinity. It appears that the energy $E(t)$ of the unstable
state behaves similarly: It tends to the minimal energy $E_{min}$ of the system
as $t \to \infty$. Within the model considered we find two first leading time
dependent elements of late time asymptotic expansions of $E(t)$ and $\gamma
(t)$. We discuss also possible implications of such a late time asymptotic
properties of $E(t)$ and $\gamma (t)$ and cases where these properties may
manifest themselves.

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