# 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.