# Survival amplitude, instantaneous energy and decay rate of an unstable system: Analytical results Source: [Survival amplitude, instantaneous energy and decay rate of an unstable system: Analytical results]() Collection index: [llms.txt]() Data updated: 2025-10-24T23:03:42.949Z Authors: K. Raczynska, K. Urbanowski Published: 2018-02-02T15:51:20.000Z ## Abstract 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()