General properties of the evolution of unstable states at long times

TL;DR

This paper investigates the long-time behavior of unstable quantum states, revealing that their survival amplitude decays more slowly than exponential and significantly affects their energy at extended times.

Contribution

It provides a general, model-independent asymptotic analysis of the survival amplitude's long-time behavior for unstable states.

Findings

Survival amplitude decays slower than exponential at long times

Energy of unstable states diminishes significantly at extended times

Asymptotic form derived using asymptotic analysis methods

Abstract

An effect generated by the nonexponential behavior of the survival amplitude of an unstable state at the long time region is considered. It is known that this amplitude tends to zero as $t$ goes to the infinity more slowly than any exponential function of $t$. Using methods of asymptotic analysis we find the asymptotic form of this amplitude in the long time region in a general model independent case. We find that the long time behavior of this amplitude affects the form of the energy of unstable states: This energy should be much smaller for suitably long times $t$ than the energy of this state for $t$ of the order of the lifetime of the considered unstable state.