Free initial wave packets and the long-time behavior of the survival and nonescape probabilities

Abstract

The behavior of both the survival $S(t)$ and the nonescape $P(t)$ probabilities at long times for the one-dimensional free-particle system is shown to be closely connected to that of the initial wave packet at small momentum. We prove that both $S(t)$ and $P(t)$ asymptotically exhibit the same power-law decrease at long times, when the initial wave packet in momentum representation behaves as ${k}^{m}$ with $m=0$ or $1$ at small momentum. On the other hand, if the integer m becomes greater than $1,$ $S(t)$ and $P(t)$ decrease in different power laws at long times.