Difference between the decay forms of the survival and nonescape probabilities

The behavior of both the survival $S(t)$ and nonescape $P(t)$ probabilities at long times for one and three dimensional free particle system is shown to be closely connected to that of initial states at small momentum. We show that $S(t)$ and $P(t)$ exhibit similar asymptotic decay forms at long times, when the initial state $\psi$ satisfies $\hat{\psi} (k)=O(k^m)$ with $m =0$ or 1 at small momentum. However, if $\hat{\psi} (k)=O(k^m)$ with an integer $m \geq 2$, $S(t)$ and $P(t)$ decay in different ways at long times