  We investigate the survival probability of a localized 1-d quantum particle
subjected to a time dependent potential of the form $rU(x)\sin{\omega t}$ with
$U(x)=2\delta (x-a)$ or $U(x)= 2\delta(x-a)-2\delta (x+a)$. The particle is
initially in a bound state produced by the binding potential $-2\delta (x)$. We
prove that this probability goes to zero as $t\to\infty$ for almost all values
of $r$, $\omega$, and $a$. The decay is initially exponential followed by a
$t^{-3}$ law if $\omega$ is not close to resonances and $r$ is small; otherwise
the exponential disappears and Fermi's golden rule fails. For exceptional sets
of parameters $r,\omega$ and $a$ the survival probability never decays to zero,
corresponding to the Floquet operator having a bound state. We show similar
behavior even in the absence of a binding potential: permitting a free particle
to be trapped by harmonically oscillating delta function potential.
