Continuum transition probability at long times
= Continuum transition probability at long times
For a harmonic perturbation and continuum detuning $\Omega(k)$, first-order amplitudes contain
$$
\frac{\sin(\Omega(k)t/2)}{\Omega(k)/2}.
$$
The distributional limit
$$
\frac{\sin^2(\Omega t/2)}{(\Omega/2)^2}
\longrightarrow 2\pi t\,\delta(\Omega)
$$
selects energy-conserving continuum states and produces a transition probability linear in time.