  The decay rate $\gam$ of an excited dipole molecule inside a waveguide is
evaluated for the strongly coupled matter-field case near a cutoff frequency
$\ome_c$ without using perturbation analysis. Due to the singularity in the
density of photon states at the cutoff frequency, we find that $\gam$ depends
non-analytically on the coupling constant $\ggg$ as $\ggg^{4/3}$. In contrast
to the ordinary evaluation of $\gam$ which relies on the Fermi golden rule
(itself based on perturbation analysis), $\gam$ has an upper bound and does not
diverge at $\ome_c$ even if we assume perfect conductance in the waveguide
walls. As a result, again in contrast to the statement found in the literature,
the speed of emitted light from the molecule does not vanish at $\ome_c$ and is
proportional to $c\ggg^{2/3}$ which is on the order of $10^3 \sim 10^4$ m/s for
typical dipole molecules.
