  Using the Eliashberg theory of superconductivity we have examined several
properties of a model in which electrons are coupled only to rattling phonon
modes represented by a sharp peak in the electron-phonon coupling function. Our
choice of parameters was guided by experiments on $\beta$-pyrochlore oxide
superconductor KOs$_{2}$Os$_{6}$. We have calculated the temperature dependence
of the superconducting gap edge, the quasiparticle decay rate, the NMR
relaxation rate assuming that the coupling between the nuclear spins and the
conduction electrons is via a contact hyperfine interaction which would be
appropriate for the O-site in KOs$_{2}$Os$_{6}$, and the microwave
conductivity. We examined the limit of very strong coupling by considering
three values of the electron-phonon coupling parameter $\lambda=$ 2.38, 3, and
5 and {\em did not} assume that the rattler frequency $\Omega_{0}$ is
temperature dependent in the superconducting state. We obtained a very unusual
temperature dependence of the superconducting gap edge $\Delta(T)$, very much
like the one extracted from photoemission experiments on KOs$_2$O$_6$.
