Superconductivity in $\beta$-pyrochlore superconductor KOs${}_2$O${}_6$: treatment within strong-coupling Eliashberg theory

We study the influence of the rattling phonons on superconductivity in $\beta$-pyrochlore KOs${}_2$O${}_6$ compound based on the strong-coupling Eliashberg approach. In particular, analyzing the specific heat data we find that the rattling phonon frequency changes discontinuously at the critical temperature of the first order phase transition. Solving the strong-coupling Eliashberg equations with temperature dependent ${\alpha }^{2}F(\omega )$, we investigate the consequence of this first order phase transition for the anomalous temperature dependence of the superconducting gap. We discuss our results in context of the recent experimental data.