Title: Quantitative analysis of nonadiabatic effects in dense H$_3$S and PH$_3$ superconductors,
Abstract: The comparison study of high pressure superconducting state of recently
synthesized H$_3$S and PH$_3$ compounds are conducted within the framework of
the strong-coupling theory. By generalization of the standard Eliashberg
equations to include the lowest-order vertex correction, we have investigated
the influence of the nonadiabatic effects on the Coulomb pseudopotential,
electron effective mass, energy gap function and on the $2\Delta(0)/T_C$ ratio.
We found that, for a fixed value of critical temperature ($178$ K for H$_3$S
and $81$ K for PH$_3$), the nonadiabatic corrections reduce the Coulomb
pseudopotential for H$_3$S from $0.204$ to $0.185$ and for PH$_3$ from $0.088$
to $0.083$, however, the electron effective mass and ratio $2\Delta(0)/T_C$
remain unaffected. Independently of the assumed method of analysis, the
thermodynamic parameters of superconducting H$_3$S and PH$_3$ strongly deviate
from the prediction of BCS theory due to the strong-coupling and retardation
effects.