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The high-temperature superconducting state in sulfur trihydride($T_{C}=203$~K) has been investigated in the context of the non-adiabatic andanharmonic effects. The Migdal-Eliashberg equations and the extended Eliashbergequations, which include the lowest-order vertex corrections, have been solvednumerically in the self-consistent way. For $R3m$ crystal structure, thelowest-order vertex corrections decrease the value of the Coulombpseudopotential from $0.123$ to $0.108$. The anharmonic effects workantagonistically in relation to the vertex corrections shifting the value of$\mu^{\star}$ to $0.156$. The studies conducted for the structure$Im\overline{3}m$, where the Eliashberg function includes both thenon-adiabatic and anharmonic effects, prove the even higher value of$\mu^{\star}=0.185$. Independently of the assumed method of the analysis, thenearly identical no mean-field dependence of the order parameter on thetemperature was obtained: $2\Delta(0)/k_{B}T_{C}\sim 4.7$ - due to thesignificant strong-coupling and retardation effects: $\lambda\sim 2$ and$k_{B}T_{C}\slash \omega_{\rm \ln}\sim 0.15$-$0.19$. It means that theclassical equations of Migdal-Eliashberg can be treated as a correct effectivemodel for the superconducting state in $\rm H_{3}S$. This paper has shown thatthe McMillan or Allen-Dynes formulas substantially lower the value of thecritical temperature in relation to the result obtained with the Eliashbergequations.

Migdal-Eliashberg equations - the effective model for superconducting state in H3S