Metal hydrogen critical temperature at the pressure 500 GPa
  \'Eliashberg theory, being generalized to the case of the electron-phonon (EP) systems with the not constant density of electronic states, as well as to the frequency behavior of the renormalization of both the electron mass and the chemical potential, is used to study the normal properties and of the metal hydrogen phase under pressure of P = 500 GPa. The phonon contribution to the anomalous electron Green's function (GF) is considered. The pairing is considered within the overall width of the electron band, and not only in a narrow layer at the Fermi surface. The frequency dependence of the electron mass renormalization ReZ , the density of electronic states , the spectral function of the electron-phonon interaction, obtained by calculation are used to calculate the electronic abnormal GF. The \'Eliashberg generalized equations with the variable density of electronic states are resolved. The dependence of both the real and the imaginary parts of the order parameter on the frequency in the phase is obtained. The value for the metal hydrogen phase at the pressure P = 500 GPa has been defined. 
Nikolay Kudryashov, Alexander Kutukov, Evgeny Mazur
cond-mat.supr-con