EFFECTS OF THE VERTEX CORRECTION BY THE METHOD OF NAMBU TO THE ELIASHBERG-MIGDAL EQUATION IN THE STRONG-COUPLING SUPERCONDUCTIVITY UNDER THE ULTRA-HIGH MAGNETIC-FIELD

Beyond Migdal's theorem, the theoretical study on the strong coupling superconductivity under the quantizing magnetic field in the system of isotropic 3-dimensional electron gas is reported. The vertex correcti on is taken into account to the formulation, previously developed by t he present authors in the expression of Dyson-Eliashberg theory, which appears in J. Phys. Soc. Jpn. 63 (1994) 1465. Under the condition tha t Ward-Takahashi identity is exactly satisfied, the set of equations i ncluding the vertex correction up to infinite order is proposed in the context of the theoretical investigation by Nambu in Phys. Rev. 117 ( 1960) 648. Numerical results for demanded strength of the field H for the given concentration of the electron density N are presented in rel ation with the investigation of the problem of how the Nambu-Goldstone mode appears. The possibility of new superconducting phase under the ultra-high field is also discussed.