The conserving approximation scheme due to Baym and Kadanoff (BK) provides a fundamental algorithm for a quantitative treatment of the many-body effects in the framework of the Green's-function method. Takada has noticed that the repeated applications of the BK algorithm produce the exact solution and thus this is regarded as the finding of a basic prescription for the exact self-energy [1]. Takada has also proposed a simplified version of the prescription using the Ward identity and named it GISC [2]. The usefulness of GISC together with the importance of the vertex corrections is illustrated in the polaron problem [3,4]. Based on GISC, Takada has embarked for the construction of a strong-coupling theory of superconductivity with the vertex corrections included properly [5]. 
  Takada and Hotta have obtained a concept of dynamical localization in the course of the investigation into the relation between GISC and the mean-field theory in infinite dimensions [7-9]. Currently, we are developing a method which goes beyond GISC. Employing the method, we begin with an intensive study on the microscopic mechanism of high-temperature superconductivity.