Correcting the fatal flaw in BCS theory

Cooper pair is formed by two electrons outside E_F due to the phonon and the screened Coulomb fields[1]. The screened Coulomb field is very weak, which makes the electron density n_N outside E_F small. Thus, the Cooper pair density, n_Cooper, becomes very small, the pair coherence length ξ becomes very large, the double potential structure is made. We assert the weak screened Coulomb field is the fatal flaw in BCS theory not explaining superconductivity[2]. To correct the flaw, we apply the on-site Coulomb interaction U in the Brinkman-Rice(BR) picture[3]. When U increases, discontinuity[=m/m*=1-(U/U_c)²)] at E_F decreases, then n_N rapidly increases, n_Cooper becomes much larger, ξ becomes much smaller, a single potential well structure is generated. Thus T_{c_BR_BCS}[2] can explain the superconductivity. [1]PRL 104(1956)1189. [2]ScientificReport 11(2021)10329. [3]PRB 2(1970)4302.