Analytical determination of the Cooper pair condensation using linearized solutions of the BCS Hamiltonian

• Analytical solution of the BCS Hamiltonian at the first-order perturbation theory.
• Determination of the number of single electrons, ground-state and excited pairs.
• Superconductivity as a Bose–Einstein condensate of Cooper pairs for weak coupling.

In this paper, we find analytically the first order solutions of the Bardeen, Cooper and Schrieffer (BCS) Hamiltonian with degenerated single-electron energy levels. The results are compared to the Richardson exact solutions calculated numerically, showing good agreement in the weak interaction limit. Using this first-order solution, we further calculate the number of pairs at the ground state as a function of temperature. In particular, the Bose–Einstein condensation (BEC) temperature is found when the population of ground-state pairs starts growing. This study provides a BEC analysis of the superconductivity for weak coupling regime, which traditionally belongs to the BCS side of the BCS–BEC crossover picture.