When both two-electron \textit{ and} two-hole Cooper-pairing are treated on an equal footing in the ladder approximation to the Bethe-Salpeter (BS) equation, the zero-total-momentum Cooper-pair energy is found to have two \textit{ real} solutions $\mathcal{ E}_{ 0}^{ BS}=\pm 2\hbar \omega_{ { D}%}/\sqrt{ { e}^{ 2/\lambda }+{ 1}}$ which coincide with the zero-temperature BCS energy gap $\Delta =\hbar \omega_{ D}/\sinh (1/\lambda) $ in the weak coupling limit. Here, $\hbar \omega_{ D}$ is the Debye energy and $\lambda \geq 0$ the BCS model interaction coupling parameter. The interpretation of the BCS energy gap as the binding energy of a Cooper-pair is often claimed in the literature but, to our knowledge, never substantiated even in weak-coupling as we find here. In addition, we confirm the two purely-\textit{ imaginary} solutions assumed since at least the late 1950s as the \textit{ only} solutions, namely, $\mathcal{ E}_{ 0}^{ BS}=\pm i2\hbar \omega_{ { D}}/\sqrt{ { e}^{ 2/\lambda}{ -1}}.$
Categories: cond-mat.supr-con