Pairing in the Bogoliubov-de Gennes equations

It is shown that the Bogoliubov-de Gennes equations pair the electrons in states which are linear combinations of the normal states. Accordingly, the BCS-like reduction procedure is required to choose a correct pairing. For a homogeneous system, we point out that the kernel of the self-consistency equation derived from the Bogoliubov-de Gennes equations needs to be constrained by the BCS pairing condition. In the presence of ordinary impurities, on the other hand, the Bogoliubov-de Gennes equations should be supplemented by Anderson's pairing condition to obtain the correct vacuum state by the corresponding unitary transformation. This results in localization correction to the phonon-mediated interaction.