# Energy gap in BCS theory

In the context of the BCS theory of superconductivity, which energy gap is one referring to when using the BCS gap equation

$$\Delta_{\textbf{k}}=-\sum_{\textbf{k'}}V_{\textbf{k},\textbf{k'}}\text{g}_{\textbf{k'}}$$

with the Gorkov amplitude $$\text{g}_{\textbf{k}}=\left\langle \hat{c}_{-\textbf{k}\uparrow}\hat{c}_{\textbf{k}\downarrow}\right\rangle$$? In which energy spectrum does this gap occur? Is it the one of Cooper pairs or Bogoliubov quasiparticles? I am somewhat confused at the moment, so any handwaving explanation would be really helpful.

It's the energy gap of the single-particle spectrum. If one tries to probe the system with a STM, for example, you will get no tunneling below this gap. It can be read from the retarded Green function which in matrix form is $$g^r(\omega) = -i\frac{\pi\rho_0}{\sqrt{(\omega^2-\Delta^2)}}\left(\begin{array}{cc}\omega & \Delta \\ \Delta & \omega \end{array} \right)$$ where the basis is $$[\psi(x)_\uparrow, -\psi^{\dagger}(x)_\downarrow]$$. So the single-particle density of states, when read off the green function $$\rho(\epsilon) = -{\rm Im}\{g^r(\epsilon)\}/\pi$$ has a gap for $$|\omega|<\Delta$$.