Integration range in BCS theory In two different ways of finding the Cooper pair energy gap, the limits of integration are different, yet both give the same result.
In the first case, when working out the energy $E_{pair}$ of a single Cooper pair, the integral is given by $$1=V_{eff}g(E_F)\int_{E_F}^{E_F+\hbar \omega_D}\frac{dE}{2E-E_{pair}}\Rightarrow E_{pair}=2E_F-2\hbar \omega_D e^{-2/(V_{eff}g(E_F))}\tag{1}\label{1}$$
and so the integral extends to $\hbar \omega_D$ above the Fermi energy $E_F$.
On the other hand, when deriving the energy gap $\Delta$ from the Bogoliubov transformation, the integral is now (with $E$ relative to $E_F$) $$1=\frac{1}{4}V_{eff}g(E_F)\int_{-\hbar \omega_D}^{\hbar \omega_D}\frac{dE}{\sqrt{\Delta^2+E^2}}\Rightarrow\Delta=2\hbar \omega_De^{-2/(V_{eff}g(E_F))}\tag{2}\label{2}$$
and so it extends to $\hbar \omega_D$ either side of $E_F$.
Thus the energy gap $\Delta$ from ($\ref{2})$ agrees with the binding energy in ($\ref{1})$ but both were derived with different limits. What is the explanation for this? All sources I can find make no mention of it.
 A: The reason is that those are two different calculations with two different integrals but the hypothesis that link them is made before arriving at the final form.
In the Cooper Problem the hypothesis on the potential states that it is non-zeri and attractive when the energies of the electrons are in $[E_F ; E_F+ \hbar \omega_D ]$ and comes from the rigid Fermi sphere hypothesis plus the phonon induced attractive potential.
In the Bogoliubov approach the same hypothesis on the potential is made but in this approach you get an equation for the energy gap parameter that is taken non-zeri for $|\epsilon_k|< \hbar \omega_D$ because you don't have the rigid Fermi sphere hypothesis since all the electrons participate.
Now, this being said, one could also state that the interval are made such that the exponentials give the same results, indeed $\hbar \omega_D$ gives you an order of magnitude of the interaction and it's meant to give results that are qualitatively correct more than quantitatively. Think about the fact that those two approaches are talking about systems that are also different, why would the interaction be the same for a system with a rigid Fermi sphere and another where this assumption isn't made?
I think that you must watch at those results qualitatively more than quantitatively and, if you want, you can say that the intervals are taken such that they give the same results
