Energy gap in the nearly free electron I am trying to understand the visualization of the energy gap in the nearly free electron approximation.

Context
The general equation to solve in a periodic potential $V=\sum_G V_G e^{i\mathbf{G} \cdot \mathbf{r}}$ is:
$$\tag{1} \left ( E - \frac{\hbar^2}{2m} |\mathbf{k}-\mathbf{G}|^2\right ) C_{\mathbf{k}-\mathbf{G}} = \sum_{\mathbf{G}'} V_{\mathbf{G}'} C_{\mathbf{k}-\mathbf{G}-\mathbf{G}'}$$
In the nearly-free approximation we only have contributions from levels in which $ E \approx E^0 \equiv \frac{\hbar^2}{2m} k^2$. In the one dimensional case, we have that waves going into opposite directions have the same $E^0$, and therefore they can be considered as a separate problem in which the solutions are standing waves and the energy is:
$$ E^\pm(k)=\frac{1}{2}E^0_k \pm \sqrt{V_G}$$
Which correspond to the wavefunctions: $\psi^\pm \propto e^{ikx} \pm e^{-ikx}$
Question:
I don't understand why we only have the gaps at the Brillouin boundary ($k=\pm \pi/a$), I would expect that we have 2 energy levels per $|k|$ and have the following energy diagram:

What am I getting wrong or misinterpreting?
 A: The first approach is called the empty-lattice approximation. It means neglecting all Fourier components of the periodic potential, $V_G$. If you do so, you will find
$$E=\frac{\hbar^2}{2m} (\vec{k}-\vec{G})$$
And if you plot $E$ vs $k$, you find that you have a parabola for a fixed value of $G$, but $G$ is a parameter, $G$ can be any vector of teh recirprocal lattice. So, you have one parabola per $G$ value.
All those parabolas, by symemtry, intersect at a distance of $G$.

But, you know that, for Bloch's theorem, you can restrict yourself to the first Brillouin's zone, FBZ. Cosnequently, you should only look from minus pi/a to +pi/a.
As you can see, those  are also teh only points which share the same energy. Keep that in mind.

Now, back to the quasi-free electrno model, it is logical to think that having a weak potential will be a perturbation of "having no potential", so we can start with perturbation theory. This does not work, but it is a good approach that few books do. What actually works is solving the central equation, but here is a qualitative argument of why considering only two components.
PErturbation theory to second ordenr ives something like
$$E=E_0+ \langle k|V|k\rangle + \sum_m\sum_n \frac{|\langle k'|V|k\rangle|^2}{E_n-E_m} $$
The first matrix element is a constant, namely, the reference of your potential. If you set $V=0$ in the center of FBZ, it is 0. Otherwise it is jsut the reference potential, which is an irrelevant constant. It obviously affects the energy because it is the reference for all energies. IT's an optional energy-shifting.
So the thing is in the 2nd term. This term will be negligible... unless the energies are close!
So, that's why we can say that the energies are those from the free electons... except at points with simialr energies.
Now, check that only points in FBZ are degenerate in energies. That's why you should take into the central equation  as many components as they are degenerate in energies
