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I have been trying to understand Ashcroft's take on electrons in a weak periodic potential, and his approach by Fourier analysis is proving to be extremely challenging. I understand how to get to the master equation:

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and then he assumes this (which by itself is odd, but I assume it is to isolate a band and study it isolated from the others): enter image description here

But this is what I can't wrap my head around:

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I can't understand the behaviour of $O(U^2)$. How does he get to that conclusion. I have read the text several times, but I can't make sense of it. Can someone help me? I have looked into parseval's theorem to see if it would help relate the coefficients $c_{k-K}$ but that theorem relates the square of the coefficients to the energy, so it didn't quite help.

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Well, this is easy. Simple math, I suppose.

We have $$\mathbf{c_{k-K} = \frac{U_{K_1-K}\, c_{k-K_1}}{\epsilon-\epsilon^{\circ}_{k-K}} + \sum_{K'\ne K_1} \frac{U_{K'-K}\, c_{k-K'}}{\epsilon-\epsilon^{\circ}_{k-K}}}$$

Now in place of $\mathbf{c_{k-K'}}$ in the numerator of the 2nd expression in RHS, you plug in the above expression for $\mathbf{c_{k-K}}$.

Then you get the following expression:

$$\mathbf{c_{k-K} = \frac{U_{K_1-K}\, c_{k-K_1}}{\epsilon-\epsilon^{\circ}_{k-K}} + \sum_{K'\ne K_1} \frac{U_{K'-K}}{\epsilon-\epsilon^{\circ}_{k-K}}} \left(\mathbf{\frac{U_{K_1-K'} \, c_{k-K_1}}{\epsilon-\epsilon^{\circ}_{k-K'}} + \sum_{K''\ne K_1} \frac{U_{K''-K'}\, c_{k-K''}}{\epsilon-\epsilon^{\circ}_{k-K'}}}\right)$$

which on further simplification gives:

$$\mathbf{c_{k-K} = \frac{U_{K_1-K}\, c_{k-K_1}}{\epsilon-\epsilon^{\circ}_{k-K}} + \sum_{K'\ne K_1} \frac{U_{K'-K}\, U_{K_1-K'} \, c_{k-K_1}}{(\epsilon-\epsilon^{\circ}_{k-K})(\epsilon-\epsilon^{\circ}_{k-K'})} + \sum_{K'\ne K_1} \sum_{K''\ne K_1} \frac{U_{K'-K}\, U_{K''-K'}\, c_{k-K''}}{(\epsilon-\epsilon^{\circ}_{k-K})(\epsilon-\epsilon^{\circ}_{k-K'})}}$$

Now, if you look at the second and third terms in the RHS, it involves quadratic terms of $U_k$. In other words, in big O notation, $\mathcal{O}(U^2)$.

In this way, the author arrives at:

$$\mathbf{c_{k-K} = \frac{U_{K_1-K}\, c_{k-K_1}}{\epsilon-\epsilon^{\circ}_{k-K}} + \mathcal{O}(U^2) }$$

Since he is talking about weak potentials here, the author later neglects quadratic terms in U to quantify the effect of weak periodic potentials to create Wannier functions, leading to the formation of the s-band, p-band so on, for different overlap integrals.

Keep on reading Ashcroft and Mermin, it is a very good book for understanding the basics of solid-state physics and condensed matter physics.

Hope this helps you.

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It is because of this passage before Eq. (9.10):

Since we are examining that solution for which $c_{k-K}$ vanishes when $K\ne K_1$ in the limit of vanishing $U$

In other words, in the sum in Eq. (9.10) the coefficient $c_{k-K'}$ are also of order $\mathcal{O}(U)$, which multiplied by $U$ (and assuming no degeneracy, so that the denominator cannot be small) gives $\mathcal{O}(U^2)$.

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