Just so we don't all have to download the file the question is

If

\(\displaystyle \psi (x) = \sum_{q}a_q e^{i q x}\)

Then does the Schrodinger equation become:

\(\displaystyle \sum_{q'} \left ( \dfrac{\hbar ^2 q'^2}{2m} e^{i q' x} + U e^{i (q' + g) x} + U e^{i (q' - g) x} \right ) a_{q'} = E \sum_{q} a_{q'} e^{i q' x}\)

A few points. Why are you using q' instead of just q? Where did the g come from? And finally I'm presuming your potential energy function is U(x) but I can't see how you would have got that form. It should be

\(\displaystyle U \psi = U ~ \sum_{q'} a_{q'} e^{i q' x}\)

-Dan