= Solution
{c}
For zero coupling, $\sqrt{4y^2}=2|y|$, and the two folded <energy bands> are
$$
E_-(y)=E_{\rm ZB}(1-|y|)^2,\qquad
E_+(y)=E_{\rm ZB}(1+|y|)^2.
$$
They touch at $y=0$, where the underlying free <plane wave> branches cross. A nonzero <Fourier coefficient> replaces this crossing by an <avoided crossing>, with $E_\pm(0)=E_{\rm ZB}(1\pm\epsilon)$. The lower <energy band> moves downward and the upper <energy band> upward. Both branches are smooth near $y=0$ for $\epsilon>0$.
\Image[/past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2022-nearly-free-electron-bands.png]
{title=Folded free-electron bands and the gap opened by a periodic potential}
The figure uses the boundary-centred interval $-1\leq y\leq1$ required in part (b). With the printed definition $y=1-ka/\pi$, this is the translated <Brillouin zone> $0\leq k\leq2\pi/a$. The printed $0\leq y\leq2$ instead corresponds to $-\pi/a\leq k\leq\pi/a$; the local two-wave formula must be folded appropriately before treating it as a dispersion over that entire zone. It must not be integrated over $0\leq y\leq2$ as though it were the filled first <energy band> there. The subsequent calculation follows the explicit continuation prescribed in part (b).
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