Solution
= Solution
The Floquet multipliers satisfy
$$
\lambda^2-(\operatorname{tr}F)\lambda+1=0.
$$
If $(\operatorname{tr}F)^2<4$, they are complex conjugates on the unit circle, $e^{\pm ika}$, and bounded Bloch waves exist: $E$ lies in an allowed band. If $(\operatorname{tr}F)^2>4$, the multipliers are real reciprocal numbers, one growing and one decaying, so there is no extended bounded Bloch state and $E$ lies in a gap.