Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 34E / a / iii / Solution

The Floquet multipliers satisfyIf $(\mathrm{tr}F{)}^2<4$, they are complex conjugates on the unit circle, $e^{\pm ika}$, and bounded Bloch waves exist: $E$ lies in an allowed band. If $(\mathrm{tr}F{)}^2>4$, 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.