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

For two solutions of the Schrodinger equation, their Wronskian is constant because the equation has no first-derivative term. The determinant of the Floquet matrix is the ratio of the Wronskian at $x+a$ to that at $x$, hence

For real $E$ and real $V$, choose a real fundamental pair of solutions. The corresponding Floquet matrix has real entries, so $\mathrm{tr}F(E)$ is real. The trace is basis independent, so this holds for every fundamental pair.

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.