Floquet matrix for a one-dimensional periodic potential

For a potential of period $a$, the Floquet matrix $M(E)$ maps Cauchy data through one cell:For a real Schrodinger equation, $\det M=1$.

The Floquet multipliers solveBounded Bloch waves occur when $|\mathrm{tr}M|\le 2$, with $\mu =e^{\pm iKa}$. Band edges satisfy $\mathrm{tr}M=\pm 2$; outside the bands the multipliers are real reciprocal numbers and one solution grows exponentially.