A particle of mass $m$ moves in one dimension in a periodic potential $V(x)$ satisfying $V(x+a)=V(x)$. Define the Floquet matrix $F$. Show that $\operatorname{det} F=1$ and explain why Tr $F$ is real. Show that allowed bands occur for energies such that $(\operatorname{Tr} F)^{2}<4$. Consider the potential

$$V(x)=-\frac{\hbar^{2} \lambda}{m} \sum_{n=-\infty}^{+\infty} \delta(x-n a)$$

For states of negative energy, construct the Floquet matrix with respect to the basis of states $e^{\pm \mu x}$. Derive an inequality for the values of $\mu$ in an allowed energy band.

For states of positive energy, construct the Floquet matrix with respect to the basis of states $e^{\pm i k x}$. Derive an inequality for the values of $k$ in an allowed energy band.

Show that the state with zero energy lies in a forbidden region for $\lambda a>2$.