A particle of mass $m$ and energy $E=-\hbar^{2} \kappa^{2} / 2 m<0$ moves in one dimension subject to a periodic potential

$$V(x)=-\frac{\hbar^{2} \lambda}{m} \sum_{\ell=-\infty}^{\infty} \delta(x-\ell a) \quad \text { with } \quad \lambda>0 .$$

Determine the corresponding Floquet matrix $\mathcal{M}$. [You may assume without proof that for the Schrödinger equation with potential $\alpha \delta(x)$ the wavefunction $\psi(x)$ is continuous at $x=0$ and satisfies $\left.\psi^{\prime}(0+)-\psi^{\prime}(0-)=\left(2 m \alpha / \hbar^{2}\right) \psi(0) .\right]$

Explain briefly, with reference to Bloch's theorem, how restrictions on the energy of a Bloch state can be derived from $\mathcal{M}$. Deduce that for the potential $V(x)$ above, $\kappa$ is confined to a range whose boundary values are determined by

$$\tanh \left(\frac{\kappa a}{2}\right)=\frac{\kappa}{\lambda} \quad \text { and } \quad \operatorname{coth}\left(\frac{\kappa a}{2}\right)=\frac{\kappa}{\lambda} .$$

Sketch the left-hand and right-hand sides of each of these equations as functions of $y=\kappa a / 2$. Hence show that there is exactly one allowed band of negative energies with either (i) $E_{-} \leqslant E<0$ or (ii) $E_{-} \leqslant E \leqslant E_{+}<0$ and determine the values of $\lambda a$ for which each of these cases arise. [You should not attempt to evaluate the constants $E_{\pm} .$]

Comment briefly on the limit $a \rightarrow \infty$ with $\lambda$ fixed.