Electronic band structure

Remarks

If $u$ is an eigenfunction of $H(k)$, then $\psi(x) = e^{i k \cdot x} u(x)$ solves $H\psi = E\psi$ and satisfies $\psi(x + R) = e^{ik \cdot R} \psi(x)$; this is Bloch's theorem. Each $E_m$ is continuous and $\Lambda^*$-periodic, since $H(k + G)$ is unitarily equivalent to $H(k)$ through multiplication by $e^{-iG \cdot x}$. The spectrum of $H$ is the union of the bands $E_m(B)$, which are closed intervals, and the open intervals between them that are not covered are the spectral gaps.

Example: the free electron in dimension $1$, with $\Lambda = \Z$ and $V = 0$. Then $H(k) = (-i\, d/dx + k)^2$ on $L^2(\R/\Z)$ has eigenfunctions $e^{2\pi i m x}$, $m \in \Z$, with eigenvalues $(k + 2\pi m)^2$. Sorting these for $k \in [-\pi, \pi]$ gives the bands $[0, \pi^2], [\pi^2, 4\pi^2], [4\pi^2, 9\pi^2], \dots$, with no gaps, and their union is $[0, \infty)$.

Non-example: if $V$ is not periodic, for instance a random potential, the operators $H(k)$ are not defined, and there is no band structure.