Energy Bands and Projection Operators in a Crystal: Analytic and Asymptotic Properties

In an $n$-dimensional crystal, an energy band is usually made of several branches which are connected with each other. Accordingly, the Bloch states of wave vector K which are eigenfunctions of a one-electron Hamiltonian $H=-\Delta +V$ and which belong to a given band $\mathcal{B}$, define a subspace $\mathcal{S}\left(\mathrm{K}\right)$ of finite dimensionality. For a large class of potentials, two properties concerning the subspaces $\mathcal{S}\left(\mathrm{K}\right)$ which are associated with a fixed band $\mathcal{B}$ have been proved for $n$-dimensional crystals. (1) The projection operator $P\left(\mathrm{K}\right)$ on $\mathcal{S}\left(\mathrm{K}\right)$ can be defined for complex values of K, and its matrix elements $〈\mathrm{r}\left|P\right(\mathrm{K}\left)\right|{\mathrm{r}}^{\prime }〉$ are analytic in a strip of the complex K space; this strip is centered on the real K space and is independent of r and r'. (2) The projection operator $P=\int d^h\mathrm{K}P\left(\mathrm{K}\right)$ (integration on the Brillouin zone) has matrix elements $〈\mathrm{r}\mathrm{}\left|P\right|\mathrm{}{\mathrm{r}}^{\prime }〉$ which decrease exponentially when the length|r-${\mathbf{r}}^{\prime }$| goes to infinity.