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=-Δ+V and which belong to a given band B, define a subspace S(K) of finite dimensionality. For a large class of potentials, two properties concerning the subspaces S(K) which are associated with a fixed band B have been proved for n-dimensional crystals. (1) The projection operator P(K) on S(K) can be defined for complex values of K, and its matrix elements 〈r|P(K)|r^'〉 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=∫dʰKP(K) (integration on the Brillouin zone) has matrix elements 〈r|P|r^'〉 which decrease exponentially when the length|r-r^'| goes to infinity.