Analytical Properties of $n$-Dimensional Energy Bands and Wannier Functions

If, in an $n$-dimensional crystal, the structure of a simple ($d=1\mathrm{}$) or complex ($d>\mathrm{}1\mathrm{}$) energy band fulfills proper symmetry conditions, the band can be spanned by a set of Wannier functions and, in many cases, the following statements can be established. (1) There exists a set of Bloch waves ($d=1\mathrm{}$) or quasi Bloch waves ($d>\mathrm{}1\mathrm{}$) which are periodic and analytic functions of the complex wave vector $\mathrm{K}={\mathrm{K}}^{\prime }+i{\mathrm{K}}^{\prime \prime }$ in a domain of the complex K space defined by an equation of the form $\left|{\mathrm{K}}^{\prime \prime }\right|<A$ where $A$ is a positive constant. (2) The corresponding Wannier functions fall off exponentially at infinity.