Analytic Properties of Bloch Waves and Wannier Functions

摘要: The one-dimensional Schr\"odinger equation with a periodic and symmetric potential is considered, under the assumption that energy bands do not intersect. Bloch waves, ${\ensuremath{\phi}}_{n,k}$, bands, ${E}_{n,k}$, are studied as functions of complex variable, $k$. In plane, they branches multivalued analytic functions, ${\ensuremath{\phi}}_{k}$, ${E}_{k}$, branch points, ${k}^{\ensuremath{'}}$, off real axis. A simple procedure described for locating points. Application made to power series Fourier developments these functions. analyticity periodicity ${\ensuremath{\phi}}_{n,k}$ has some consequences form Wannier particular, it shown each band there exists one only function which real, or antisymmetric an appropriate reflection, falling exponentially distance. rate falloff determined by distance points ${k}^{\ensuremath{'}}$ from