Wannier Functions for Quasiperiodic Finite-Gap Potentials

We consider Wannier functions of quasiperiodic $g$-gap ($g\geq1$) potentials and investigate their main properties. In particular, we discuss the problem of averaging that underlies the definition of the Wannier functions for both periodic and quasiperiodic potentials and express Bloch functions and quasimomenta in terms of hyperelliptic $\sigma$-functions. Using this approach, we derive a power series for the Wannier function for quasiperiodic potentials valid for $|x|\simeq0$, and an asymptotic expansion valid at large distances. These functions are important in a number of applied problems.