Wannier functions for quasi-periodic finite-gap potentials

In this paper we consider Wannier functions of quasi-periodic g-gap ($g\geq 1$) potentials and investigate their main properties. In particular, we discuss the problem of averaging underlying the definition of Wannier functions for both periodic and quasi-periodic potentials and express Bloch functions and quasi-momenta in terms of hyperelliptic $\sigma$ functions. Using this approach we derive a power series expansion of the Wannier function for quasi-periodic potentials valid at $|x|\simeq 0$ and an asymptotic expansion valid at large distance. These functions are important for a number of applied problems.