  Wannier functions of the one dimensional Schroedinger equation with elliptic
one gap potentials are explicitly constructed. Properties of these functions
are analytically and numerically investigated. In particular we derive an
expression for the amplitude of the Wannier function in the origin, a power
series expansion valid in the vicinity of the origin and an asymptotic
expansion characterizing the decay of the Wannier function at large distances.
Using these results we construct an approximate analytical expression of the
Wannier function which is valid in the whole spatial domain and is in good
agreement with numerical results.
