Wannier functions of elliptic one-gap potentials

Wannier functions of the one-dimensional Schrodinger equation with an elliptic one-gap potential 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.