Wannier functions of elliptic one-gap potentials

TL;DR

This paper explicitly constructs and analyzes Wannier functions for a specific class of elliptic one-gap potentials in the Schrödinger equation, providing analytical expressions and numerical validation.

Contribution

The paper presents the first explicit construction and detailed analysis of Wannier functions for elliptic one-gap potentials, including analytical and numerical characterizations.

Findings

Derived an expression for Wannier function amplitude at the origin

Developed a power series expansion near the origin

Established an asymptotic decay expansion at large distances

Abstract

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.