Wannier functions for quasi-periodic finite-gap potentials

TL;DR

This paper studies Wannier functions for quasi-periodic finite-gap potentials, deriving new series expansions and expressing key functions via hyperelliptic sigma functions, advancing understanding of their properties.

Contribution

It introduces a novel approach to express Wannier functions for quasi-periodic potentials using hyperelliptic sigma functions and derives new series expansions.

Findings

Power series expansion near zero for Wannier functions

Asymptotic expansion at large distances

Representation of Bloch functions via hyperelliptic sigma functions

Abstract

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.