This article is cited in 6 papers

Wannier Functions for Quasiperiodic Finite-Gap Potentials

E. D. Belokolos^a, V. Z. Ènol'skii^b, M. Salerno<sup>c

a</sup> Institute of Magnetism, National Academy of Sciences of Ukraine
^b Concordia University, Department of Mathematics and Statistics
^c INFM — Istituto Nazionale di Fisica della Materia

Abstract: We consider Wannier functions of quasiperiodic $g$-gap ($g\geq1$) potentials and investigate their main properties. In particular, we discuss the problem of averaging that underlies the definition of the Wannier functions for both periodic and quasiperiodic potentials and express Bloch functions and quasimomenta in terms of hyperelliptic $\sigma$-functions. Using this approach, we derive a power series for the Wannier function for quasiperiodic potentials valid for $|x|\simeq0$, and an asymptotic expansion valid at large distances. These functions are important in a number of applied problems.

Keywords: Wannier functions, finite-gap potentials, theta functions, hyperelliptic curves.

DOI: 10.4213/tmf1850

 English version: Theoretical and Mathematical Physics, 2005, 144:2, 1081–1099

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