Wannier Functions for Quasiperiodic Finite-Gap Potentials

Abstract

In this paper we consider Wannier functions of quasi-periodic g-gap (g ≥ 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 σ functions. Using this approach we derive a power series expansion of the Wannier function for quasi-periodic potentials valid at |x| ≃ 0 and an asymptotic expansion valid at large distance. These functions are important for a number of applied problems.

The Addition Law Attached to a Stratification of a Hyperelliptic Jacobian Variety

“…for 0 ≤ i ≤ g, 0 ≤ j ≤ g, and u = (u 1 , · · · , u g ) ∈ C g , which is natutal generalizations of Weierstrass ℘ function. As we mentioned before 4.2 the following special case of (m, n) = (g, 1) in 4.2 appeared in [BES,(3.21)], which was the motivation of this paper.…”

“…for u (1) , u (2) , and v varying on the canonical universal Abelian covering of C 2 in C 2 . This appeared in [G] at the first time, and a generalization of this for any hyperellitpic curve was reported in [BES,(3.21)], without proof. The main result (Theorem 4.2 below) of this paper seems to be a unification of (1.5) and (1.6).…”

Bilinear recurrences and addition formulae for hyperelliptic sigma functions

“…More generally Morgan Ward [45,46] introduced a family of such antisymmetric sequences defined by recurrences of the form 4) which are derived by considering sequences of rational points nP on an elliptic curve E over Q. To obtain integer sequences of this kind it is required that…”

“…(In fact this Somos sequence is just obtained by selecting the odd index terms of the elliptic divisibility sequence (2.1), up to an alternating sign.) More generally, following the terminology of [39,43], we refer to any sequence defined by a bilinear recurrence of the form (1.1) as a Somos 4 sequence, while the particular sequence above is denoted Somos (4). It turns out that any such sequence is associated to a sequence of points P 0 + nP on an associated elliptic curve E: this fact was proved by algebraic means in the thesis of Swart [43], which refers to unpublished results established independently by both Nelson Stephens and Noam Elkies.…”

“…The addition formula (3.8) is a special case of the generalized Frobenius-Stickelberger addition formula in genus two considered in [12,33]. Onishi has further generalized the Frobenius-Stickelberger formula to hyperelliptic sigma functions for all genera [34], and the special case of the formula corresponding to addition of one point has been applied to the problem of construction of Wannier functions for quasi-periodic finite-gap potentials in [4]. Cantor has constructed the division polynomials for hyperelliptic curves, and obtained certain recurrence relations for them in the paper [11], where in particular an eighth order bilinear recurrence is found in genus two.…”