Lam\'e polynomials, hyperelliptic reductions and Lam\'e band structure

TL;DR

This paper investigates the band structure of the Lamé equation using hyperelliptic reductions, providing new formulas, correcting previous results, and exploring degeneracies and spectral polynomials in elliptic and hyperelliptic contexts.

Contribution

It introduces a general formula for hyperelliptic to elliptic curve coverings, corrects prior dispersion relation results, and analyzes degeneracies in Lamé band structures.

Findings

Reduced hyperelliptic integrals to elliptic ones.

Corrected previous dispersion relation tables.

Derived a formula for degeneracy points in elliptic moduli space.

Abstract

The band structure of the Lam\'e equation, viewed as a one-dimensional Schr\"odinger equation with a periodic potential, is studied. At integer values of the degree parameter l, the dispersion relation is reduced to the l=1 dispersion relation, and a previously published l=2 dispersion relation is shown to be partially incorrect. The Hermite-Krichever Ansatz, which expresses Lam\'e equation solutions in terms of l=1 solutions, is the chief tool. It is based on a projection from a genus-l hyperelliptic curve, which parametrizes solutions, to an elliptic curve. A general formula for this covering is derived, and is used to reduce certain hyperelliptic integrals to elliptic ones. Degeneracies between band edges, which can occur if the Lam\'e equation parameters take complex values, are investigated. If the Lam\'e equation is viewed as a differential equation on an elliptic curve, a formula is conjectured for the number of points in elliptic moduli space (elliptic curve parameter space) at which degeneracies occur. Tables of spectral polynomials and Lam\'e polynomials, i.e., band edge solutions, are given. A table in the older literature is corrected.