On the spectral curve of the difference Lam\'e operator

TL;DR

This paper characterizes the spectral curve of the difference Lamé operator, revealing its hyperelliptic structure and connections to elliptic curves and Sklyanin algebra representations.

Contribution

It provides two complementary descriptions of the spectral curve, highlighting its hyperelliptic nature and relation to algebraic structures.

Findings

The spectral curve is a hyperelliptic curve of genus 2ℓ.

The curve covers an elliptic curve with parameter τ.

Connections between the spectral curve and Sklyanin algebra are established.

Abstract

We give two "complementary" descriptions of the curve $\Gamma$ parametrizing double-Bloch solutions to the difference analogue of the Lam\'e equation. The curve depends on a positive integer number $\ell$ and two continuous parameters: the "lattice spacing" $\eta$ and the modular parameter $\tau$. Apart from being a covering of the elliptic curve with the modular parameter $\tau$, $\Gamma$ is a hyperelliptic curve of genus $2\ell$. We also point out connections between the spectral curve and representations of the Sklyanin algebra.