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

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.