Difference Krichever-Novikov operators

TL;DR

This paper investigates commuting difference operators of rank two, introduces an equation relating their potentials, and provides initial examples linked to higher genus spectral curves.

Contribution

It presents the first known examples of rank two commuting difference operators associated with higher genus spectral curves, advancing the understanding of their structure.

Findings

Derived an equation for potentials V(n), W(n) of the difference operator

Constructed initial examples of rank two commuting difference operators

Connected these operators to spectral curves of higher genus

Abstract

In this paper we study commuting difference operators of rank two. We introduce an equation on potentials $V(n),W(n)$ of the difference operator $L_4=(T+V(n)T^{-1})^2+W(n)$ and some additional data. With the help of this equation we find the first examples of commuting difference operators of rank two corresponding to spectral curves of higher genus.