One-point commuting difference operators of rank one

TL;DR

This paper investigates rank one one-point commuting difference operators with a focus on hyperelliptic spectral curves, constructing examples with polynomial and trigonometric coefficients, and linking difference operators to differential ones.

Contribution

It introduces a novel approach to constructing commutative subalgebras in the Weyl algebra via difference operators with polynomial coefficients.

Findings

Constructed examples with polynomial and trigonometric coefficients.

Embedded difference operators into differential operators with polynomial coefficients.

Provided a new method for creating commutative subalgebras in the Weyl algebra.

Abstract

We consider one-point commuting difference operators of rank one. The coefficients of these operators depend on a functional parameter, shift operators being included only with positive degrees. We study these operators in the case of hyperelliptic spectral curve when the marked point coincides with the branch point. We construct examples of operators with polynomial and trigonometric coefficients. Moreover, difference operators with polynomial coefficients can be embedded in the differential ones with polynomial coefficients. This construction provides a new way of constructing commutative subalgebras in the first Weyl algebra.