Commuting differential operators of rank 2 with rational coefficients

TL;DR

This paper constructs new self-adjoint commuting differential operators of rank 2 with rational coefficients and establishes their spectral curves as elliptic and hyperelliptic curves of genus 2 and 3.

Contribution

It introduces new classes of commuting operators with rational coefficients and characterizes their spectral curves for genus 2 and 3 cases.

Findings

Spectral curves of genus 2 are elliptic and hyperelliptic.

Operators of rank 2 with rational coefficients are explicitly constructed.

Genus 3 curves can also serve as spectral curves for such operators.

Abstract

In this paper we find new self-adjoint commuting operators of rank 2 with rational coefficients and prove that any elliptic and hyperelliptic curves of genus 2 are spectral curves of commuting operators with rational coefficients. Also the case when curves of genus 3 are spectral curves of commuting operators with rational coefficients is studied.