Commuting ordinary differential operators of arbitrary genus and arbitrary rank with polynomial coefficients

TL;DR

This paper constructs explicit examples of commuting differential operators with polynomial coefficients for any genus and rank, solving a longstanding existence problem in the spectral theory of differential operators.

Contribution

It provides a complete solution to the existence problem for commuting operators of arbitrary genus and rank with polynomial coefficients, using Chebyshev polynomials.

Findings

Explicit examples of commuting operators for any genus and rank

Operators generated by Chebyshev polynomials

Complete resolution of the existence problem

Abstract

In this paper we construct examples of commuting ordinary scalar differential operators with polynomial coefficients that are related to a spectral curve of an arbitrary genus g>0 and to an arbitrary rank r>1 of the vector bundle of common eigenfunctions of the commuting operators over the spectral curve. This solves completely the well-known existence problem for commuting operators of arbitrary genus and arbitrary rank with polynomial coefficients. The constructed commuting operators of arbitrary rank r>1 and arbitrary genus g>0 are given explicitly, they are generated by the Chebyshev polynomials T_r (x).