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

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