任意亏格任意秩多项式系数的可交换普通微分算子
Commuting ordinary differential operators of arbitrary genus and arbitrary rank with polynomial coefficients
摘要：
本文构造了可交换的普通纯量微分算子的例子，这些算子的系数与任意亏格g>0的谱曲线和谱曲线上可交换算子的公共本征函数的向量丛的任意秩r>1有关.这完全解决了著名的存在性问题的交换算子的任意亏格和任意秩多项式系数。给出了由切比雪夫多项式T_r（x）生成的任意秩r>1、任意亏格g>0的交换算子.
英文摘要：
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).
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