分数阶微分算子的 Burchnall-Chaundy-Krichever 理论
A Burchnall-Chaundy-Krichever Theory for Fractional Differential Operators
AI总结：
本文将关于交换微分算子代数的 Burchnall-Chaundy 定理、极大交换代数分类和 Krichever 对应三大经典结果扩展至分数阶微分算子，并引入谱场和无穷 jet 丛等新工具给出了创新证明。
摘要：
分数阶微分（及差分）算子在许多不同领域中发挥作用：可积系统、镜像对称、Hurwitz 数、Bethe ansatz 方程。我们将关于交换（常）微分算子代数的三个主要结果扩展到分数阶微分算子的情形：(1) Burchnall-Chaundy 定理，即一对交换的微分算子是代数相关的；(2) 基于 Sato 理论对极大交换微分算子代数的分类；(3) 以代数几何方式构造秩为 1 的算子的 Krichever 对应。现有的 Burchnall-Chaundy 定理证明利用了一个微分算子作用于另一个算子的核，与此不同，我们对分数阶情形的扩展利用了分数阶微分算子阶数的界和代数的增长性，这也为原始结果提供了一个全新且简短得多的证明。第二个主要定理是通过开发 Sato Grassmannian 中点的谱场这一新工具来实现的，它比广泛使用的 KP 解的谱曲线概念包含更多信息。我们针对分数阶微分算子的 Krichever 型对应是基于无穷 jet 丛的。
英文摘要：
Fractional differential (and difference) operators play a role in a number of diverse settings: integrable systems, mirror symmetry, Hurwitz numbers, the Bethe ansatz equations. We prove extensions of the three major results on algebras of commuting (ordinary) differentials operators to the setting of fractional differential operators: (1) the Burchnall-Chaundy theorem that a pair of commuting differential operators is algebraically dependent, (2) the classification of maximal commutative algebras of differential operators in terms of Sato's theory and (3) the Krichever correspondence constructing those of rank 1 in an algebro-geometric way. Unlike the available proofs of the Burchnall-Chaundy theorem which use the action of one differential operator on the kernel of the other, our extension to the fractional case uses bounds on orders of fractional differential operators and growth of algebras, which also presents a new and much shorter proof of the original result. The second main theorem is achieved by developing a new tool of the spectral field of a point in Sato's Grassmannian, which carries more information than the widely used notion of spectral curve of a KP solution. Our Krichever type correspondence for fractional differential operators is based on infinite jet bundles.
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