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arXiv 1503.00485math-phmath.AGmath.CAmath.MPmath.RAnlin.SI
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多项式系数的可交换常微分算子与第一Weyl代数的自同构

Commuting ordinary differential operators with polynomial coefficients and automorphisms of the first Weyl algebra

Andrey E. Mironov, Alexander B. Zheglov

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15 pages
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摘要：

本文研究了多项式系数的秩为2的可交换常微分算子及其上的第一Weyl代数的自同构群的轨道空间。证明了对于亏格为1的任意固定谱曲线，轨道空间是无限的。此外，我们证明了在这种情况下，对于任何$n\ge 1$有一对自伴交换常微分算子秩二$L_4=(\partial_x^2+V(x))^2+W(x)$，$L_{6}$，其中$W(x),V(x)$是多项式的次数$n$和$n+2$。我们还证明了存在具有无穷轨道空间的超椭圆谱曲线。

英文摘要：

In this paper we study rank two commuting ordinary differential operators with polynomial coefficients and the orbit space of the automorphisms group of the first Weyl algebra on such operators. We prove that for arbitrary fixed spectral curve of genus one the space of orbits is infinite. Moreover, we prove in this case that for for any $n\ge 1$ there is a pair of self-adjoint commuting ordinary differential operators of rank two $L_4=(\partial_x^2+V(x))^2+W(x)$, $L_{6}$, where $W(x),V(x)$ are polynomials of degree $n$ and $n+2$. We also prove that there are hyperelliptic spectral curves with the infinite spaces of orbits.

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