Abstract

In a 2004 paper by V. M. Buchstaber and D. V. Leikin, published in “Functional Analysis and Its Applications,” for each \(g > 0\), a system of \(2g\) multidimensional Schrödinger equations in magnetic fields with quadratic potentials was defined. Such systems are equivalent to systems of heat equations in a nonholonomic frame. It was proved that such a system determines the sigma function of the universal hyperelliptic curve of genus \(g\). A polynomial Lie algebra with \(2g\) Schrödinger operators \(Q_0, Q_2, \dots, Q_{4g-2}\) as generators was introduced.

In this work, for each \(g > 0,\) we obtain explicit expressions for \(Q_0\), \(Q_2\), and \(Q_4\) and recurrent formulas for \(Q_{2k}\) with \(k>2\) expressing these operators as elements of a polynomial Lie algebra in terms of the Lie brackets of the operators \(Q_0\), \(Q_2\), and \(Q_4\).

As an application, we obtain explicit expressions for the operators \(Q_0, Q_2, \dots, Q_{4g-2}\) for \(g = 1,2,3,4\).

中文翻译：

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薛定谔算子的西格玛函数和李代数

摘要

在 VM Buchstaber 和 DV Leikin 于 2004 年发表于“Functional Analysis and Its Applications”的一篇论文中，对于每个\(g > 0\)，定义了具有二次势的磁场中的\(2g\)多维薛定谔方程组. 这样的系统等价于非完整坐标系中的热方程系统。证明了这样的系统确定了属\(g\)的通用超椭圆曲线的sigma函数。引入了以\(2g\)薛定谔算子\(Q_0, Q_2, \dots, Q_{4g-2}\)作为生成元的多项式李代数。

在这项工作中，对于每个\（克> 0，\）我们得到明确的表达式\（Q_0 \） ，\（Q_2 \） ，和\（C 1-4 \）和复发性公式\（Q_ {2K} \）与\(k>2\)根据运算符\(Q_0\)、\(Q_2\)和\(Q_4\)的李括号将这些运算符表示为多项式李代数的元素。

作为一个应用程序，我们为\(g = 1,2,3,4\)获得运算符\(Q_0, Q_2, \dots, Q_{4g-2}\) 的显式表达式。