Sigma functions and Lie algebras of Schr\"odinger operators

Abstract

In the work by V. M. Buchstaber and D. V. Leikin for any $g > 0$ is defined a system of $2g$ multidimensional Schr\"odinger equations in magnetic fields with quadratic potentials. This systems are equivalent to systems of heat equations in nonholonomic frame. It is proved that such a system determines the sigma function of the universal hyperelliptic curve of genus $g$. A polynomial Lie algebra with $2g$ Schr\"odinger operators $Q_0, Q_2, \ldots, Q_{4g-2}$ as generators was introduced. In this work for any $g > 0$ we obtain explicit expressions for $Q_0$, $Q_2$, $Q_4$, and recurrent formulas for $Q_{2k}$ with $k>2$ expressing this operators as elements of the polynomial Lie algebra using Lie brackets of the operators $Q_0$, $Q_2$, and $Q_4$. As an application we obtain explicit expressions for the operators $Q_0, Q_2, \ldots, Q_{4g-2}$ for $g = 1,2,3,4$.