Sigma Functions and Lie Algebras of Schrödinger Operators

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 $g = 1,2,3,4$.