Sigma functions and Lie algebras of Schr\"odinger operators

TL;DR

This paper links multidimensional Schr"odinger equations with hyperelliptic sigma functions, providing explicit operator formulas and Lie algebra structures that deepen understanding of integrable systems and algebraic curves.

Contribution

It derives explicit formulas for Schr"odinger operators and their Lie algebra relations for all genera, extending previous theoretical frameworks.

Findings

Explicit formulas for operators Q_0, Q_2, Q_4 for g=1,2,3,4

Recurrent formulas for higher Q_{2k} operators

Connection between Schr"odinger systems and hyperelliptic sigma functions

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