Sigma Functions and Lie Algebras of Schrödinger Operators

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