论文标题
二阶线性微分方程的liouvillian解决方案具有lourent多项式系数
Liouvillian solutions for second order linear differential equations with Laurent polynomial coefficient
论文作者
论文摘要
本文致力于对具有劳伦（Laurent）多项式系数的一般无痕量二阶微分方程的Liouvillian溶液进行完整的参数研究。这个方程式的家族，适用于$ 0 $ $ 0 $和$ \ infty $的固定订单，被视为仿射代数品种。我们证明，家族中的Picard-vessiot一组可集成的微分方程是代数亚变化的一个列举的结合。我们明确计算其组件的代数方程。我们为众所周知的亚家族提供了一些应用，作为双重汇合和双色heun方程，以及Shrödinger方程的代数可解决势能的理论。同样，作为一种辅助工具，我们改善了二阶线性微分方程的先前已知标准，以接纳多项式溶液。
This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at $0$ and $\infty$ of the Laurent polynomial, is seen as an affine algebraic variety. We proof that the set of Picard-Vessiot integrable differential equations in the family is an enumerable union of algebraic subvarieties. We compute explicitly the algebraic equations of its components. We give some applications to well known subfamilies as the doubly confluent and biconfluent Heun equations, and to the theory of algebraically solvable potentials of Shrödinger equations. Also, as an auxiliary tool, we improve a previously known criterium for second order linear differential equations to admit a polynomial solution.
