论文标题
Heun通用方程和椭圆形Darboux方程的解决方案
Solutions of Heun's general equation and elliptic Darboux equation
论文作者
论文摘要
椭圆形Darboux方程的新解决方案是为HEUN通用方程式构建的特定解决方案的。我们考虑了两组功率系列扩展，并考虑了一系列高斯高几幅功能中的两组扩展组。一组功率序列的收敛是通过无限序列比率测试的方式来确定的，而其他组则旨在解决允许有限系列解决方案的问题。实际上，我们设想固定的一维schrödinger方程降低为darboux方程的周期性准确解决的势。通常，有限和无限系列解决方案是从HEUN方程的功率系列扩展中获得的。但是，我们表明，Schrödinger方程在固体理论中使用的相关Lamé电位家族的高几何功能方面接受了其他有限序列的扩展。对于每个有限序列解决方案，我们发现了四个无限序列的扩展，这些扩展是对自变量的所有值进行界定和收敛的。最后，我们发现可以使用变量的转换来为HEUN方程的一系列超小几幅函数中的其他扩展生成新的解决方案。
New solutions for the elliptic Darboux equation are obtained as particular cases of solutions constructed for Heun's general equation. We consider two groups of power series expansions and two new groups of expansions in series of Gauss hypergeometric functions. The convergence of one group in power series is determined by means of ratio tests for infinite series, while the other groups are designed to solve problems which admit finite-series solutions. Actually, we envisage periodic quasi-exactly solvable potentials for which the stationary one-dimensional Schrödinger equation is reduced to the Darboux equation. In general, finite- and infinite-series solutions are obtained from power series expansions for Heun's equation. However, we show that the Schrödinger equation admits additional finite-series expansions in terms of hypergeometric functions for a family of associated Lamé potentials used in band theory of solids. For each finite-series solution we find as well four infinite-series expansions which are bounded and convergent for all values of the independent variable. Finally we find that it is possible to use transformations of variables in order to generate new solutions for the Darboux equation out of other expansions in series of hypergeometric functions for the Heun equation.
