关于具有非解析系数的线性二阶常微分方程连接矩阵的一个新的精确关系
On a new exact relation for the connection matrices in case of a linear second-order ODE with non-analytic coefficients
AI总结：
本文基于Frobenius方法提出一种通用技术，用于推导具有非解析系数的线性二阶常微分方程连接矩阵之间的新精确关系，并给出Stokes常数的精确及近似方程，以Budden问题为例验证其有效性。
摘要：
我们考虑将相位积分方法应用于具有非解析系数的任意线性二阶常微分方程。我们提出一种基于Frobenius方法的通用技术，该技术允许获得与其通解相关联的连接矩阵之间的新的精确关系。该技术使读者能够在微分方程于复平面有限区域内至多有一个正则奇点的情况下，写出Stokes常数的精确代数方程。我们还提出了一种在多个正则奇点彼此相距很远的情况下写出Stokes常数之间近似关系的方法。作为该技术应用的一个示例，著名的Budden问题借助此方法得到求解。如需访问本文的HTML版本并与作者讨论，请访问https://enabla.com/pub/607。
英文摘要：
We consider the phase-integral method applied to an arbitrary linear ordinary second-order differential equation with non-analytical coefficients. We propose a universal technique based on the Frobenius method which allows to obtain new exact relation between connection matrices associated with its general solution. The technique allows the reader to write an exact algebraic equation for the Stokes constants provided the differential equation has at most one regular singular point in a finite area of the complex plane. We also propose a way to write approximate relations between Stokes constants in case of multiple regular singular points located far away from each other. The well-known Budden problem is solved with help of this technique as an illustration of its usage. To access the HTML version of the paper & discuss it with the author, visit https://enabla.com/pub/607.
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