On a new exact relation for the connection matrices in case of a linear second-order ODE with non-analytic coefficients

TL;DR

This paper introduces a universal Frobenius-based method to derive exact relations between connection matrices for second-order linear ODEs with non-analytic coefficients, improving understanding of Stokes phenomena.

Contribution

It presents a new algebraic approach to connect matrices in second-order ODEs with non-analytic coefficients, including exact and approximate relations for Stokes constants.

Findings

Derived exact algebraic equations for Stokes constants with at most one regular singular point.

Proposed approximate relations for multiple distant singular points.

Successfully applied the method to solve the Budden problem.

Abstract

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.