Asymptotic properties of the solutions of a differential equation appearing in QCD

TL;DR

This paper analyzes the asymptotic behavior of solutions to a specific differential equation relevant in QCD, focusing on the ratio of derivatives at zero as the parameter grows large.

Contribution

It establishes the asymptotic properties of solutions to a differential equation in QCD, including the ratio of derivatives at zero for large parameters.

Findings

Asymptotic behavior of the ratio h'(0)/h(0) as λ→∞

Results applicable to more general ω functions

Insights into solutions vanishing at infinity in QCD context

Abstract

We establish the asymptotic behaviour of the ratio $h^\prime(0)/h(0)$ for $\lambda\rightarrow\infty$, where $h(r)$ is a solution, vanishing at infinity, of the differential equation $h^{\prime\prime}(r) = i\lambda \omega (r) h(r)$ on the domain $0 \leq r <\infty$ and $\omega (r) = (1-\sqrt{r} K_1(\sqrt{r}))/r$. Some results are valid for more general $\omega$'s.