Hi I am studying some physics that in a liminal case produce the Bessel equation:
$ U''_k(\tau)+\omega_k^2(\tau)U_k(\tau)=0 $
Where in the liminal case $\omega_k^2(\tau)=k^2 - \frac{C}{\tau^2}$ with $C$ being a constant.
I am studying a slight perturbed variant i.e. $\omega^2(\tau)=k^2-\frac{C+f(\tau)}{\tau^2}$ where the relative magnitude of $f(\tau)$ relative to $C$ is very very small.
Does anybody know of a stability analysis of the Bessel equation in that respect, or otherwise have some experience and can share some insight into this problem?
BTW the actual solutions for the unperturbed problem is some modified Bessel functions $U_k = \sqrt{-\tau}J_{\tilde{C}}(k\tau)$ Where $\tilde{C}$ is some constant that is related to $C$.
Thanks in advance.